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# Concentration and quantitative regularity in homogenization and hydrodynamics

Christophe Prange

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Christophe Prange. Concentration and quantitative regularity in homogenization and hydrodynamics. Analysis of PDEs [math.AP]. CY Cergy Paris Université, 2022. tel-03814044

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Distributed under a Creative Commons Attribution 4.0 International License
Mémoire présenté par

Christophe Prange

en vue de l'obtention de

l'Habilitation à Diriger des Recherches

Concentration and quantitative regularity in
homogenization and hydrodynamics

Soutenue le 4 octobre 2022 à la Maison Internationale de la Recherche, CY Cergy Paris
Université, devant le jury composé de :

<table><tr><td>Mme Isabelle Gallagher</td><td>Professeure</td><td>ENS Paris</td><td>Examinatrice</td></tr><tr><td>M. Thierry Gallay</td><td>Professeur</td><td>Université Grenoble Alpes</td><td>Examenateur</td></tr><tr><td>M. Patrick Gérard</td><td>Professeur</td><td>Paris Saclay</td><td>Rapporteur</td></tr><tr><td>M. David Gérard-Varet</td><td>Professeur</td><td>Université Paris Cité</td><td>Examenateur</td></tr><tr><td>M. Emmanuel Hebey</td><td>Professeur</td><td>CY Cergy Paris Université</td><td>Examenateur</td></tr><tr><td>M. Pierre-Gilles Lemarié-Rieusset</td><td>Professeur</td><td>Université Evry-Val d'Essonne</td><td>Examenateur</td></tr><tr><td>M. Zhongwei Shen</td><td>Professor</td><td>University of Kentucky, USA</td><td>Rapporteur</td></tr><tr><td>M. Vladimír Šverák</td><td>Professor</td><td>University of Minnesota, USA</td><td>Rapporteur</td></tr></table>
**Abstract**

This habilitation thesis is about a selection of my works concerned with the study of regularity for Partial Differential Equations, with a focus on equations stemming from fluid dynamics. In a broad sense, regularity theory is the study of the local behavior of solutions. In our view the main objectives are: (i) to identify a range of scales where there is a certain self-similar behavior, (ii) to find basic objects i.e. building blocks, that represent well the solutions at these scales, (iii) to prove scaling laws for excess quantities, i.e. local error estimates, at these scales. Our analysis is motivated by physics: concentration effects in composite materials, fluids slipping over rough surfaces, generation of turbulence near boundaries in fluids. Our aim is to contribute to analyzing such phenomena from the perspective of regularity theory.

There are two main parts in this thesis. The first part is concerned with large-scale regularity and quantitative homogenization of three-dimensional stationary Navier-Stokes and elliptic equations. With Higaki (former postdoctoral researcher, now at Kobe University) and Zhuge (The University of Chicago), we prove large-scale regularity results in bumpy regions possibly as rough as fractals. With Armstrong (NYU), Kuusi (University of Helsinki) and Mourrat (ENS de Lyon), we obtain near-optimal error estimates for the homogenization of boundary layer correctors. The second part is concerned with the three-dimensional non stationary Navier-Stokes equations. With Maekawa (Kyoto University) and Miura (Tokyo Institute of Technology), we find new pressure estimates that enable us to control the strong nonlocality in the half-space. With Albritton (IAS, Princeton) and Barker (University of Bath), we investigate norm and geometric concentration near potential singularities. We also establish a connection between concentration and quantitative regularity in the critical case that leads to a slight breaking of the criticality barrier.
# Résumé

## Concentration et régularité quantitative en homogénéisation et en hydrodynamique

Cette thèse d'habilitation à diriger des recherches porte sur une sélection de mes travaux sur la régularité des solutions d'Équations aux Dérivées Partielles, provenant en particulier de la mécanique des fluides. En un sens, l'étude de la régularité est l'analyse du comportement local des solutions. Dans cette optique, les objectifs principaux sont : (i) d'identifier une gamme d'échelles avec une certaine auto-similarité, (ii) de trouver des briques de base qui représentent les solutions à ces échelles, (iii) d'établir des lois pour l'erreur locale à ces échelles. Notre travail est motivé par la physique : concentration dans les composites, glissement de fluides sur des surfaces rugueuses, génération de la turbulence près de bord. Notre objectif est de contribuer à l'analyse de ces phénomènes à travers le prisme de la régularité.

Il y a deux parties principales dans cette thèse. La première est dédiée à la régularité dite améliorée ou aux grandes échelles et à l'homogénéisation quantitative pour les équations de Navier-Stokes tri-dimensionnelles stationnaires et elliptiques. Avec Higaki (ancien postdoctorant, à présent à Kobe University) et Zhuge (The University of Chicago), nous démontrons des estimations de régularité à grande échelle au-dessus de bord rugueux, possiblement aussi irréguliers que certaines fractales, mais asymptotiquement plats. Avec Armstrong (NYU), Kuusi (University of Helsinki) et Mourrat (ENS de Lyon), nous obtenons des estimations quantitatives presque optimales pour l'homogénéisation de couches limites. La seconde partie est consacrée à l'étude des équations de Navier-Stokes tri-dimensionnelles non stationnaires. En collaboration avec Maekawa (Kyoto University) et Miura (Tokyo Institute of Technology), nous trouvons de nouvelles estimations pour la pression qui nous permettent de contrôler la forte nonlocalité due à l'incompressibilité et à la condition de non-glissement dans le demi-espace. Avec Albritton (IAS, Princeton) et Barker (University of Bath) nous étudions les phénomènes de concentration en norme et de concentration géométrique au voisinage de singularités potentielles. Nous explorons aussi les liens entre concentration et régularité quantitative dans le cas critique, ce qui nous permet de faire sauter dans une certaine mesure le verrou de la criticalité.
# List of publications after Ph.D.

## Research articles and preprints presented in this habilitation thesis

The following papers are the main papers that are discussed in this habilitation thesis. The papers are ranked in the reverse chronological order taking into account the first release on arXiv. All the papers are available on arXiv and on my webpage.

- [10], with Dallas Albritton and Tobias Barker, *Localized smoothing and concentration for the Navier-Stokes equations in the half space*, submitted (2021).

- [184], with Mitsuo Higaki and Jinping Zhuge, *Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries*, to appear in *Analysis & PDE* (2021).

- [37], with Tobias Barker, *Mild criticality breaking for the Navier-Stokes equations*, *J. Math. Fluid Mech.* (2021).

- [38], with Tobias Barker, *Quantitative regularity for the Navier-Stokes equations via spatial concentration*, *Comm. Math. Phys.* (2021).

- [183], with Mitsuo Higaki, *Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law*, *Calc. Var. Partial Differential Equations* (2020).

- [36], with Tobias Barker, *Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions*, *Arch. Ration. Mech. Anal.* (2020).

- [35], with Tobias Barker, *Localized Smoothing for the Navier-Stokes Equations and Concentration of Critical Norms Near Singularities*, *Arch. Ration. Mech. Anal.* (2020).

- [247], with Yasunori Maekawa and Hideyuki Miura, *Local energy weak solutions for the Navier-Stokes equations in the half-space*, *Comm. Math. Phys.* (2019).

- [249], with Yasunori Maekawa and Hideyuki Miura, *Estimates for the Navier-Stokes equations in the half-space for non localized data*, *Analysis & PDE* (2020).

- [23], with Scott Armstrong, Tuomo Kuusi and Jean-Christophe Mourrat, *Quantitative analysis of boundary layers in periodic homogenization*, *Arch. Ration. Mech. Anal.* (2017).

## Other research articles and preprints posterior to Ph.D. thesis

The following papers are posterior to my Ph.D. thesis, but are not presented in this habilitation thesis. They are ranked in the reverse chronological order taking into account the first release on arXiv. All the papers are available on arXiv and on my webpage.

- [187], with Richard Höfer and Franck Sueur, *Motion of several slender rigid filaments in a Stokes flow*, *Journal de l'École polytechnique – Mathématiques* (2022).

- [54], with Edoardo Bocchi and Francesco Fanelli, *Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids*, to appear in *Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire* (2022).

- [248], with Yasunori Maekawa and Hideyuki Miura, *On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain*, *J. Math. Fluid Mech.* (2019).
[214], with Carlos Kenig, *Improved regularity in bumpy Lipschitz domains*, *J. Math. Pures Appl.* (2018).

[213], with Carlos Kenig, *Uniform Lipschitz estimates in bumpy half-spaces*, *Arch. Ration. Mech. Anal.* (2015).
# Contents

List of Figures 9

1 Introduction 11

1.1 A few transversal themes 13

1.2 Presentation of our main contributions 18

1.3 Outline of the thesis 21

I Boundary layers and regularity for steady problems in disordered environments 27

2 Compactness and quantitative methods 29

2.1 Context: state of the art and obstacles 29

2.2 Main results 33

2.3 New ideas and strategy for the proofs 42

3 Quantitative homogenization of boundary layers 49

3.1 Context: state of the art and obstacles 50

3.2 Main result 54

3.3 New ideas and strategy for the proofs 57

II Regularity and concentration for unsteady fluids 63

4 Geometric concentration 65

4.1 Context: state of the art and obstacles 67

4.2 Main results 73

4.3 New ideas and strategy for the proofs 78

5 Norm concentration 87

5.1 Context: state of the art and obstacles 88

5.2 Main results 92

5.3 New ideas and strategy for the proofs 101

6 Quantitative regularity 109
<table><tr><td>6.1 Context: state of the art and obstacles</td><td>110</td></tr><tr><td>6.2 Main results</td><td>113</td></tr><tr><td>6.3 New ideas and strategy for the proofs</td><td>118</td></tr><tr><td><b>7 Bibliography</b></td><td><b>129</b></td></tr></table>
List of Figures

<table>
  <tbody>
    <tr>
      <td>1.1</td>
      <td>Quantitative estimates via concentration</td>
      <td>18</td>
    </tr>
    <tr>
      <td>1.2</td>
      <td>Chronology of results, Part I</td>
      <td>23</td>
    </tr>
    <tr>
      <td>1.3</td>
      <td>Chronology of results, Part II</td>
      <td>24</td>
    </tr>
    <tr>
      <td>1.4</td>
      <td>Main results ordered according to 'complexity of the equation' against 'complexity of the domain'</td>
      <td>25</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Some results ordered according to 'criticality'</td>
      <td>26</td>
    </tr>
    <tr>
      <td>2.1</td>
      <td>A bumpy John domain</td>
      <td>35</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Regularity of the homogenized boundary data</td>
      <td>57</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Non-uniform boundary layer size</td>
      <td>59</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Stopping time argument for the decomposition of the boundary layer</td>
      <td>60</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Calderón-Zygmund decomposition of the boundary layer</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Continuous alignment on concentrating sets</td>
      <td>76</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Interpolation for L&#x0302;<sub>t</sub>L&#x0302;&#x0307;<sub>x</sub>(Q<sub>+</sub>(1))</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Quantitative local-in-space short-time smoothing and concentration near potential Type I singularities</td>
      <td>95</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Blow-up of the L<sup>3</sup> norm on concentrating sets</td>
      <td>114</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Quantification of Seregin's 2012 result</td>
      <td>117</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Quantitative regularity via concentration: a toy model</td>
      <td>120</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Quantitative regularity via concentration: quantitative boundedness</td>
      <td>121</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Quantitative regularity via concentration: propagation of concentration</td>
      <td>122</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Quantitative regularity via concentration: summation of scales</td>
      <td>123</td>
    </tr>
  </tbody>
</table>
# Chapter 1

# Introduction

## Contents

<table><tr><td>1.1</td><td>A few transversal themes</td><td>13</td></tr><tr><td>1.1.1</td><td>Steady vs. unsteady incompressible viscous fluids</td><td>13</td></tr><tr><td>1.1.2</td><td>Interplay of incompressibility and no-slip</td><td>14</td></tr><tr><td>1.1.3</td><td>An enlarged point of view of regularity theory: building blocks.</td><td>14</td></tr><tr><td>1.1.4</td><td>An enlarged point of view of regularity theory: improved regularity</td><td>15</td></tr><tr><td>1.1.5</td><td>Self-similarity and building blocks</td><td>16</td></tr><tr><td>1.1.6</td><td>Scaling laws for excess quantities</td><td>16</td></tr><tr><td>1.1.7</td><td>Concentration</td><td>17</td></tr><tr><td>1.1.8</td><td>Concentration and quantitative regularity</td><td>17</td></tr><tr><td>1.1.9</td><td>Quantitative asymptotics and regularity</td><td>17</td></tr><tr><td>1.1.10</td><td>Regularity and spatial decay</td><td>18</td></tr><tr><td>1.2</td><td>Presentation of our main contributions</td><td>18</td></tr><tr><td>1.2.1</td><td>Overview of the main new methods</td><td>19</td></tr><tr><td>1.2.2</td><td>Chronology of the results in this thesis</td><td>21</td></tr><tr><td>1.2.3</td><td>Two graphs to represent our main results</td><td>21</td></tr><tr><td>1.3</td><td>Outline of the thesis</td><td>21</td></tr></table>

This habilitation thesis is about a selection of my works carried out since my recruitment as a CNRS researcher in the fall of 2015. These works are concerned with the study of regularity for Partial Differential Equations, with a focus on equations stemming from fluid dynamics. In a broad sense, regularity theory is the study of the local behavior of solutions. We promote an extended point of view of regularity, that goes beyond the one of classical regularity in two aspects. First regularity in not necessarily measured in the sense of $C^k$ or $H^s$ function spaces. Second regularity may hold at certain scales but not all, which is emphasized by the terminology ‘improved’, ‘mesoscopic’ or ‘large-scale’ regularity. Therefore, in our view the main objectives are: (i) to identify a range of scales where there is a certain self-similar behavior, (ii) to find basic objects i.e. building blocks, that represent well the solutions at these scales (polynomials, piecewise polynomials, singular functions in corners, oscillating polynomials in homogenization...), (iii) to prove scaling laws for excess quantities, i.e. local error estimates, at these scales. Our analysis is motivated by physics (concentration effects in composite materials, fluids slipping over rough surfaces, generation of turbulence near boundaries in fluids). It is our hope that the tools that I developed together with my collaborators can further mature and contribute to analyzing such
phenomena from the perspective of regularity theory (scales relevant for turbulence, shape of potential singularities, breaking of the criticality barrier, cascade of energy, dissipation near boundaries, effect of rough or flat walls on the generation of vorticity).

There are two main parts in this thesis. The first part (2 chapters) is concerned with large-scale regularity and quantitative homogenization of stationary Navier-Stokes and el-liptic equations. The second part (3 chapters) is concerned with the three-dimensional non stationary Navier-Stokes equations. There is a gap in terms of difficulty between the station- ary and the non stationary Navier-Stokes equations. In the latter case, the nonlocality due to the incompressibility constraint is stronger especially near boundaries, and the only known controlled quantity, the energy, is supercritical. This explains that two major questions re- main essentially open for the ‘finite-energy weak’ or ‘turbulent’ solutions constructed by Leray in 1934 [238]: (a) the uniqueness of finite-energy solutions relevant to our ability to predict the behavior of fluids, see the paper ‘The real butterfly effect’ by Palmer, Döring and Seregin [278], (b) the question of the regularity of solutions, which is believed to be important to the understanding of turbulence, see the description of the Millenium problem by Fefferman [131]. As a result, for stationary fluids our program is much more advanced than for non stationary fluids.

As for the first part of the thesis, our main results are on the following topics. First, in collaboration with Higaki (former postdoctoral researcher, now at Kobe University) and Zhuge (The University of Chicago), we prove large-scale Lipschitz and higher-order $C^{1,\mu}$, $C^{2,\mu}$ regularity in bumpy (i.e. asymptotically flat) Lipschitz and John domains, that are very rough in two regards: the small-scale oscillations of the boundary are arbitrary and the boundary may be as rough as that of certain fractals. Second, in collaboration with Armstrong (NYU), Kuusi (University of Helsinki) and Mourrat (ENS de Lyon), we prove quantitative error estimates for the homogenization of boundary layer correctors and obtain near-optimal error estimates that outperform previous bounds.

As for the second part of the thesis, let us point to our main contributions. First, we in- vestigate the impact of unbounded boundaries on the nonlocal effects due to the incompres- sibility constraint. In collaboration with Maekawa (Kyoto University) and Miura (Tokyo Institute of Technology), we find new pressure formulas that on the one hand emphasize the strong nonlocality of the Navier-Stokes equations in the half-space, but on the other hand enable to find controls that are good enough for a number of applications. With Maekawa and Miura, we prove the existence of local energy solutions in the half-space, which was an open question since similar work in the whole-space. In collaboration with Barker (Uni- versity of Bath), we prove fractional pressure estimates in the half-space, that enable us to unify Type I blow-up notions and hence to develop a new strategy for the proof of regula- rity under local vorticity alignement, ‘geometric concentration’ being the other side of the coin. Second, we investigate the strength of nonlocal effects in view of local smoothing properties. With Barker, we prove local (in time and space) smoothing in the critical case and ‘norm concentration’ in the whole space. We extend these results to the half-space, where the nonlocality is much stronger, in collaboration with Albritton (IAS, Princeton) and Barker. Third, with Barker, we develop a new method for the proof of quantitative regularity that grounds on the analysis of concentration phenomena. That line of research enables us to give a partial answer of a conjecture by Tao about the blow-up of slightly supercritical Orlicz norms.

The main contributions are summarized in Section 1.2 below, mainly via tables or gra- phical representations. We now take a closer look at a few transversal themes.
# 1.1 A few transversal themes

## 1.1.1 Steady vs. unsteady incompressible viscous fluids

In this thesis we study the three-dimensional stationary Stokes and Navier-Stokes equations

$$ -\Delta U + U \cdot \nabla U + \nabla P = 0, \quad \nabla \cdot U = 0, \qquad (1.1) $$

for velocity $U = U(x) \in \mathbb{R}^3$ and pressure $P = P(x) \in \mathbb{R}$, in bumpy half-spaces, see Chapter 2. We also study the non stationary three-dimensional Navier-Stokes equations

$$ \partial_t U - \Delta U + U \cdot \nabla U + \nabla P = 0, \quad \nabla \cdot U = 0, \qquad (1.2) $$

for velocity $U = U(x, t) \in \mathbb{R}^3$ and pressure $P = P(x, t) \in \mathbb{R}$, in the whole-space and in the flat half-space. We recall that the Stokes system is the linearized version of the Navier-Stokes system around 0.

The stationary Navier-Stokes equations (1.1) and the non stationary Navier-Stokes equations (1.2) share similar scaling invariance: for $\lambda \in (0, \infty)$

$$ U \text{ solves (1.1) implies } U_{\lambda} = \lambda U(\lambda \cdot) \text{ solves (1.1),} \qquad (1.3) $$

$$ U \text{ solves (1.2) implies } U_{\lambda} = \lambda U(\lambda \cdot, \lambda^{2} \cdot) \text{ solves (1.2).} \qquad (1.4) $$

These scaling properties are fundamental for the study of the regularity or singularity of solutions, since they enable to zoom-in or zoom-out. These scaling properties enable to discriminate between norms/quantities that are invariant under the scaling, subcritical or supercritical. Scale-critical quantities play a key role for the regularity theory: $\varepsilon$-regularity results [75, 243, 342] among many other works, quantitative regularity in Chapter 6 (see also Subsection 1.1.8), etc.

One key difficulty of the non stationary three-dimensional Navier-Stokes equations is the fact that the only known controlled quantity, the global energy

$$ \mathcal{E}(U, T) = \sup_{t \in (0, T)} \frac{1}{2} \int_{\mathbb{R}^3} |U(\cdot, t)|^2 + \int_{0}^{T} \int_{\mathbb{R}^3} |\nabla U|^2 $$

is supercritical with respect to the scaling (1.1) i.e.

$$ \mathcal{E}(U_{\lambda}, T/\lambda^2) = \frac{1}{\lambda} \mathcal{E}(U, T). $$

Hence the energy does not control well the nonlinearity, or in other words the smoothing from the parabolic part of the equation is not strong enough to bootstrap the regularity starting from the energy class. As a consequence, most of the regularity results in this thesis, and in the mathematical literature concerned with (1.2), are under critical or slightly supercritical a priori bounds, in scale-critical regimes (self-similar solutions), or assume certain special structure (axisymmetry without swirl, local vorticity alignment).

A second key difficulty is related to the incompressibility that introduces non local effects via the pressure. These effects can already be seen at the level of the linear Stokes system. They are stronger in the non stationary Stokes equations than in the stationary Stokes equations. In the latter case, one can directly estimate the pressure in terms of the velocity

$$ \|P - (P)_{B(1)}\|_{L^2(B(1))} \leq C \|\nabla U\|_{L^2(B(1))} $$
via the Bogovskii estimate (see (2.14)), while in the unsteady Stokes system we have

$$ \|P - (P)_{B(1)}\|_{L^2(Q(1))} \le C \|\partial_t U - \Delta U\|_{L^2(-1,0;H^{-1}(B(1)))}. $$

We see that the pressure competes with the time derivative of the velocity. This is emphasized by the Serrin examples, see (5.1) and [311]. The nonlocal effects are also stronger in the half-space with no-slip boundary condition than in the whole-space or the half-space with perfect slip boundary condition; see (1.1.2). Finally, these effects are obstacles to local smoothing properties for the non stationary systems, see [347, 225] for the interior regularity and [295, 208, 301] for the boundary regularity.

We finish this very concise overview by defining ‘regular’ and ‘singular’ space-time points in the whole-space (resp. the half-space). We say that $(\bar{x}, t) \in \mathbb{R}^3 \times (0, \infty)$ (resp. $(\bar{x}, t) \in \mathbb{R}_+^3 \times (0, \infty)$) is a ‘regular point’ of $U$, if there exists $r \in (0, \infty)$ such that $U \in L^\infty(B_{\bar{x}}(r) \times (t-r^2, t))$ (resp. $U \in L^\infty(B_{\bar{x},+}(r) \times (t-r^2, t)))$. A contrario, a point $(\bar{x}, t) \in \mathbb{R}^3 \times (0, \infty)$ (resp. $(\bar{x}, t) \in \overline{\mathbb{R}}_+^3 \times (0, \infty)$) is a ‘singular point’, or a ‘blow-up point’ if it is not regular. A time $T^* \in (0, \infty)$ is called a blow-up time if there exists $\bar{x} \in \mathbb{R}^3$ such that $(\bar{x}, T^*)$ is a singular point.

### 1.1.2 Interplay of incompressibility and no-slip

Boundaries break the structure of free flows by introducing additional constraints. In the case of the no-slip boundary condition the fluid sticks to the boundary $U = 0$, which results in the creation of small scales in the vicinity of the boundary. These small scales can be further amplified by the nonlinearity of the fluid equation. The nonlocal effects due to the incompressibility are stronger near the boundary, which is reflected by the fact that the pressure, its harmonic component, depends on the history of the flow and involves the initial data, see Subsection 5.3.2. This makes the analysis of the regularity near the boundary particularly challenging. From the mathematical point of view, some of the difficulties are:

(1) the nonlinear and nonlocal boundary condition for the vorticity, see [246, 159] and (4.1);

(2) the lack of local smoothing in spatial variables, see the examples in [208, 301, 210, 91] that demonstrate that local boundedness of the solution near the flat boundary does not imply boundedness of its gradient, see Subsection 4.1.4;

(3) the fact that one cannot estimate the pressure in terms of the nonlinearity $U \otimes U$ contrary to the whole-space, see [220] and Subsection 4.1.2.

We investigate these effects thoroughly in Section 4.1.2 and Subsection 4.1.4. Some of the open problems in the half-space are also mentioned in Subsection 4.1.6. Let us mention that we manage to obtain certain results in a scale-critical (so-called Type I) regime where the nonlinear effects compete directly with the diffusion. Finally, the relationship between boundary effects and potential singularity formation is discussed in [232].

### 1.1.3 An enlarged point of view of regularity theory: building blocks

The goal of the regularity theory is to describe the local behavior of solutions, which means finding functions that represent well the solutions locally, at certain scales, and measuring quantitatively the local error, or excess decay. This goal can be very different from proving regularity in the sense of $C^k$ or $H^s$ function spaces. Let us list several examples:
(i) **Homogenization**

The idea is already present in the work of Avellaneda and Lin [29], where 'a(·/ε)-harmonic linear functions', i.e. linear functions perturbed by oscillating cell correctors of the form

$$x + \varepsilon\chi(\frac{x}{\varepsilon})$$

are the building blocks of the $C^{1,\mu}$ regularity theory of elliptic operators $-\nabla \cdot a(\frac{x}{\varepsilon})\nabla \cdot a(\frac{x}{\varepsilon})$ with highly oscillating $\varepsilon$-periodic coefficients. These building blocks retain some of the oscillations of the solutions. Notice the highly singular behavior of derivatives in $\varepsilon$, which allows to say that in that sense these building blocks are not smooth. Further developments include the following works: Armstrong, Kuusi and Smart [24] higher-order a-harmonic polynomials in periodic homogenization, Armstrong, Gloria and Kuusi [20] for almost-periodic homogenization, Armstrong, Kuusi and Mourrat [22, 21] and Gloria, Otto and Neukamm [167] for stochastic homogenization.

(ii) **Transmission problems**

In the paper [355], Zhuge introduces 'piecewise linear functions' that play the role of building blocks for the $C^{1,\mu}$ regularity theory for transmission problems between periodic structures separated by a flat interface.

(iii) **Corners**

In the paper [205], Josien, Raithel and Schäffner introduce 'a-harmonic singular functions' that play the role of building blocks for the higher-order boundary regularity theory of elliptic equations with highly-oscillating random coefficients near corners. These building blocks contain the singularities of the function due to the corner.

(iv) **Wall laws**

In our work with Higaki [183], see Theorem 2.7 and Remark 2.8, we introduce 'Navier polynomials' that describe a local wall law for fluids above a periodic Lipschitz boundary. Higher-order correctors were constructed in a similar context in collaboration with Higaki and Zhuge [184], see Theorem 2.9 and Theorem 2.10, and by Higaki and Zhuge in [185].

The building blocks are 'blow-up limit' or 'blow-down limit' solutions depending on the context. Their role for the regularity is emphasized by Liouville-type results such as Theorem 2.11.

1.1.4 An enlarged point of view of regularity theory: improved regularity

The terminology 'improved' regularity refers to the fact that we are able to prove, in different contexts, regularity estimates at certain scales that are false at other scales. This is typically the case in homogenization, where improved regularity is inherited at large-scales from the convergence to a homogenized, constant coefficient operator, with improved regularity. This improvement of flatness at large scales was successfully exploited (by compactness or more quantitative arguments) in many works concerned with periodic homogenization by Avellaneda and Lin [29], almost-periodic homogenization by Armstrong and Shen [27], Zhuge [352] and stochastic homogenization by Armstrong and Smart [25], Armstrong, Kuusi and Mourrat [22, 21], Gloria, Otto and Neukamm [167] of elliptic operators $-\nabla \cdot a(\frac{x}{\varepsilon})\nabla \cdot a(\frac{x}{\varepsilon})$ or variants. If one considers elliptic operators with barely bounded coefficients, the Lipschitz regularity that one proves at large scales $\gtrsim \varepsilon$ may certainly fail at small scales $\lesssim \varepsilon$. Hence the terminology 'improved regularity'. We also sometimes refer to 'large-scale regularity' or 'mesoscopic regularity'.
Our leitmotiv is that of a scale-dependent notion of regularity. We pursue that objective in our works concerned with improved regularity above rough boundaries that are asymptotically flat. This program was initiated while I was a postdoctoral researcher at the University of Chicago working with Kenig [214], and continued first in collaboration with Higaki [183] and then with Higaki and Zhuge [184]; see also the related works of Zhuge [356], Gu and Zhuge [178]. A major objective is to develop tools that enable us to free ourselves from the lack of smoothness at small scales, see Subsection 1.2.1 below and Chapter 3.

### 1.1.5 Self-similarity and building blocks

Here we allude to the fact that the building blocks described in Subsection 1.1.3 above ensure a certain ‘self-similarity’ across the scales where one proves ‘improved’ regularity. They are essential to iteration arguments, where the control of excess quantities at scale $\theta^{k+1}$ by excess quantities at scale $\theta^k$ is similar, by rescaling, to the control of excess quantities at scale $\theta$ by excess quantities at scale 1. This idea is further developed in Remark 2.1 and Remark 2.12 below.

### 1.1.6 Scaling laws for excess quantities

By the Morrey-Campanato characterization of Hölder continuity [157, Theorem 5.5], Hölder norms $C^{0,\mu}$, $\mu \in (0, 1]$, can be measured in terms of the decay of ‘excess quantities’ or ‘oscillation’

$$ \inf_{a \in \mathbb{R}} \int_{B(r)} |U - a|^2 = \int_{B(r)} |U - (U)_r|^2 \lesssim r^{2\mu} $$

where $(U)_r$ denotes the mean of $U$ over $B(r)$ and $f = |B(r)|^{-1} \int$. This characterization is very useful since $L^2$ or $L^p$ norms are easy to accessed for solutions of elliptic equations, Stokes or Navier-Stokes, by energy estimates. Caccioppoli’s inequality for solutions of divergence-form elliptic equations and Poincaré-Wirtinger’s inequality show that Lipschitz regularity is equivalently measured in terms of the boundedness of the ‘energy density’ [22]

$$ \int_{B(r)} |\nabla U|^2 \lesssim 1. $$

Let us give two further examples to illustrate the previous ideas. In order to measure classical $C^{1,\mu}$ regularity, $\mu \in (0, 1]$, one considers the decay of the ‘excess quantities’ or ‘oscillation’

$$ \inf_{a \in \mathbb{R}, b \in \mathbb{R}^d} \int_{B(r)} |U - a - b \cdot x|^2 = \int_{B(r)} |U - (U)_r - (\nabla U)_r \cdot x|^2 \lesssim r^{2\mu}. $$

The linear polynomials $a + b \cdot x$ are for instance the building blocks for the $C^{1,\mu}$ regularity theory of constant coefficients elliptic equations. The large-scale $C^{1,\mu}$ regularity for the operator $-\nabla \cdot a(\frac{x}{\epsilon})\nabla$, is measured in terms of the decay of the ‘excess quantities’ or ‘oscillation’

$$ \inf_{a \in \mathbb{R}, b \in \mathbb{R}^d} \int_{B(r)} |U^{\epsilon} - a - b \cdot (x + \epsilon \chi(\frac{x}{\epsilon}))|^2 = \int_{B(r)} |U^{\epsilon} - (U^{\epsilon})_r - (\nabla U^{\epsilon})_r \cdot (x + \epsilon \chi(\frac{x}{\epsilon}))|^2 \lesssim r^{2\mu}. $$

Finally, let us note that there is also a version of this characterization for parabolic Hölder spaces, see [237, Lemma 13.2].
1.1.7 Concentration

In this thesis, concentration refers to three phenomena:

(1) **Concentration in boundary layers**

These boundary layers describe the strong gradients of solutions near bumpy boundaries in Chapter 2 or near the boundary of a domain with a singular, highly oscillating, boundary data in Chapter 3. In Chapter 2 the main difficulty is to construct first-order and second-order boundary layers in a context where there is no smoothness of the boundary. In Chapter 3 the main difficulty is to paste together boundary layers with very different sizes (due to resonance or non-resonance properties) in a curved geometry.

(2) **Norm concentration**

This refers to the concentration/accumulation of certain critical norms on concentrating sets near potential Type I singularities of the unsteady Navier-Stokes equations in Chapter 5 and the use of norm concentration to obtain quantitative estimates in Chapter 6.

(3) **Geometric concentration**

This refers to breaking of smoothness of the vorticity direction on concentrating sets near potential Type I singularities of the unsteady Navier-Stokes equations in Chapter 4. Such results were previously known only globally.

1.1.8 Concentration and quantitative regularity

In Chapter 6 we develop a scheme that enables us to obtain explicit quantitative estimates of the form

$$
\text{a subcritical norm of the solution} \lesssim \mathcal{G}(\text{a critical norm of the solution}) \quad (1.5)
$$

where $\mathcal{G}$ is an explicit function; see (6.1) for a more precise form. Our scheme is a physical
space analog of the one developed by Tao in 2019 [334].

To illustrate the strategy we give a physical space version for $U \in L_{t,x}^5$ and a Fourier
space version for $U \in L_t^\infty L_x^3$, see Figure 1.1 for a summary of the method. First the
critical standing assumption, $U \in L_{t,x}^5$ or $U \in L_t^\infty L_x^3$ implies that certain scale-critical
quantities are small (in terms of the size of $U$ in $L_{t,x}^5$ or $L_t^\infty L_x^3$) near final time or for high
frequencies. The threshold scales can be interpreted as certain Kolmogorov dissipation
scales where diffusion starts to dominate the nonlinearity. One then, this is the second step,
infers regularity via $\varepsilon$-regularity or tools from the mild solution theory. The outcome is the
exponential bound (6.24) ($\mathcal{G} = \exp(\cdot^5)$) for the control of the $L_{t,x}^\infty$ norm in terms of the
$L_{t,x}^5$ norm. For the other case handled by Tao [334], one gets the triple exponential bound
(6.12), with $\mathcal{G} = \exp \exp \exp(\cdot^C)$, for the control of the $L_{t,x}^\infty$ norm in terms of the $L_t^\infty L_x^3$
norm.

1.1.9 Quantitative asymptotics and regularity

There is a close link between quantitative estimates and regularity. As far as we know,
this link was first used by Caffarelli and Peral [76] and then brought to the field of stochastic
homogenization by Armstrong and Smart [25]. The idea is simple. If one has a family of
functions $U^\varepsilon$ that is quantitatively close (at any scale) to a family of functions $\bar{U}$ that have
<table><caption>Figure 1.1 – Quantitative regularity via concentration: a summary</caption><tr><td colspan="3">the global scale-critical standing assumption</td></tr><tr><td>U ∈ L<sup>5</sup><sub>t,x</sub><br>see Subsection 6.3.1</td><td>U ∈ L<sup>∞</sup><sub>t</sub>L<sup>3,∞</sup><sub>x</sub> and U(·, 0) ∈ L<sup>3</sup><br>see Subsection 6.3.2</td><td>U ∈ L<sup>∞</sup><sub>t</sub>L<sup>3</sup><sub>x</sub><br>see Subsection 6.3.3</td></tr><tr><td colspan="3">prevents the following scale-critical quantities</td></tr><tr><td>(-t)<sup>1/5</sup> ||U(·, t)||<sub>L<sup>5</sup></sub><br>to concentrate to close to final time</td><td>√-t ∫<sub>B<sub>O(||U||<sub>L<sup>∞</sup><sub>t</sub>L<sup>3,∞</sup><sub>x</sub>)√-t</sub></sub> |ω(·, t)|<sup>2</sup></td><td>N<sup>-1</sup> |P<sub>N</sub>U(t', x')|<br>to concentrate for large<br>frequencies N ≥ N<sub>0</sub></td></tr><tr><td colspan="3">no concentration, i.e. smallness, implies regularity</td></tr></table>

improvement of integrability or improvement of flatness, one can use that error estimate in a Schauder-type iteration to infer regularity of the first family of functions $U^ε$. It is enough to have suboptimal error estimates on $U^ε - \bar{U}$ to make this scheme work. Regularity in turn implies improved error estimates. We describe this ‘quantitative’ scheme in full details in Subsection 2.1.3 and compare it to a ‘compactness’ scheme, see Subsection 2.1.2, initiated by Avellaneda and Lin [29] in the context of periodic homogenization.

Such a quantitative scheme is particularly useful to us, see Subsection 2.3.2, for the regularity in bumpy John domains. By a careful bootstrap of the integrability of the gradient of the solution (Caccioppoli, Meyers, Calderón-Zygmund and Lipschitz), we manage to go around the lack of boundary layer correctors.

### 1.1.10 Regularity and spatial decay

By using scaling properties, large-scale regularity can be transferred to large-scale decay estimates for Green and Poisson kernels associated to elliptic or Stokes operators, see the pioneering work by Avellaneda and Lin [30], and the subsequent papers [216] by Kenig, Lin and Shen, the paper [282] from my Ph.D. work, [177] by Gu and Zhuge. This property also plays a role in our paper [184, Appendix B] with Higaki and Zhuge.

For the non stationary Navier-Stokes equations, scaling properties were used in combination with Hölder regularity to obtain a priori estimates for forward self-similar solutions in the work by Jia and Šverák that constructs forward self-similar solutions for arbitrarily large data [197].

## 1.2 Presentation of our main contributions

To facilitate the reading, we summarize our results in three different manners in the next subsections. We emphasize different aspects: new methods that we develop in Subsection 1.2.1, chronology of the results (main works that inspire us, chronology of our results and some further developments that followed our papers) in Subsection 1.2.2, location of our results on ‘complexity of equation’, ‘criticality’ and ‘complexity of domain’ axes in Subsection 1.2.3.
### 1.2.1 Overview of the main new methods

We list here the main new methods that I developed with my collaborators for **Part I** of the thesis:

#### (1) Tools to decouple large-scale from small scale regularity I

In the context of bumpy Lipschitz domains, in collaboration with Higaki we adapt a method that enables us to build boundary layer correctors over (rough) Lipschitz boundaries; see Subsection 2.3.1. The method is based on a domain decomposition and local energy estimates (so-called Saint-Venant estimates) near the boundary.

As a consequence we prove *improved regularity in bumpy Lipschitz domains* (Theorem 2.5, Theorem 2.7), i.e. large-scale regularity estimates (Lipschitz, $C^{1,\mu}$) that are false at the small scales.

#### (2) Tools to decouple large-scale from small scale regularity II

In the context of bumpy John domains (that include certain fractal boundaries such as Koch's snowflake), in collaboration with Higaki and Zhuge we develop a method that circumvents the construction of boundary layer correctors for the proof of large-scale Lipschitz estimates; see Subsection 2.3.2. Indeed the method of construction of boundary layers for bumpy John domains breaks down. Therefore, one has to perseveringly bootstrap the integrability of the gradient of the solution via an interplay between quantitative suboptimal error estimates, regularity estimates (Caccioppoli inequalities, regularity in smooth domains) and arguments oblivious to the equation (iterations or real variable arguments).

As a consequence we prove *improved Lipschitz regularity in bumpy John domains* (Theorem 2.6).

#### (3) Construction of higher-order boundary layers

In collaboration with Higaki and Zhuge, we pioneer a new method to build second-order boundary layers in bumpy John domains, i.e. with data growing linearly in the horizontal direction; see Subsection 2.3.2. Our construction uses the first-order boundary layer corrector in an Ansatz for the second-order boundary layer. The method can be made more systematic to address the existence of correctors of arbitrary high order.

As a consequence we prove *higher-order $C^{2,\mu}$ regularity* (Theorem 2.10) and *Liouville theorems for entire solutions of the stationary Stokes* in bumpy John domains (Theorem 2.11).

#### (4) Calderón-Zygmund decomposition of a boundary layer

In order to handle the homogenization of systems with highly oscillating boundary conditions, so-called boundary layers, that are strongly heterogeneous and anisotropic, in collaboration with Armstrong, Kuusi and Mourrat we initiate a new stopping time argument to decompose the boundary layer; see Subsection 3.3.2. This enables us to construct the boundary layer by a careful pasting of local boundary layers that are thick near bad almost resonant directions and thin near good non-resonant directions.

As a consequence we prove *near optimal error estimates* for the quantitative homogenization of elliptic systems with highly oscillating Dirichlet data in uniformly convex domains (Theorem 3.1). The method can be successfully extended to more general domains or different boundary conditions.

We list here the main new methods that I obtained with my collaborators for **Part II** of the thesis:
(5) ε-regularity for a Navier-Stokes equation with scale-critical drifts

In collaboration with Barker, we adapt the method of Caffarelli-Kohn-Nirenberg for the proof of ε-regularity for a perturbed Navier-Stokes system in the whole-space with scale-critical drift terms. Our method is flexible enough to handle drifts in critical (Lebesgue) and ultra-critical (Lorentz or Besov) spaces.

As a consequence we prove *local-in-space short-time smoothing* in the critical case (Theorem 5.1) and initiate the study of ‘*norm concentration*’ for critical norms near potential singularities in the whole-space (Theorem 5.5).

(6) New method for pressure estimates in the half-space

In collaboration with Maekawa and Miura, we find a new decomposition of the pres-
sure for the unsteady Navier-Stokes equations in the half-space. The interplay be-
tween the incompressibility condition and the no-slip boundary condition is responsi-
ble for strong non local effects, that are concentrated in a part of the pressure that we
call the ‘harmonic pressure’. We obtain bounds for the harmonic pressure near initial
time that enable us to use local energy methods.

As a consequence we prove the existence of *global-in-time local energy solutions* in the half-space (Theorem 5.10) that was mentioned as an open question in several works. This provides then a good framework to study local smoothing properties and norm concentration near potential singularities in the half-space. Our formulas are also useful outside the uniformly locally finite energy framework, for instance for Morrey-type local energies and ‘intermittent data’.

(7) **A new scheme for regularity under local coherence of the vorticity field** In collaboration with Barker, we introduce a new strategy for the proof of regularity for the three-dimensional Navier-Stokes equations under continuous alignment of the vorticity; see Subsection 4.3.3. That strategy is based on compactness in a scale-critical regime and persistence of singularities. It avoids the use of Liouville theorems that are particularly complicated to prove in the half-space. Our method works in the whole-space and in the half-space.

As a consequence we prove ‘geometric concentration’ results (Theorem 4.3), i.e. breaking of the continuity of the vorticity direction on concentrating sets. Such results are new even for the whole-space.

(8) Quantitative regularity via concentration of scale-critical quantities

In collaboration with Barker, we develop a method based on the concentration of
scale-invariant quantities to prove quantitative regularity under the boundedness of a
standing critical assumption (such as $L_{t,x}^5$, $L_t^\infty L_x^3$ or a Type I condition); see Subsec-
tion 6.3.1 for a toy model and Subsection 6.3.2 for the presentation of the method.
Our method is the pendant in physical space of the Fourier space method pioneered
by Tao in 2019; see Subsection 6.3.3. Working directly in physical space, with a
scale-invariant local enstrophy, as we do, enables us to have localized results. Our
method as well as Tao’s rely on the quantitative estimate of a ‘Kolmogorov scale’
(either a sufficiently high frequency in Tao’s work, or a sufficiently small space/time
length in our work), at which diffusion overtakes the nonlinearity.

As a consequence we prove the *localized blow-up of the critical $L^3$ norm* in a Type I blow-up scenario (Theorem 6.1), and *quantify Seregin's 2012 regularity criteria* (Theorem 6.3).

(9) Transferring subcriticality of the data forward in time

In collaboration with Barker, we bring a method from the field of dispersive equations
to the field of fluid mechanics; see Subsection 6.3.5. The method is taken from a work
by Bulut for nonlinear defocusing Schrödinger that was pointed out to us by Patrick
Gérard. Subcritical energy estimates can be combined with quantitative regularity
estimates as obtained by Tao in 2019. Hence the growth of the subcritical norm along
the evolution can be estimated.

As a consequence we prove *mild criticality breaking* (Lemma 6.7) and the *blow-up of slightly supercritical Orlicz norms* (Theorem 6.5), which answers an open question mentioned by Tao in his 2019 work.

Let us emphasize that the schemes developed in points (7), (8) and (9) above are non per-
turbative.

1.2.2 Chronology of the results in this thesis

We refer to Figure 1.2 on page 23 that describes the chronology of the results in Part I of the thesis and to Figure 1.3 on page 24 that describes the chronology of the results in Part II of the thesis.

1.2.3 Two graphs to represent our main results

We refer to the Figure 1.4 on page 25 that orders our results according to ‘complexity of the equation’ against ‘complexity of the domain’ and to the Figure 1.4 on page 26 that order the results of Part II according to ‘criticality’.

**1.3 Outline of the thesis**

There are two parts in the thesis. The first part (Chapter 2 and Chapter 3) is devoted to steady problems: elliptic systems and the steady three-dimensional Stokes and Navier-Stokes systems. The second part (Chapter 4, Chapter 5 and Chapter 6) is devoted to the unsteady three-dimensional Navier-Stokes equations. To increase the readability of the manuscrit, each chapter has the exact same structure with four sections: ‘context: state of the art and obstacles’, ‘main results’ (with a paragraph underlining the ‘novelty of our results’ and another ‘further developments’ if relevant) and ‘new ideas and strategy for the proofs’.

In Chapter 2 we study the large-scale local boundary regularity of the steady Navier-Stokes system over very rough Lipschitz or John boundaries, that are asymptotically flat; we call such domains ‘bumpy Lipschitz’ or ‘bumpy John domains’. This is work in collaboration with Higaki (Kobe University) and Zhuge (The University of Chicago).

In Chapter 3, we study the homogenization of elliptic systems with a highly oscillating Dirichlet boundary condition and obtain near optimal error estimates. This is work with Armstrong (NYU), Kuusi (University of Helsinki) and Mourrat (ENS de Lyon).

In Chapter 4, we study the regularity for the three-dimensional Navier-Stokes equations under continuous alignement of the vorticity. We introduce a new method in the scale-critical regime (Type I scenario) that enables us to obtain ‘geometric concentration’ results in the whole-space and the half-space. This is work in collaboration with Barker (University of Bath).
In Chapter 5, we study local smoothing properties for the three-dimensional Navier-Stokes equations in the whole-space and the half-space. As a corollary of ‘local-in-space short-time smoothing’ we get ‘norm concentration’ results near potential Type I singularities. This is work with Albritton (IAS, Princeton) and Barker (University of Bath). We also obtain new pressure estimates that enable us to study solutions with uniformly locally (but not necessarily globally) finite energy in the half-space. This is work with Maekawa (Kyoto University) and Miura (Tokyo Institute of Technology).

In Chapter 6, we study quantitative regularity for the three-dimensional Navier-Stokes equations in three different scenarios: (i) in a Type I blow-up scenario, (ii) in a slightly supercritical scenario (boundedness of an Orlicz norm), (iii) under boundedness of a critical norm along a sequence of times. This is work with Barker (University of Bath).
<table><caption>Figure 1.2 – Chronology of results, Part I: boundary layers and regularity for steady problems in disordered environments; the dates of the theorems are those of first release on arXiv</caption><thead><tr><th></th><th>quantitative large-scale regularity in bumpy domains</th><th>concentration in homogenization</th></tr></thead><tbody><tr><th>main sources of inspiration</th><td>- Avellaneda, Lin [29] (compactness methods in homogenization)<br>- Caffarelli, Peral [76] (quantitative method for regularity)<br>- Kenig, Prange [213] (elliptic equations, smooth boundaries with no structure)<br>- Kenig, Prange [214] (elliptic equations, Lipschitz boundaries with no structure)<br>- Gérard-Varet, Masmoudi [154] (boundary layers for Stokes in rough domains)</td><td>- Gérard-Varet, Masmoudi [155] (boundary layers in polygonal domains)<br>- Gérard-Varet, Masmoudi [156] (boundary layers in convex domains)<br>- Kenig, Lin, Shen [216] (homogenization of Poisson kernels)</td></tr><tr><th>main results of the thesis</th><td>for the stationary Navier-Stokes equations in bumpy domains<br>- Theorem 2.7 with Higaki in 2019 (local wall law in Lipschitz domains)<br>- Theorem 2.6 with Higaki and Zhuge in 2021 (improved regularity in bumpy John domains)<br>- Theorem 2.10 with Higaki and Zhuge in 2021 (higher-order regularity in John domains)</td><td>for boundary layers in periodic homogenization<br>- Theorem 3.1 with Armstrong, Kuusi and Mourrat in 2016 (near-optimal error estimates)</td></tr><tr><th>some further developments</th><td>- Gu, Zhuge [178] (system of elasticity in large-scale C<sup>1,α</sup> domains)<br>- Zhuge [356] (elliptic equations in large-scale Reiffenberg flat domains)<br>- Higaki, Zhuge [185] (arbitrary order regularity in bumpy John domains)</td><td>- Shen, Zhuge [316] (optimal estimates in low dimensions)<br>- Shen, Zhuge [317] (almost Lipschitz regularity of the homogenized boundary data)<br>- Shen, Zhuge [353] (relaxation of uniform convexity)</td></tr></tbody></table>
<table><caption>Figure 1.3 – Chronology of results, Part II: regularity and concentration for unsteady fluids; the dates of the theorems are those of first release on arXiv</caption><thead><tr><th></th><th>boundary regularity and pressure estimates</th><th>local smoothing and norm concentration</th><th>concentration and quantitative regularity</th></tr></thead><tbody><tr><td>main sources of inspiration</td><td>- Desch, Hieber, Prüss [124] (Stokes resolvent problem in the half-space)<br>- Lemarié-Rieusset [235] (local energy solutions)<br>- Chang, Kang [89] (fractional pressure estimates)<br>- Constantin, Fefferman [107] (regularity under vorticity alignment)<br>- Giga, Miura [162] (regularity under vorticity alignment in Type I case)</td><td>- Caffarelli, Kohn, Nirenberg [75] (ε-regularity)<br>- Lin [243] (ε-regularity via compactness)<br>- Jia, Šverák [197] (local-in-space smoothing for subcritical data)<br>- Li, Ozawa, Wang [240] (concentration)</td><td>- Escauriaza, Seregin, Šverák [129] (qualitative regularity for L<sup>∞</sup><sub>t</sub>L<sup>3</sup><sub>x</sub>)<br>- Seregin [294] (blow-up of the L<sup>3</sup> norm)<br>- Tao [334] (quantitative regularity for L<sup>∞</sup><sub>t</sub>L<sup>3</sup><sub>x</sub>)<br>- Bulut [73] (mild criticality breaking for nonlinear Schrödinger)</td></tr><tr><td>main results of the thesis</td><td colspan="3">for the unsteady 3D Navier-Stokes equations</td></tr><tr><td></td><td>in the half-space<br>- Theorem 5.10 with Maekawa and Miura in 2017 (global-in-time local energy solutions)<br>- Theorem 4.1 with Barker in 2019 (fractional pressure estimates)<br>- Theorem 4.2 with Barker in 2019 (unification of Type I blow-ups)<br>- Theorem 4.3 with Barker in 2019 (geometric concentration under Type I)</td><td>in the whole-space or the half-space<br>- Theorem 5.1 with Barker in 2018 (local-in-space smoothing in R<sup>3</sup> for critical data)<br>- Theorem 5.5 with Barker in 2018 (critical norm concentration in R<sup>3</sup>)<br>- Theorem 5.11 with Albritton and Barker in 2021 (local-in-space smoothing in R<sup>3</sup><sub>+</sub> for critical and subcritical data)<br>- Theorem 5.13 with Albritton and Barker in 2021 (critical norm concentration in R<sup>3</sup><sub>+</sub>)</td><td>in the whole-space<br>- Theorem 6.1 with Barker in 2020 (localized quantitative blow-up of the L<sup>3</sup> norm near a Type I singularity)<br>- Theorem 6.3 with Barker in 2020 (quantification of Seregin's 2012 result about the blow-up of the L<sup>3</sup> norm)<br>- Theorem 6.5 with Barker in 2021 (blow-up of a slightly supercritical Orlicz norm)</td></tr><tr><td>some further developments</td><td>- Bradshaw, Kukavika, Ożański [63] (global-in-time existence for 'intermittent' data)</td><td>- Kang, Miura, Tsai [211] (local version of local-in-space smoothing)<br>- Kang, Miura, Tsai [212] (concentration for the supercritical L<sup>2</sup> norm)</td><td>- Palasek [276] (quantitative regularity axisymmetric)<br>- Palasek [277] (quantitative regularity 4D Navier-Stokes)<br>- Feng, He, Wang [133] (quantitative regularity for critical Lorentz norms)</td></tr></tbody></table>
<table><caption>Figure 1.5 – Selected results of Part II and from the literature ordered according to ‘criticability’</caption><thead><tr><th colspan="6">increasing level of criticality</th></tr><tr><th>borderline critical</th><th>borderline endpoint critical</th><th>slightly supercritical</th><th colspan="3">supercritical</th></tr><tr><th>L<sub>t</sub><sup>&infin;</sup> L<sub>x</sub><sup>3</sup></th><th>Type I</th><th>'log' breaking</th><th>sup<sub>k</sub> ||U(·, t<sub>k</sub>)||<sub>L<sup>3</sup></sub> &lt; &infin;</th><th>geometric breaking</th><th>energy</th></tr></thead><tbody><tr><td>Escauriaza, Seregin, Šverák [129] (qualitative)</td><td></td><td>Pan [279], Seregin [299], Chen, Tsai, Zhang [95] (axisymmetric)</td><td>Seregin [294] (qualitative regularity)</td><td>Constantin, Fefferman [107] (nonlinearity depletion by vorticity alignment)</td><td>?</td></tr><tr><td>Tao [334] (quantitative)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>- Theorem 4.3 (geometric concentration under Type I)</td><td>- Theorem 6.5 (quantitative blow-up of a slightly supercritical Orlicz norm)</td><td>- Theorem 6.3 (quantitative regularity)</td><td></td><td></td></tr><tr><td></td><td>- Theorem 5.5 (norm concentration in the whole-space)</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>- Theorem 5.13 (norm concentration in the half-space)</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>- Theorem 6.1 (quantitative blow-up of the L<sup>3</sup> norm on concentrating sets)</td><td></td><td></td><td></td><td></td></tr></tbody></table>
# Part I

## Boundary layers and regularity for steady problems in disordered environments
<figure><img src="image_4.png" /></figure>
# Chapter 2

## Compactness and quantitative methods for regularity

This chapter relies mainly on the papers:

- [184], with Mitsuo Higaki and Jinping Zhuge, Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries, to appear in *Analysis & PDE* (2021).

- [183], with Mitsuo Higaki, Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law, *Calc. Var. Partial Differential Equations* (2020).

### Contents

<table><tbody><tr><td><strong>2.1 Context: state of the art and obstacles</strong></td><td><strong>29</strong></td></tr><tr><td>2.1.1 Asymptotic analysis in bumpy domains</td><td>30</td></tr><tr><td>2.1.2 Compactness methods for regularity</td><td>31</td></tr><tr><td>2.1.3 Quantitative methods for regularity</td><td>32</td></tr><tr><td><strong>2.2 Main results</strong></td><td><strong>33</strong></td></tr><tr><td>2.2.1 Large-scale Lipschitz regularity</td><td>36</td></tr><tr><td>Novelty of our results</td><td>37</td></tr><tr><td>2.2.2 Higher-order regularity and local wall laws</td><td>38</td></tr><tr><td>Novelty of our results</td><td>41</td></tr><tr><td>Further developments</td><td>41</td></tr><tr><td><strong>2.3 New ideas and strategy for the proofs</strong></td><td><strong>42</strong></td></tr><tr><td>2.3.1 Boundary layers and large-scale boundary regularity in Lipschitz domains</td><td>42</td></tr><tr><td>2.3.2 Large-scale boundary regularity and boundary layers in John domains</td><td>44</td></tr></tbody></table>

### 2.1 Context: state of the art and obstacles

The study of fluids over rough boundaries plays a prominent role in the field of hydro-dynamics, at least for three reasons.
First, rough, bumpy or corrugated surfaces are ubiquitous in nature and engineering. They appear at any scales from geophysics to zoology and microfluidics. At the microstructure, the geometry may be anything from fractal to periodic and crenellated. No surface is perfectly smooth, and the lack of smoothness may actually enable us to resolve certain oddities, such as the no-collision paradox for a sphere dropped in a viscous fluid under the action of gravity [318, 202, 121, 152, 191]. Moreover, certain roughness patterns are either selected by biological processes and environmental pressure such as scales of sharks for their drag reduction properties, or designed for industrial applications especially in aeronautics, microfluidics and for the transport of fluids in pipes.

Second, the study of roughness is strongly tied to the derivation of boundary conditions in fluid mechanics. The question whether fluids slip or not over surfaces is still a matter of active debate. Experiments show that there is no universal answer and that the slip behavior depends a lot on the geometry and microstructure of the surface [55, 233]. A widespread idea is that roughness favors slip. To give one specific example where finding the most accurate boundary condition is critical, let us cite the field of glaciology. The assessment of various friction laws for the flow of a glacier over a rough bedrock is crucial in order to understand the speed of glacier discharge and eventually estimate the sea level rise as a result of global warming [206, 262].

Third, the study of the impact of roughness on the behavior of fluids accompanied the development of turbulence research, as underlined by Jiménez in [199]:

> Turbulent flows over rough walls have been studied since the early works of Hagen (1854) and Darcy (1857), who were concerned with pressure losses in water conduits. They have been important in the history of turbulence. Had those conduits not been fully rough, turbulence theory would probably have developed more slowly. The pressure loss in pipes only becomes independent of viscosity in the fully rough limit, and this independence was the original indication that something was amiss with laminar theory. Flows over smooth walls never become fully turbulent, and their theory is correspondingly harder.

Investigations of the effect of roughness on fluid flows span three distinct regimes. In the laminar regime, studies focus on the drag reducing properties of roughness elements [46, 146]. As for the onset of turbulence [292, 326], there are some indications that roughness lowers the critical Reynolds number for the transition from the laminar to the turbulent regime [341]. In the fully turbulent regime, a similarity hypothesis for the flow over flat surfaces and for the flow over rough surfaces was put forward [337]. The extent to which such a universal law holds is still being disputed [199, 82, 137, 291].

Our main (ambitious) objective can be vaguely stated as follows:

**Our goal:** Study the impact of roughness on the behavior of fluids and the possible generation of vorticity from the point of view of the regularity theory.

The results in this chapter should be seen as first steps in this longterm research program.

### 2.1.1 Asymptotic analysis in bumpy domains

First, there is an extensive body of works that deal with wall (or friction) laws, or in other words, effective or homogenized boundary conditions. One aims at replacing rough boundaries by fictitious, smooth or flat boundaries. The wall law catches an averaged effect from the $O(\varepsilon)$-scale on large-scale flows of order $O(1)$ through homogenization. In that
line of research, it is well-known that Navier-slip boundary conditions provide refined approximations for fluids above bumpy boundaries. This effective boundary condition reads for instance in two dimensions

$$u_1 = \varepsilon\alpha\partial_2 u_1, \quad u_2 = 0 \quad \text{on} \quad \partial\mathbb{R}_+^2 \qquad (2.1)$$

with a constant $\alpha$ depending only on the boundary function $\gamma$. Under some quantitative ergodicity assumptions, one can get error estimates. Historically, periodic roughness profiles were first looked at [18, 4, 192, 193]. Analysis of almost-periodic [154] or random stationary ergodic [150, 45] boundary oscillations was done more recently. Let us also mention a few works that address non-stationary fluids [71, 182], for which the analysis is less developed due to its inherent difficulties. We also point out that some authors attempted to justify boundary conditions arising in fluid mechanics starting from boundary conditions at the microscopic scale; see for instance [81, 72, 56] for the derivation of the no-slip boundary condition from a perfect slip boundary condition at the microscale, or [114] for the computation of the homogenized effect starting from Navier-slip boundary conditions at the microscale.

A second topic is the study of the effect of roughness on singular limits. The topics of rotating fluids and of the homogenized effect of bumpiness on Ekman pumping was studied in numerous papers [149, 151, 116, 115]. The paper [153] carries out an analysis of the vanishing viscosity limit in a specific scaling regime. There are also studies concerned with equations in singularly perturbed domains such as the Stokes equations in rough thin films [101] or water waves above a rough topography in the shallow regime [110].

Third, rough domains pose considerable numerical challenge. This aspect has certainly driven the development of wall laws in a model reduction perspective; see for instance [4, 123]. Other approaches are being elaborated, such as Direct Numerical Simulations [80], Lattice Boltzmann Methods that are adapted to intricate geometries [341] and Large Eddy Simulations [19, 57] that in this context cause important parametrization issues of the small scales.

### 2.1.2 Compactness methods for regularity

Compactness arguments in the regularity theory originate from the works of De Giorgi [122] and Almgren [17] in the calculus of variations, and were adapted to various contexts, notably in homogenization by Avellaneda and Lin [29], for the study of partial regularity for five-dimensional stationary fluids by Struwe [327] and for three-dimensional unsteady fluids by Lin [243] or Ladyženskaja and Seregin [230]. There are far more works relating to this topic. Let us just point that our works with Kenig [213, 214], with Higaki [183] presented here, and the work with Albritton and Barker [10] presented in Chapter 5 are based on a compactness argument. Schematically the compactness argument is a two-steps process:

(Step-1) **Improvement of flatness** in certain certain small-scale (regular coefficients), large-scale (homogenization), or linear regime ($\varepsilon$-regularity) limit; one gets an improved estimate at a fixed scale using regularity for the limit system and compactness of the family of solutions (via Caccioppoli-type inequalities, local energy estimates, Aubin-Lions-type lemmas);

(Step-2) **Iteration** of the one-scale improvement of flatness estimate; this step relies on the reiterated use of the first step, which requires a sort of self-similarity of the approximation.
*Remark 2.1* (About the self-similarity property). Let us illustrate the self-similarity that is needed to iterate the one-scale improvement of flatness estimate by looking at just three different situations:

(1) In the case of the local interior regularity for the Laplace operator, harmonic polynomials are the right objects to measure the regularity of solutions; see the standard regularity for harmonic functions [157].

(2) In the case of the local interior regularity for divergence-form elliptic equations with oscillating coefficients, such as $-\nabla \cdot a(x/\epsilon)\nabla$, harmonic polynomials are not solutions of the equations. The right objects to consider are hence corrected polynomials that we dub **a**-harmonic polynomials. If the coefficients **a** are periodic, the **a**-harmonic polynomials of degree one are the functions $y + \chi(y)$, where $\chi$ is the usual cell corrector; see [29] and [24] for the higher-order **a**-harmonic polynomials.

(3) In the case of the local boundary regularity for the Laplace equation near a bumpy boundary $x_3 = \epsilon\gamma(x'/\epsilon)$, the trace of harmonic polynomials does not vanish on the bumpy boundary. The right objects are hence harmonic polynomials corrected with boundary layer correctors, that vanish on the boundary; see [213, 214] and Subsection 2.3.1 below.

Some other recent works rely on the compactness method to prove uniform estimates in homogenization. In the work of Niu and Zhuge [272], the compactness method is used to prove uniform Hölder estimates for elliptic systems oscillating with multiple non separated scales. Notice that in the case of Hölder estimates, no correctors are needed because the right objects to measure the regularity are just constants, which are trivial solutions to the equation with oscillating coefficients. The Lipschitz estimate is stated there as an open problem. In [314], Shen studies the large-scale regularity for the Stokes equation in periodically perforated domains. The compactness method is effective in this setting because the size of the holes is comparable to the size of the periodicity cell. In certain situations, the compactness method reaches its limits and a quantitative method is more successful.

### 2.1.3 Quantitative methods for regularity

In some situations (stochastic homogenization, homogenization of high-contrast materials, homogenization of porous media with dilute holes) there is a need for quantitative versions of the compactness method of Subsection 2.1.2. The main idea is that the original problem with variable coefficients or small scales is quantitatively close to a homogeneous limit problem, which has improved regularity. The scheme is reminiscent of the perturbation argument in Schauder's theory for solutions of equations with smooth coefficients at the small scale. The general framework was laid down by Caffarelli and Peral [76] for the proof of $W^{1,p}$ estimates, $p$ finite, for divergence-form elliptic equations. There the method is dubbed ‘$W^{1,p}$ estimates by approximation’ and encompasses the case of regular coefficients, where the perturbation is at the small scales around a fixed regular point, as well as the case of homogenization, where the perturbation is at the large scales around the homogenized equation. The scheme was summarized in [76] as follows:

(1) **Improvement of flatness** for a fixed operator $-\nabla \cdot a_0 \nabla$ (either the operator frozen at a fixed point when the coefficients are smooth, or the homogenized operator);

(2) **A local (sub-optimal) error estimate at any scale** to approximate the solutions of the variable coefficient operator $-\nabla \cdot a \nabla$ by solutions of the fixed operator $-\nabla \cdot a_0 \nabla$;
(3) **A real variable argument** which is a version of the Calderón-Zygmund lemma; see the original paper by Caffarelli and Peral [76, Lemma 1.3] and the book by Shen [313, Theorem 4.2.3] for a refined version.

Lipschitz and higher-order $C^{k,\mu}$ estimates can be obtained following a similar strategy. The last step, which requires a Calderón-Zygmund type argument to get on top of the $L^p$ integrability in the case of $W^{1,p}$ estimate, is less subtle in the case of higher-order estimates because one can afford to loose a bit on the exponent $\mu$.

Such a scheme is effectively implemented in the context of homogenization. In periodic homogenization, quantitative arguments may be needed to treat:

(i) borderline cases where there is no improvement of regularity as in the proof of boundary Rellich estimates by Shen [312],

(ii) multiple-scales problems as in the paper by Geng and Shen [147] for the homogenization of parabolic equations $\partial_t \cdot \nabla \cdot a(x/\epsilon, t/\epsilon^k)\nabla \cdot$ with $k \neq 2$ i.e. time and space scaling in a non self-similar way, or in the paper [271] by Niu, Shen and Xu for divergence-form elliptic equations with multiple separated scales,

(iii) singularly perturbed problems, such as the fourth-order elliptic system studied by Niu and Shen [270], or elliptic equations with high-contrast coefficients studied by Russel [288] (contrast parameter $\delta = 0$), [289] (contrast parameter $\delta \in (0, 1)$) and Shen [315] (contrast parameter $\delta \in (0, \infty)$).

In stochastic and almost-periodic homogenization, the existence of correctors with certain growth at most sub-linear at space infinity, or even bounded as in the periodic case, is subtle and related to the large-scale regularity theory, see the book by Armstrong, Kuusi and Mourrat [22, Chapter 3]. A starting point for the regularity theory of such operators is therefore to consider approximate correctors, that enable to get an approximation of solutions at any scales and sub-optimal error estimates; see [22, page (viii) and (iv)] for a general description of the method. These quantitative approximations, along with improvement of flatness for the homogenized problem, can then be used in a classical Schauder $C^{1,\mu}$ regularity scheme similarly as explained above for the $W^{1,p}$ estimate. The method was pioneered by Armstrong and Smart [25] and developed in the papers by Gloria, Neukamm and Otto [167] and by Armstrong, Kuusi and Mourrat [21] for the random case. The method was also successfully applied in almost-periodic homogenization to the large-scale interior regularity by Armstrong and Shen [27] and boundary regularity by Zhuge [352].

## 2.2 Main results

Our results are concerned with the local large-scale quantitative regularity for the steady incompressible three-dimensional Navier-Stokes equations in bumpy domains

$$ egin{cases} -\Delta U^{\epsilon} + \nabla P^{\epsilon} = -U^{\epsilon} \cdot \nabla U^{\epsilon} & \text{in } B_{+}^{\epsilon}(1), \\ \nabla \cdot U^{\epsilon} = 0 & \text{in } B_{+}^{\epsilon}(1), \\ U^{\epsilon} = 0 & \text{on } \Gamma^{\epsilon}(1), \end{cases} \quad (NS^{\epsilon}) $$

where the functions $U^{\epsilon} = U^{\epsilon}(x) \in \mathbb{R}^3$ and $P^{\epsilon} = P^{\epsilon}(x) \in \mathbb{R}$ denote respectively the velocity and the pressure fields of the fluid and $\epsilon \ll 1$ is the typical size of the microscopic roughness of the boundary, see below.
The following two general objectives in regularity theory motivate our results: (i) decouple the large-scale regularity question from the small-scale regularity (or lack of regularity) of the boundary, (ii) handle very rough or singular boundaries (Lipschitz, possibly fractals), (iii) remove structure assumptions on the microscopic oscillations of the boundary, (iv) identify building blocks that describe the local behavior of solutions, (iv) quantify local errors at mesoscopic scales, i.e. estimate the decay of certain excess quantities at various scales. Our longterm research program stated in the introduction of this chapter is to investigate the impact of roughness on regularity for steady or unsteady fluids.

Our research program was started with the works [213, 214] in collaboration with Kenig. These papers are concerned with uniform regularity estimates above highly oscillating boundaries for elliptic equations. The improved regularity results obtained at the large scales are generally false at the small scales due to the roughness of the boundary. Our results are in the spirit of large-scale regularity estimates pioneered in [29] for periodic homogenization, and later extended to stochastic homogenization; see for instance [25, 26, 167, 166] and [21] for the higher-order large-scale regularity theory.

The tools we develop in the papers [213, 214, 183, 184] in collaboration with Kenig, Higaki and Zhuge enable us to decouple the large-scale regularity from the small-scale properties of the boundary. We prove that fluids above bumpy boundaries, that are very rough at the microscopic scale but asymptotically flat, have improved regularity at large scales.

As for the regularity of the domain, we study two cases:

(1) in the work [183] with Higaki, we take a boundary given by the graph of a Lipschitz function;

(2) in the work [184] with Higaki and Zhuge, we consider John domains, which in some sense that we explain below are the most general domains in which we can handle the regularity for the Stokes system.

**Lipschitz domains** This is the setting of the paper [183]. We say that $\Omega$ is a 'bumpy Lipschitz half-space' if it is defined by

$$ \Omega := \{y \in \mathbb{R}^3, y_3 > \gamma(y')\}, \qquad (2.2) $$

where the boundary graph $\gamma \in W^{1,\infty}(\mathbb{R}^2)$ is assumed to satisfy $\gamma(y') \in (-1, 0)$ for all $y' \in \mathbb{R}^2$. For $\varepsilon \in (0, 1]$, we set $\Omega^\varepsilon := \varepsilon\Omega$, which is the highly oscillating bumpy Lipschitz half-space. For $r \in (0, 1]$,

$$ B^\varepsilon_+(r) := B(r) \cap \Omega^\varepsilon \quad \text{and} \quad \Gamma^\varepsilon(r) := B(r) \cap \partial\Omega^\varepsilon. \qquad (2.3) $$

**John domains** In the work [184] we consider bumpy John domains that are very rough in two aspects: (i) the boundary may be fractal or have inward cusps, (ii) the boundary is highly oscillating. Hence, these boundaries get closer to the modeling of real surfaces found in nature (for instance porous media), that in particular do not need to be graphs, see Figure 2.1. John domains have in a broad sense the minimal properties for the analysis of incompressible fluids. Indeed, we rely on a Bogovskii operator to estimate the pressure. As far as we know, John domains are the most irregular ones for which we have the existence of such a tool.

These domains were introduced by John in [201] and named after John in [253].
<figure><img src="image_5.png" /><figcaption>Figure 2.1 – A bumpy John domain</figcaption></figure>

**Definition 2.2** ([184, Definition 1.1] taken from [5]). Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded set and $\bar{x} \in \Omega$. We say that $\Omega$ is a John domain (or a bounded John domain) with respect to $\bar{x}$ and with constant $L$ if for any $y \in \Omega$, there exists a Lipschitz mapping $\rho: [0, |y - \bar{x}|] \to \Omega$ with Lipschitz constant $L \in (0, \infty)$, such that $\rho(0) = y$, $\rho(|y - \bar{x}|) = \bar{x}$ and $\text{dist}(\rho(t), \partial\Omega) \ge t/L$ for all $t \in [0, |y - \bar{x}|]$.

Our analysis takes advantage of a key property of John domains, namely the existence of a right inverse of the divergence operator, see [5]. Such an operator is usually called a Bogovskii operator.

Examples of John domains are: Lipschitz domains, NTA domains, domains with inward cusps or certain fractals such as Koch's snowflake. Notice that domains with outward cusps are not John domains. For our work, we generalize the above definition from bounded domains to a class of unbounded domains.

**Definition 2.3.** Let $\Omega$ be a domain containing the upper half-space $\mathbb{R}_+^3$ and assume that $\partial\Omega \subseteq \{-1 < x_3 < 0\}$. We say that $\Omega$ is a 'bumpy John domain' with constant $L \in (0, \infty)$, if for any $x \in \partial\mathbb{R}_+^3$ and any $R \ge 1$, there exists a bounded John domain $\Omega_R(x)$ with respect to $\bar{x}_R = x + Re_d$ and with constant $L$ according to Definition 2.2 such that

$$B_{x,+}(R) \subseteq \Omega_R(x) \subseteq B_{x,+}(2R). \qquad (2.4)$$

The above definition guarantees that the constants of John domains are rescaling- and translation-invariant. This is a natural requirement as we are considering unbounded domains. Let $\Omega^\epsilon := \epsilon\Omega = \{x \in \mathbb{R}^3 | \epsilon^{-1}x \in \Omega\}$. We refer to $\Omega^\epsilon$ as a 'highly oscillating bumpy John domain'. Note that

$$\partial\Omega^\epsilon \subseteq \{x \in \mathbb{R}^3, -\epsilon < x_3 < 0\}. \qquad (2.5)$$

A key fact about $\Omega^\epsilon$ is that it is still a John domain with the same constants as in Definition 2.3, as these constants are scale-invariant.

**Definition 2.4.** We say that $\Omega$ is a 'periodic bumpy John domain' if the following holds:

(i) $\Omega$ is a bumpy John domain with constant $L$,
(ii) and $\Omega$ is $(2\pi\mathbb{Z})^2$-translation invariant, namely $2\pi\xi + \Omega = \Omega$ for any $\xi \in \mathbb{Z}^2 \times \{0\}$.

Periodicity is only needed at one place, to construct second-order boundary layer correctors in Theorem 2.10 below.

Finally, we define $B_+^\epsilon(r)$ and $\Gamma^\epsilon(r)$ in the same way as (2.3) above.

### 2.2.1 Large-scale Lipschitz regularity

We state first a large-scale (in other words ‘mesoscopic’, ‘improved’) Lipschitz estimate in the highly oscillating bumpy Lipschitz domain defined above, see Theorem 2.5. Second, we state a large-scale Lipschitz estimate in the highly oscillating bumpy John domain defined above, see Theorem 2.6. Notice that the John boundary case includes of course the case of Lipschitz boundaries. The statement of the theorems in both the Lipschitz and the John case may therefore seem a bit redundant. There are two main reasons for that: (i) historically we understood the case of boundaries given by Lipschitz graphs first and we were only able to handle John domains later, (ii) the proofs of these two theorems rely on completely different approaches, see Subsection 2.3.1 and Subsection 2.3.2 below, and therefore the estimates that we get are also a bit different.

**Theorem 2.5** (improved Lipschitz regularity in Lipschitz domains; [183, Theorem 1], in collaboration with Higaki). For all $\epsilon \in (0, 1/2)$ and for all $M \in (0, \infty)$ the following statement holds. Let $\gamma$ be a Lipschitz graph as defined above. If $(U^\epsilon, P^\epsilon) \in H^1(B_+^\epsilon(1))^3 \times L^2(B_+^\epsilon(1))$ is a weak solution of (NS$^\epsilon$) we have

$$ \left( \int_{B_+^\epsilon(1)} |\nabla U^\epsilon|^2 \right)^{\frac{1}{2}} \le M \quad \text{implies} \quad \sup_{r \in (\epsilon, 1/2)} \left( \int_{B_+^\epsilon(r)} |\nabla U^\epsilon|^2 \right)^{\frac{1}{2}} \le C(M), \quad (2.6) $$

where the constant $C(M)$ is independent of $\epsilon$ and $r$, and depends on $\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}$ and $M$. Moreover, $C(M)$ is a monotone increasing function of $M$ and converges to zero as $M$ goes to zero.

Notice that $C(M)$ grows polynomially in $M$. Of course, if one considers the linear Stokes equation, then $C(M)$ grows linearly in $M$.

**Theorem 2.6** (improved Lipschitz regularity in John domains; [184, Theorem A], in collaboration with Higaki and Zhuge). For all $\epsilon \in (0, 1/2)$, $L \in (0, \infty)$, $M \in (0, \infty)$ and $\delta \in (0, 1)$, the following statement holds. Let $\Omega$ be a bumpy John domain with constant $L$ according to Definition 2.3. If $(U^\epsilon, P^\epsilon) \in H^1(B_+^\epsilon(1))^3 \times L^2(B_+^\epsilon(1))$ is a weak solution of (NS$^\epsilon$), we have

$$ \left( \int_{B_+^\epsilon(1)} |\nabla U^\epsilon|^2 \right)^{\frac{1}{2}} \le M \quad \text{implies} \quad \sup_{r \in (\epsilon, 1/2)} \left( \int_{B_+^\epsilon(r)} |\nabla U^\epsilon|^2 \right)^{\frac{1}{2}} \le C(M + M^{4+\delta}), \quad (2.7) $$

where the constant $C$ is independent of $\epsilon$, $M$ and $r$, and depends on $L$ and $\delta$.

Notice that in both results above, we also have the large-scale boundedness of the pressure

$$ \left( \int_{B_+^\epsilon(r)} |P^\epsilon - \int_{B_+^\epsilon(1/2)} P^\epsilon|^2 \right)^{1/2} . $$
This is explicitly stated in [184, Theorem A].

In the vein of the seminal works [29, 30] and of [216, 177], we provide pointwise estimates for the large-scale decay of the velocity and pressure parts of the Green function associated to the Stokes system in bumpy John half-spaces, [184, Appendix B]. These estimates are pivotal to construct the first-order boundary layers in Subsection 2.3.2.

Let us insist on the fact that we separate the small-scale regularity, i.e. at scales $\lesssim \varepsilon$, from the mescopic- or large-scale regularity, i.e. at scales $\varepsilon \lesssim r \le 1$. Concerning the small scales, the classical Schauder regularity theory for the Stokes and the Navier-Stokes equations was started by Ladyženskaja [229] using potential theory and by Giaquinta and Modica [158] using Campanato spaces. These classical estimates require some smoothness of the boundary and typically depend on the modulus of continuity of $\nabla\gamma$ when the boundary is given by the graph $x_3 = \gamma(x')$. Therefore, these estimates degenerate for highly oscillating boundaries $x_3 = \varepsilon\gamma(x'/\varepsilon)$ with $\varepsilon \ll 1$. As for the large scales, on the contrary, the regularity is inherited from the limit system when $\varepsilon \to 0$ posed in a domain with a flat boundary. Here no regularity is needed for the original boundary, beyond the boundedness of $\gamma$ and of its gradient. The mechanism for the regularity at small scales and at large scales is hence completely different.

As in the works [29, 150, 213] one can combine the mesoscopic regularity estimate with the classical regularity at small scales provided the boundary is regular enough, i.e. $C^{1,\kappa}$ with $\kappa \in (0, 1)$. In that case, we can prove the full Lipschitz estimate on $\|\nabla U^\varepsilon\|_{L^\infty(B_+^\varepsilon(1/2))}$. However, one cannot expect such an estimate to hold in Lipschitz domains, not mentioning John domains, even for the Laplace equation with the Dirichlet boundary condition. This fact justifies the terminology of ‘improved regularity’.

### Novelty of our results

The common novelties of these two results are that:

(1) They do not rely on any smoothness of the boundaries at the microscopic scale. Works on this topic prior to the paper [214] in collaboration with Kenig all relied on microscopic smoothness of the boundary. Let us cite the paper [150], where a uniform Hölder estimate for weak solutions of the Stokes equations is obtained when the boundary graph $\gamma \in C^{1,\omega}(\mathbb{R}^2)$ for a fixed modulus of continuity $\omega$. In the work [213] with Kenig we proved uniform Lipschitz regularity for elliptic equations above bumpy $C^{1,\kappa}$ boundaries, $\kappa \in (0, 1)$. In the paper [214] Kenig and I managed to decouple the small-scale regularity from the large-scale regularity for elliptic equations. Our paper [183] with Higaki and Theorem 2.5 in particular builds upon this progress. Theorem 2.6 from our work [184] in collaboration with Higaki and Zhuge goes even one step further since the regularity of the boundary is reduced to that of a John domain, and moreover the boundary does not need to be a graph.

(2) They do not rely on any structure assumption on the microscopic oscillations of the boundary. In other words, we do not need any periodicity, almost-periodicity or stationary ergodicity assumptions. The only important point is that the bumpy boundary is asymptotically flat.

(3) They hold outside the perturbative regime, i.e. without a priori smallness on the size of the solutions, which is due to the energy subcritical nature of the three-dimensional stationary Navier-Stokes equations. This is in stark contrast with previous works
concerned with linear equations (elliptic or Stokes systems), see [29, 150, 175, 176, 213, 214].

### 2.2.2 Higher-order regularity and local wall laws

Beyond the large-scale Lipschitz estimate above we prove:

**Higher-order $C^{1,\mu}$ and $C^{2,\mu}$ estimates for $\mu \in (0, 1)$**

For higher-order $C^{1,\mu}$ regularity results, we refer to Theorem 2.9, estimate (2.11). For higher-order $C^{2,\mu}$ regularity results, we refer to Theorem 2.10, estimate (2.12) in the John case. In these estimates we measure the oscillation of the solution with respect to modified polynomials that vanish on the bumpy boundary. These modified polynomials are polynomials of degree one and two that are corrected by the first-order and second-order boundary layers.

**Local wall laws**

Remark 2.8 establishes the connection between Theorem 2.7 and the Navier wall law. Again, as explained above, we state the case of Lipschitz domains separately, although it is mostly covered by the theorems handling John domains.

**Theorem 2.7** (a local wall law in bumpy Lipschitz domains; [183, Theorem 2], in collaboration with Higaki). Fix $M \in (0, \infty)$ and $\mu \in (0, 1)$. We assume in addition that $\gamma \in W^{1,\infty}(\mathbb{R}^2)$ is $2\pi$-periodic in each variable. Then for all weak solutions $(U^\varepsilon, P^\varepsilon) \in H^1(B_+^\varepsilon(1))^3 \times L^2(B_+^\varepsilon(1))$ to $(\text{NS}^\varepsilon)$ satisfying the a priori bound

$$ \left( \int_{B_+^\varepsilon(1)} |\nabla U^\varepsilon|^2 \right)^{\frac{1}{2}} \le M, \qquad (2.8) $$

the following statement holds. There exists a constant vector field $\alpha^{(j)} = (\alpha_1^{(j)}, \alpha_2^{(j)}, 0)^T \in \mathbb{R}^3$, $j \in \{1, 2\}$, depending only on $\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}$ such that for all $\varepsilon \in (0, 1/2)$ and $r \in (\varepsilon, 1/2)$, we have

$$ \left( \int_{B_+^\varepsilon(r)} |U^\varepsilon(x) - \sum_{j=1}^2 c_{r,j}^\varepsilon (x_3 \mathbf{e}_j + \varepsilon \alpha^{(j)})|^2 dx \right)^{\frac{1}{2}} \le C(M)(r^{1+\mu} + \varepsilon^{\frac{3}{2}} r^{-\frac{1}{2}}), \quad (2.9) $$

where the coefficient $c_{r,j}^\varepsilon$, $j \in \{1, 2\}$, is a functional of $U^\varepsilon$ depending on $\varepsilon$, $r$, $M$, $\mu$ and $\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}$. The constant $C(M) \in (0, \infty)$ is independent of $\varepsilon$ and $r$, but polynomial in $M$ and dependent on $\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}$, $M$, and $\mu$.

Remark 2.8 (relation with the wall law).

(1) Let us denote the polynomial in (2.9) by $Q_{N,j}^\varepsilon$, $j \in \{1, 2\}$:

$$ Q_{N,j}^{\varepsilon}(x) = x_3 \mathbf{e}_j + \varepsilon \alpha^{(j)}. \qquad (2.10) $$

These polynomials are the building blocks of the large-scale regularity theory of the Navier-Stokes equations above bumpy Lipschitz boundaries. Then each $Q_{N,j}^\varepsilon$ is a shear flow in the half-space $\mathbb{R}_+^3$ and is an explicit solution to the following Navier-Stokes equations in the half-space with a Navier-slip boundary condition

$$ \left\{ \begin{array}{ll} -\Delta \mathcal{U}_N^\varepsilon + \nabla \mathcal{P}_N^\varepsilon = -\mathcal{U}_N^\varepsilon \cdot \nabla \mathcal{U}_N^\varepsilon & \text{in } \mathbb{R}_+^3, \\ \nabla \cdot \mathcal{U}_N^\varepsilon = 0 & \text{in } \mathbb{R}_+^3, \\ \mathcal{U}_{N,3}^\varepsilon = 0 & \text{on } \partial \mathbb{R}_+^3, \\ (\mathcal{U}_{N,1}^\varepsilon, \mathcal{U}_{N,2}^\varepsilon)^T = \varepsilon \bar{M} (\partial_3 \mathcal{U}_{N,1}^\varepsilon, \partial_3 \mathcal{U}_{N,2}^\varepsilon)^T & \text{on } \partial \mathbb{R}_+^3, \end{array} \right. \quad (\text{NS}_N^\varepsilon) $$
with a trivial pressure $P_N^\epsilon = 0$. Here the $2 \times 2$ matrix $\overline{M} = (\alpha_i^{(j)})_{1\le i,j\le 2}$ can be proved to be positive definite. Thus the estimate (2.9) in Theorem 2.7 reads as follows: any weak solution $U^\epsilon$ to $(NS^\epsilon)$ can be approximated at any mesoscopic scale by a linear combination of the Navier polynomials $Q_{N,1}^\epsilon$ and $Q_{N,2}^\epsilon$. This is a local version of the Navier wall law, which has been widely studied in the global framework; see Subsection 2.1.1.

(2) Our result can be extended to the stationary ergodic or the almost-periodic setting. We also note that the wall law breaks down when the boundary does not have any structure at all, see [154] where it is showed that some ergodicity is needed to study the tails of boundary layer correctors that determine the Navier polynomials.

For the John domains, the analogous result reads as follows. We state the $C^{1,\mu}$ estimate with the boundary layer correctors. An asymptotic wall law in the spirit of (2.9) would also hold in periodic John domains, but we do not state it here.

**Theorem 2.9** (large-scale $C^{1,\mu}$ regularity for John domains; [184, Theorem B], in collaboration with Higaki and Zhuge). For all $\mu \in [0, 1)$, $\epsilon \in (0, 1/2)$, $L \in (0, \infty)$, $M \in (0, \infty)$ and $\delta \in (0, 1)$, the following statement holds. Let $\Omega$ be a bumpy John domain with constant $L$ according to Definition 2.3. If $(U^\epsilon, P^\epsilon) \in H^1(B_+^\epsilon(1))^3 \times L^2(B_+^\epsilon(1))$ is a weak solution of $(NS^\epsilon)$ satisfying the a priori bound (2.8), then, there exists a constant $\bar{P}_1$ depending on $P^\epsilon$ such that, for any $r \in (\epsilon, \frac{1}{2})$,

$$ \inf_{(W,P) \in \mathcal{Q}_1(\Omega)} \left\{ \frac{1}{r} \left( \int_{B_+^\epsilon(r)} |U^\epsilon - \epsilon W(x/\epsilon)|^2 dx \right)^{1/2} + \left( \int_{B_+^\epsilon(r)} |P^\epsilon - P(x/\epsilon) - \bar{P}_1|^2 dx \right)^{1/2} \right\} \le Cr^\mu(M + M^{4+2\mu+\delta}), \quad (2.11) $$

where $\mathcal{Q}_1(\Omega)$ is the class of all solutions to the Stokes equations in the bumpy John half-space $\Omega$ with linear growth at infinity that vanish on $\partial\Omega$, see [184, equation (5.1)]. The constant $C$ is independent of $\epsilon$, $M$ and $r$, but depends on $L$, $\mu$ and $\delta$.

While Theorem 2.9 holds for arbitrary bumpy John half-spaces, for the next result, we work in periodic John domains. As we outlined above, the extra periodicity assumption makes the analysis of the second-order boundary layers more manageable. However, we do not claim that this assumption is necessary. It is possible that a similar theorem can be proved in other frameworks with quantitative ergodicity properties, such as the quasiperiodic setting with some non-resonance condition; see Chapter 3 for related thoughts. The following result contains of course the case when the boundary is Lipschitz.

**Theorem 2.10** (large-scale $C^{2,\mu}$ regularity for John domains; [184, Theorem C], in collaboration with Higaki and Zhuge). For all $\mu \in [0, 1)$, $\epsilon \in (0, 1/2)$, $L \in (0, \infty)$, $M \in (0, \infty)$ and $\delta \in (0, 1)$, the following statement holds. Let $\Omega$ be a periodic bumpy John domain with constant $L$ according to Definition 2.4. If $(U^\epsilon, P^\epsilon) \in H^1(B_+^\epsilon(1))^3 \times L^2(B_+^\epsilon(1))$ is a weak solution of $(NS^\epsilon)$ satisfying (2.8), then, there exists a constant $\bar{P}_2$ depending on $P^\epsilon$ such
that, for any $r \in (\varepsilon, 1/2)$,

$$ 
\begin{aligned}
& \inf_{\substack{(W_1, P_1) \in \mathcal{Q}_1(\Omega) \\ (W_2, P_2) \in \mathcal{Q}_2(\Omega)}} \left\{ \frac{1}{r} \left( \int_{B_+^\varepsilon(r)} |U^\varepsilon - \varepsilon W_1(x/\varepsilon) - \varepsilon^2 W_2(x/\varepsilon)|^2 dx \right)^{1/2} \right. \\
& \qquad \left. + \left( \int_{B_+^\varepsilon(r)} |P^\varepsilon - P_1(x/\varepsilon) - \varepsilon P_2(x/\varepsilon) - \bar{P}_2|^2 dx \right)^{1/2} \right\} \\
& \leq Cr^{1+\mu}(M + M^{6+2\mu+\delta}),
\end{aligned} 
\quad (2.12) $$

where $\mathcal{Q}_1(\Omega)$ is used in Theorem 2.9 and defined in [184, equation (5.1)] and $\mathcal{Q}_2(\Omega)$ is the class of all solutions to the Stokes equations in the periodic bumpy John half-space $\Omega$, with quadratic growth at infinity, that vanish on $\partial\Omega$, see [184, equation (5.2)]. The constant $C$ is independent of $\varepsilon$, $M$ and $r$, but depends on $L$, $\mu$ and $\delta$.

We point out that the building blocks in $\mathcal{Q}_1(\Omega)$ and $\mathcal{Q}_2(\Omega)$ are defined through the first-order and second-order boundary layers. To emphasize the distinguished role of the classes $\mathcal{Q}_1(\Omega)$ and $\mathcal{Q}_2(\Omega)$ we state a Liouville-type theorem for the entire solutions of the Stokes system in $\Omega$

$$ 
\left\{ 
\begin{array}{ll} 
-\Delta W + \nabla P = 0 & \text{in } \Omega, \\ 
\nabla \cdot W = 0 & \text{in } \Omega, \\ 
W = 0 & \text{on } \partial\Omega. 
\end{array} 
\right. 
\quad (2.13) $$

**Theorem 2.11** (Liouville theorems for sub-linear, sub-quadratic and sub-cubic growth; [184, Corollary 3.1] and [184, Theorem 5.8] in collaboration with Higaki and Zhuge). Let $\Omega$ be a bumpy John domain according to Definition 2.3. Let $(W, \Pi)$ be a weak solution of (2.13).

**sub-linear** If

$$ \lim_{R \to \infty} \frac{1}{R} \left( \int_{B_R(0) \cap \Omega} |W|^2 \right)^{1/2} = 0, $$

then $W \equiv 0$, hence $P$ is constant.

**sub-quadratic** If for some $\sigma \in (0, 1]$

$$ \liminf_{R \to \infty} \frac{1}{R^{1+\sigma}} \left( \int_{B_+(R)} |W|^2 \right)^{1/2} = 0, $$

then $(W, P) \in \mathcal{Q}_1(\Omega)$, up to a constant for $P$.

**sub-cubic** In addition, assume $\Omega$ is periodic bumpy John domain according to Definition 2.4. If for some $\sigma \in (0, 1)$,

$$ \liminf_{R \to \infty} \frac{1}{R^{2+\sigma}} \left( \int_{B_{R,+}} |W|^2 \right)^{1/2} = 0, $$

then $(W, P) \in \mathcal{Q}_1(\Omega) + \mathcal{Q}_2(\Omega)$, up to a constant for $P$.

Notice that in the case of sub-cubic growth, contrary to the sub-quadratic case, we cannot reach the exponent $\sigma = 1$ because we do not have a large-scale $C^3$ estimate.
### Novelty of our results

As far as we know, the results stated above go far beyond the existing literature. They are the first ones concerned with the large-scale high-order regularity above bumpy boundaries. They also do not have any counterpart for elliptic equations.

Theorem 2.7 from our work [183] with Higaki is the first local justification of a wall law. It is important physically as well as mathematically since we are interested in the effects of rough boundaries on viscous fluids. Our result is a step toward a better understanding of roughness effects on the Navier-Stokes flows in view of regularity improvement.

Moreover, our work [184] with Higaki and Zhuge is the first to construct the second-order boundary layers with a linear growth in the direction tangential to the boundary. To make the analysis more tractable, we assume that the boundary is periodic. We are aware of the papers [42, 70], where a refined second-order approximation is constructed for the Stokes equations in a two-dimensional rough channel. However, the boundary layers considered in [42, 70] only involve data spanned by linear and quadratic polynomials of the vertical variable, $x_2$ and $x_2^2$ in this two-dimensional case, which are bounded on the bumpy boundary. In our three-dimensional situation, the class of ‘no-slip Stokes polynomials’ needed for the $C^{2,\mu}$ regularity theory, see [184, Subsection 4.1], is much richer and involves boundary data with linear growth at spatial infinity.

### Further developments

This line of research is currently being developed in the following directions:

#### (1) Smoothness at large scales

In the paper [178], Gu and Zhuge study the system of elasticity in domains which are not asymptotically flat, but smooth at the large scales; see [178, equation (1.22)] for the definition of the $\varepsilon$-scale $C^{1,\alpha}$ condition. Their domain is Lipschitz at the small scales, but this assumption can be relaxed to include John domains.

#### (2) Rougher boundaries

The paper [356] by Zhuge goes even beyond the framework of John domains for the study of the study of the large-scale Lipschitz regularity for linear elliptic equations. Indeed, no Bogovskii operator is needed for that analysis, which is the reason why we work in John domains for the Stokes system. In [356] the roughness of the boundary is arbitrary at small scales and satisfies a quantitative Reifenberg flatness condition at large scales.

#### (3) Higher-order boundary layers

In the paper [185], Higaki and Zhuge construct boundary layer correctors up to arbitrary order for the Stokes system in periodic bumpy John domains. They systematize the construction of the second-order boundary layers stated in Theorem 2.10 above. Such a study is in the vein of the paper of Armstrong, Kuusi and Mourrat [24] concerned with higher-order cell correctors for elliptic equations with periodic coefficients.

#### (4) Building blocks

The point of view adopted in Theorem 2.8, Remark 2.7 and Theorem 2.11 that certain functions play the role of ‘building blocks’ for the regularity theory is not new. The idea is already present in the work of Avellaneda and Lin [29], where a-harmonic linear functions (linear functions perturbed by cell correctors) are the building blocks
of the $C^{1,\mu}$ regularity theory. Several recent original papers put forward the same idea in different contexts; see [355] (piecewise linear functions), [205] (a-harmonic singular functions). These results are in line with the extended point of view of regularity theory that we aim at developing, see the introduction of this thesis, especially Subsection 1.1.3.

## 2.3 New ideas and strategy for the proofs

### 2.3.1 Boundary layers and large-scale boundary regularity in Lipschitz domains

In the work [183] with Higaki, we perform the analysis in bumpy Lipschitz domains. As in the papers with Kenig [213, 214], the proofs of Theorem 2.5 and of Theorem 2.7 are based on a compactness argument originating from the seminal work by Avellaneda and Lin [29] on uniform estimates in homogenization, see Subsection 2.1.2. The mains points are: (i) construction of a boundary layer corrector in the Lipschitz half-space, (ii) proof of the mesoscopic regularity by compactness and iteration. We focus on point (i) below.

Our analysis takes advantage of the progress made in [213] to remove structure assumptions on the oscillations of the boundary and in [214] to remove small-scale regularity assumptions. In the present case, there are additional difficulties related to:

(1) the incompressibility condition; a simplifying feature is the fact that in the case of a boundary given by a Lipschitz graph, the boundary condition in the boundary layer system (2.15) is a trace of a divergence-free function;

(2) the nonlocal pressure; since we consider the stationary Navier-Stokes equations, one can estimate the pressure directly in terms of the velocity using Bogovskii-type operators [5, 140]

$$ \|P - (P)_{B(1)}\|_{L^2(B(1))} \le C \|\nabla P\|_{H^{-1}(B(1))} \quad (2.14) $$

and the equation for $\nabla P$; similar estimates hold close to the boundary and imply Caccioppoli-type estimates [183, Appendix B];

(3) the nonlinearity of the Navier-Stokes equations and the lack of smallness of the solutions; on the one hand, the stationary three-dimensional Navier-Stokes equations are energy subcritical, hence the solutions of the limit stationary Navier-Stokes equations in the flat domain are smooth, see [183, Appendix A]; on the other hand, the nonlinearity creates a difficulty when iterating the one-scale improvement of flatness estimate; we overcome this difficulty by choosing the free parameter $\theta$ in the compactness lemma in terms of the data $\|\gamma\|_{W^{1,\infty}}$ and $M$, see [183, Section 5].

**Focus on the boundary layer corrector and a Saint-Venant estimate in a channel** The use of boundary layer correctors is key to the compactness argument. Indeed, in order to iterate the one-scale improvement of flatness estimate, one needs some sort of self-similarity, see Subsection 2.1.2 above and in particular Remark 2.1. The boundary layers correct the highly oscillating trace on the bumpy boundary of the affine functions $(a, b, 0)^T x_3$, which are the building blocks for the $C^{1,\mu}$ regularity in the flat domain. The boundary layers
correctors solve the following equations for $j \in \{1, 2\}$

$$ \left\{ \begin{array}{ll} -\Delta \mathcal{V} + \nabla \mathcal{P} = 0 & \text{in } \Omega, \\ \nabla \cdot \mathcal{V} = 0 & \text{in } \Omega, \\ \mathcal{V}(y', \gamma(y')) = -\gamma(y')\mathbf{e}_j, & \end{array} \right. \qquad (2.15) $$

where $\Omega$ is the bumpy Lipschitz half-space defined by (2.2). We emphasize that the boundary layer correctors solve the linear Stokes equations. This is expected from the following formal heuristics. Indeed, in the boundary layer $U^\epsilon \simeq \epsilon\mathcal{V}(x/\epsilon)$, so that $\mathcal{V}$ solves $-\frac{1}{\epsilon}\Delta\mathcal{V} + \epsilon\mathcal{V} \cdot \nabla\mathcal{V} + \nabla Q = 0$, $\nabla \cdot \mathcal{V} = 0$.

We prove that there exists a unique weak solution $(\mathcal{V}, Q) \in H_{\text{loc}}^1(\bar{\Omega})^3 \times L_{\text{loc}}^2(\bar{\Omega})$ to (2.15) satisfying the following energy bound, which is locally uniform in the tangential variable

$$ \sup_{\eta \in \mathbb{Z}^2} \int_{\eta+(0,1)^2} \int_{\gamma(y')}^{\infty} |\nabla \mathcal{V}(y', y_3)|^2 dy_3 dy' \le C (\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}); \qquad (2.16) $$

see [183, Proposition 8]

Our methodology for the analysis of the system (2.15) is close to the one developed in my work with Kenig [214] concerned with the boundary regularity of elliptic equations in bumpy Lipschitz domains. This strategy is inspired from the work of Gérard-Varet and Masmoudi [154] and was successfully implemented in other situations, such as the analysis of oceanic boundary layers [116, 115, 286] near a rough seabed (Ekman layer) or a rough coast (Munk layers), or the analysis of boundary layers for fluids slipping over a rough surface [114]. Their scheme enables us to: (i) compensate for the unavailability of certain tools (Fourier analysis, Green and Poisson kernel estimates) up to the boundary, (ii) circumvent the lack of structure of the boundary which implies a lack of compactness of the solution to the boundary layer problem. Concerning (ii), let us note for instance that contrary to the boundary layer system (3.5) analyzed in Chapter 3, there is no underlying quasiperiodic structure that allows to lift the system on a higher-dimensional cylinder, compact in the direction tangential to the boundary.

The basic idea of Gérard-Varet and Masmoudi [154] is inspired from domain decomposition methods in numerical analysis. We split the bumpy Lipschitz domain $\Omega$ in two parts: (i) a flat half-space $\mathbb{R}_+^3$ in which we can rely on explicit representation formulas, (ii) a bumpy channel $\Omega \cap \mathbb{R}_-^3$ in which we gain compactness in the vertical direction that allows the use of Poincaré-type inequalities. The half-space system (2.15) is then reduced (after lifting the Dirichlet boundary condition) to an equivalent system in the channel $\Omega \cap \mathbb{R}_-^3$,

$$ \left\{ \begin{array}{ll} -\Delta W + \nabla Q = 0, & y \in \Omega \cap \mathbb{R}_-, \\ \nabla \cdot W = 0, & y \in \Omega \cap \mathbb{R}_-, \\ W(y', \gamma(y')) = 0, & \\ (-\partial_3 W + Q\mathbf{e}_3)|_{y_3=0} = \text{DN}(W|_{y_3=0}) - \mathbf{e}_j, & \end{array} \right. \qquad (2.17) $$

with a transparent boundary condition on the upper boundary $\{y_3 = 0\}$ involving a Dirichlet-to-Neumann operator DN. That operator implicitly solves the Dirichlet-Stokes problem in the flat half-space.

We carry out local energy estimates in the channel $\Omega \cap \mathbb{R}_-^3$. Such local energy estimates were carried out by Ladyženskaja and Solonnikov [231] for the Navier-Stokes equations
in an unbounded channel with Dirichlet boundary conditions. Here the nonlocality of the Dirichlet-to-Neumann operator DN makes the treatment of the large-scales particularly delicate. For an auxiliary parameter $m \in \mathbb{N}$, $m \ge 1$ and $k \in \mathbb{N}$ with $k \ge m$, one covers the two-dimensional space $\mathbb{R}^2$ by square tiles of area $(2m)^2$:

$$C_{k,m} := \{T = \eta + (-m, m)^2, \ \eta \in \mathbb{Z}^2 \text{ and } T \subseteq \mathbb{R}^2 \setminus (-k-m+1, k+m-1)^2\},$$

We then estimate the local energy

$$E_k[\mathcal{W}] := \int_{(-k,k)^2} \int_{\gamma(y')}^0 |\nabla \mathcal{W}|^2 dy_3 dy'$$

This yields the following discrete differential inequality, so-called ‘Saint-Venant’ or ‘Phragmén-Lindelöf’ estimate,

$$E_k[\mathcal{W}] \le C_* \left( k^2 + E_{k+m}[\mathcal{W}] - E_k[\mathcal{W}] + \frac{k^4}{m^6} \sup_{T \in C_{k,m}} E_T[\mathcal{W}] \right), \quad (\text{SV})$$

where $C_*$ is independent of $m$, and $k$, and depends only on $\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}$. Estimate (SV) yields the a priori bound

$$\sup_{\eta \in \mathbb{Z}^2} \int_{\eta+(0,1)^2} \int_{\gamma(y')}^0 |\nabla \mathcal{W}(y', y_3)|^2 dy_3 dy' \le C (\|\gamma\|_{W^{1,\infty}(\mathbb{R}^2)}),$$

by backward induction and choosing $m$ in terms of $C_*$. This bound in turn implies existence for (2.17), hence also for the original boundary layer system (2.15). Moreover, uniqueness for the linear system (2.17) is also deduced from (SV), without the contribution $k^2$ of the source term, by a backward induction.

### 2.3.2 Large-scale boundary regularity and boundary layers in John domains

In the paper [184] with Higaki and Zhuge, we perform the analysis of the large-scale regularity in John domains. As we mentioned above in Subsection 2.3.1, in the works [214] with Kenig and [183] with Higaki, several elementary tools were developed for the analysis of the first-order boundary layer correctors in bumpy Lipschitz domains without structure. Here, the analysis in bumpy John domains requires to push the techniques even further, to the limit of what seems technically possible.

**Analysis in John domains** The analysis in bumpy John domains relies on two elementary tools.

First, we can rely on the existence of a right inverse for the divergence operator with Dirichlet boundary conditions. The existence of such a Bogovskii operator in John domains is stated in [5]. This operator is required in order to estimate the pressure (as in the interior pressure estimate (2.14)) and hence prove a weak Caccioppoli inequality for the Stokes system [184, Lemma A.4], which then implies the reverse Hölder inequality as a starting point of the large-scale regularity theory.

Second, we have a Poincaré inequality for functions $H^1(B^\epsilon(r))$ vanishing on $\Gamma^\epsilon(r)$. We can for instance use [157, Proposition 3.15] to get that for all fixed bumpy John domain
$\Omega$ with constant $L \in (0, \infty)$ according to Definition 2.3, for all fixed $r \ge \varepsilon$, and for all $U \in H^1(B_{r,+}^\varepsilon)$ such that $U = 0$ on $\Gamma_r^\varepsilon$,

$$ \int_{B_{r,+}^{\varepsilon}} |U|^2 \le Cr^2 \int_{B_{r,+}^{\varepsilon}} |\nabla U|^2, \qquad (2.18) $$

where $C$ is an absolute constant independent of $\varepsilon$ and $r$. Notice that this estimate is only valid at scales $r \ge \varepsilon$. Indeed, below that scale the constant in (2.18) may degenerate due to possible inward cusps of highly oscillating bumpy John domains that lead to $|B(r) \setminus B^\varepsilon(r)| \ll r^3$, which is the measure of the region where we extend $U$ by zero.

As a consequence of the fact that the Poincaré inequality fails at small scales, all the boundary estimates of our work are mesoscopic estimates in the sense that they involve averaged quantities such as

$$ \mathcal{M}_t^p[\nabla U^\varepsilon](x) = \left( \int_{Q_t(x)} |\nabla U^\varepsilon|^p \right)^{1/p} . $$

for $t \ge \varepsilon$ smoothing out the possibly rough microscales.

**Quantitative method for the large-scale Lipschitz regularity** There is one particular point, where we are completely unable to transfer the techniques used above Lipschitz graphs to the present context. Indeed, in [214, 183] we used a domain decomposition method pioneered in [154] to study the well-posedness of the Stokes system for the first-order boundary layer correctors, see Subsection 2.3.1. We do not manage to adapt this strategy to our current situation, because of difficulties to estimate the pressure in the local energy estimates in the bumpy channel; see below the paragraph ‘Construction of boundary layers’. As a consequence, we cannot rely on boundary layer correctors to prove Lipschitz estimates directly as was done in [214, 183]. This requires to patiently bootstrap the regularity via quantitative approximation arguments:

(1) Caccioppoli's inequality in combination with the Poincaré-Sobolev inequality implies a large-scale reverse Hölder inequality.

(2) Combining the large-scale reverse Hölder inequality with Gehring's lemma, we obtain a large-scale Meyers type estimate, which is a first improvement of integrability for $\mathcal{M}_t^2[\nabla U^\varepsilon]$, see [184, Lemma 2.2].

(3) The Meyers estimate enables us to prove at any scale a sub-optimal error estimate between $U^\varepsilon$ and an approximation $\bar{U}$ in the flat domain, see [184, Lemma 2.6]. Along with the real variable argument of [313, Theorem 4.2.3] and improved flatness of $\bar{U}$, this enables us to prove a large-scale $W^{1,p}$ estimate for finite $p$ for the Stokes system, see [184, Theorem 2.4], following the scheme of Caffarelli and Peral [76] explained above in Subsection 2.1.3.

(4) In order to handle the nonlinear Navier-Stokes system, we consider the nonlinear term as a perturbation of the linear Stokes system. Hence we use the linear large-scale $W^{1,p}$ estimate along with a large-scale Sobolev embedding result [184, Theorem 2.7] to bootstrap the integrability of the nonlinear source term $-U^\varepsilon \otimes U^\varepsilon$; see [184, Theorem 2.8].

(5) Finally, we prove the large-scale Lipschitz estimate stated in Theorem 2.6 by an approximation argument of $U^\varepsilon$ by $\bar{U}$ [184, Lemma 3.2] and improved flatness for $\bar{U}$.
**Construction of boundary layers** In the work [183] the well-posedness of the first-order boundary layer system (2.15) was proved over Lipschitz graphs by a domain decomposition method: coupling of the Stokes problem in a bumpy channel $\Omega \cap \mathbb{R}^3_-$ with the Stokes problem in the flat half-space $\mathbb{R}^3_+$ via a nonlocal Dirichlet to Neumann boundary condition at the interface $\partial \mathbb{R}^3_+$. We face considerable technical difficulties when trying to adapt this strategy to the case of bumpy John domains. Indeed, the local energy estimates in the bumpy channel require to estimate the pressure, or to work with divergence-free test functions. In either case, we need to construct a Bogovskii operator for a sequence of exhausting domains containing $\Omega \cap \{|x'| \le k, x_3 < 0\}$ with a constant uniform in $k$. The construction of the Bogovskii operator of [5] relies on connecting any point in the bumpy John domain to a fixed neighborhood of a reference point $\bar{x}$. Such a procedure gives, for a slim domain such as $\Omega \cap \{|x'| \le k, x_3 < 0\}$, a constant in the Bogovskii estimate that scales proportionately to the horizontal size $k$ of the domain. We are unable to take advantage of the small vertical extent of the domain to provide a modified construction of the Bogovskii operator. This would be needed to carry out the downward iteration on the local energy estimates, also called Saint-Venant estimates, in [183].

This obstacle related to the existence of boundary layer correctors in bumpy John domains lead us to the new strategy explained above, that is inspired from almost-periodic and random homogenization where similar issues for the existence of correctors with good properties arise; see Subsection 2.1.3 above.

Now that the large-scale Lipschitz estimate of Theorem 2.6 is proved without any use of boundary layers, we develop a new strategy for the existence of solutions to the boundary layer system (2.15). In fact, the large-scale Lipschitz regularity in Theorem 2.6 makes it possible to construct the velocity and pressure parts of the Green function in bumpy John domains, and to estimate their decay at large scales, see [184, Appendix B]. Similar arguments were used for instance in [30, 216] for elliptic equations and in [177] for the Stokes equations. These estimates are the key for our new proof of the existence of the first-order boundary layer correctors; see [184, Theorem 4.1]. In this way we are able to completely by-pass the difficulties posed by the domain decomposition method used in [154, 116, 115, 214, 183].

To the best of our knowledge, our present work is also the first to carry out a thorough analysis of the second-order boundary layer correctors, allowing for linear growth of the boundary data in the tangential direction. Our key observation is an algebraic connection between the first-order and second-order boundary layers on the boundary, which allows us to use the first-order boundary layer correctors in an Ansatz for the second-order boundary layers, see [184, Section 4.3]. Unlike the first-order boundary layers (which form a two-dimensional vector space), the space of second-order boundary layers is six-dimensional and needs three different ways of construction, based on the structures of the associated Stokes polynomials. For our analysis to go through, we also need some good quantitative convergence/decay of the first-order boundary layers away from the boundary. Hence we work in a periodic framework, according to Definition 2.4, but this is by no means an optimal assumption. Other structures, such as almost-periodic structures with a non-resonance condition, or random ergodic with quantitative decorrelation properties at large scales, would certainly be manageable.

**Quantitative method for the higher-order regularity** Let us summarize the chain of results that we obtain in the paper [184]:
(1) proof of a large-scale Lipschitz estimate by a quantitative method, see Theorem 2.6,

(2) proof of large-scale Green kernel bounds, see [184, Proposition B.3-B.5],

(3) proof of the existence of first-order boundary layer correctors, see [184, Theorem 4.1],

(4) large-scale $C^{1,\mu}$ estimates, see Theorem 2.9,

(5) proof of the existence of second-order boundary layer correctors see [184, Theorem 4.3],

(6) large-scale $C^{2,\mu}$ estimates, see Theorem 2.10.

We emphasize that we rely on a quantitative perturbation method inspired from the classical Schauder regularity theory. In the vein of Caffarelli and Peral [76], the method follows the lines given in Subsection 2.1.3. In a nutshell, we use the improved regularity of the approximate problem, to get the scale-by-scale decay of excess quantities (measuring for instance, Hölder continuity, Lipschitz, $C^{1,\mu}$, $C^{2,\mu}$, or higher regularity) for the original rough problem, up to a small error. We then conclude by a real variable argument or an iteration lemma, which are in some sense black boxes oblivious to the equations.

In the context of homogenization, the homogenized limit problem with constant coefficients is the approximate problem. Here, the approximate problem is a Stokes problem in a domain with a flat boundary. Both problems have improved regularity, in the sense that the solutions are basically as smooth as one wishes.

*Remark 2.12 (about the self-similarity).* This remark is the pendant of Remark 2.1 above. For the $C^{1,\mu}$ estimate, a crucial fact is that $(W, P)$ belongs to $Q_1(\Omega)$, where the class $Q_1(\Omega)$ is defined in Theorem 2.9. In other words, the building blocks of the $C^{1,\mu}$ regularity, which are linear polynomials corrected by boundary layers, are solutions to the Stokes system that vanish on the bumpy boundary. Hence, by rescaling, $(U^\varepsilon, P^\varepsilon) - (\varepsilon W(x/\varepsilon), P(x/\varepsilon))$ is still a weak solution with a no-slip boundary condition. This observation allows us to capture the regularity beyond the Lipschitz estimate. A similar remark holds for the case of the $C^{2,\mu}$ regularity.
# Chapter 3

## Quantitative homogenization of boundary layers

This chapter relies mainly on the paper:

— [23], with Scott Armstrong, Tuomo Kuusi and Jean-Christophe Mourrat, *Quantitative analysis of boundary layers in periodic homogenization*, *Arch. Ration. Mech. Anal.* (2017).

### Contents

<table><tbody><tr><td><strong>3.1 Context: state of the art and obstacles</strong></td><td style="text-align: right;"><strong>50</strong></td></tr><tr><td style="padding-left: 2em;">3.1.1 Boundary layers in the half-space and polygonal domains</td><td style="text-align: right;">50</td></tr><tr><td style="padding-left: 2em;">3.1.2 Boundary layers in convex domains</td><td style="text-align: right;">52</td></tr><tr><td><strong>3.2 Main result</strong></td><td style="text-align: right;"><strong>54</strong></td></tr><tr><td style="padding-left: 2em;">Novelty of our result</td><td style="text-align: right;">55</td></tr><tr><td style="padding-left: 2em;">Further developments</td><td style="text-align: right;">55</td></tr><tr><td><strong>3.3 New ideas and strategy for the proofs</strong></td><td style="text-align: right;"><strong>57</strong></td></tr><tr><td style="padding-left: 2em;">3.3.1 High level strategy</td><td style="text-align: right;">58</td></tr><tr><td style="padding-left: 2em;">3.3.2 The size of the boundary layer is not uniform</td><td style="text-align: right;">59</td></tr><tr><td style="padding-left: 2em;">3.3.3 Regularity estimates for the homogenized boundary data</td><td style="text-align: right;">61</td></tr></tbody></table>

This chapter is concerned with the quantitative analysis of a boundary layer problem that arises in the study of periodic homogenization. We consider the oscillating Dirichlet problem for uniformly elliptic systems with periodic coefficients, taking the form

$$ \begin{cases} -\nabla \cdot (a(\frac{x}{\epsilon})\nabla U^{\epsilon}(x)) = 0 & \text{in } \Omega, \\ U^{\epsilon}(x) = g(x, \frac{x}{\epsilon}) & \text{on } \partial\Omega. \end{cases} \qquad (BL^{\epsilon}) $$

Here $0 < \epsilon \ll 1$, the dimension $d \ge 2$ and $\Omega \subseteq \mathbb{R}^d$. We will require in addition that $\Omega$ is polygonal or smooth, convex or uniformly convex, although this last two assumptions can be relaxed. The coefficients are given by a tensor $a = (a_{ij}^{\alpha\beta})_{i,j=1,...,L}^{\alpha,\beta=1,...,d}$ and the unknown function $U^{\epsilon} = (U_j^{\epsilon})_{j=1,...,L}$ takes values in $\mathbb{R}^L$. The coefficients are assumed to satisfy, for some fixed constant $\lambda \in (0, 1)$, the uniformly elliptic condition, for all $\xi = (\xi_i^{\alpha}) \in \mathbb{R}^d \times \mathbb{R}^L$, for all $y \in \mathbb{R}^d$,

$$ \lambda |\xi|^2 \le a_{ij}^{\alpha\beta}(y) \xi_i^{\alpha} \xi_j^{\beta} \le \lambda^{-1} |\xi|^2. \qquad (3.1) $$
Both the coefficients $\mathbf{a}$ and the Dirichlet boundary condition $g : \partial\Omega \times \mathbb{R}^d \to \mathbb{R}$ are assumed to be smooth functions,

$$ \mathbf{a} \in C^\infty(\mathbb{R}^d; \mathbb{R}^{L \times L \times d \times d}) \quad \text{and} \quad g \in C^\infty(\partial\Omega \times \mathbb{R}^d) \qquad (3.2) $$

and periodic in the fast variable, that is,

$$ \mathbf{a}(y) = \mathbf{a}(y + \xi) \quad \text{and} \quad g(x, y) = g(x, y + \xi) \quad \forall x \in \partial\Omega, y \in \mathbb{R}^d, \xi \in \mathbb{Z}^d. \qquad (3.3) $$

The goal is to understand the asymptotic behavior of the system (BL$^{\epsilon}$) as $\epsilon \to 0$.

## 3.1 Context: state of the art and obstacles

The problem (BL$^{\epsilon}$) arises naturally in the theory of elliptic homogenization when one attempts to obtain a two-scale expansion of solutions of the Dirichlet problem with non-oscillating boundary condition near the boundary

$$ \left\{ \begin{array}{ll} -\nabla \cdot \mathbf{a}\left(\frac{x}{\epsilon}\right) \nabla V^{\epsilon} = 0 & \text{in } \Omega, \\ V^{\epsilon}(x) = g(x) & \text{on } \partial\Omega. \end{array} \right. \qquad (3.4) $$

Indeed, the oscillating interior first-order corrector in the two-scale expansion induces a locally periodic perturbation of the boundary condition of order $O(\epsilon)$:

$$ V^{\epsilon}(x) = \bar{V}(x) + \epsilon \left( \chi\left(\frac{x}{\epsilon}\right) \cdot \nabla \bar{V}(x) + U_{bl}^{\epsilon}(x) \right) + \dots $$

where $\chi = \chi(y)$ is the periodic cell corrector and $U_{bl}^{\epsilon}$ solves (BL$^{\epsilon}$) with $g(x, \frac{x}{\epsilon}) = -\chi(\frac{x}{\epsilon}) \cdot \nabla \bar{V}(x)$. For more details, we refer to [50, 313]. In other words, when examining the fine structure of solutions of the Dirichlet problem with oscillating coefficients, one expects to find a boundary layer in which the solutions behave qualitatively differently than they do in the interior of the domain, for which we have a complete understanding to arbitrary order [24]. The study of this boundary layer can be reduced to a problem of the form (BL$^{\epsilon}$). This problem was identified at the start of the homogenization theory in the seventies by Bensoussan, Lions and Papanicolaou [50]. Despite this, major progress on this question was only achieved about ten years ago with the works of Gérard-Varet and Masmoudi [154, 155, 156]. As a matter of fact, the analysis of (BL$^{\epsilon}$) is particularly challenging because of: (i) the high oscillations that create strong gradients in the vicinity of the boundary $\partial\Omega$, (ii) the boundary that breaks the periodic structure and (iii) the possible resonances between the periodic coefficients and the non-periodic oscillations on the boundary. This chapter is devoted to the quantitative analysis of these concentration phenomena.

### 3.1.1 Boundary layers in the half-space and polygonal domains

The analysis of the boundary layer in the domain $\Omega$ and the definition of the homogenized boundary condition $\bar{g}$ are based on an approximation procedure involving half-space boundary layer problems

$$ \left\{ \begin{array}{ll} -\nabla \cdot \mathbf{a}(y) \nabla \mathcal{U} = 0, & y \cdot n(x) > a, \\ \mathcal{U} = g(x, y), & y \cdot n(x) = a, \end{array} \right. \qquad (3.5) $$
where $n(x) \in \partial B_1$, $a \in \mathbb{R}$, $\mathcal{U} = \mathcal{U}(y; a, n(x))$. Full understanding of these boundary layers was achieved in the works [155, 156, 282], after earlier works [128, 267, 53, 266, 15, 268] concerned with rational half-spaces for which $n \in \mathbb{R}\mathbb{Z}^d$.

Convergence of $\mathcal{U}(y; a, n)$ when $y \cdot n \to \infty$ to a 'boundary layer tail' i.e. a constant vector field $\mathcal{U}^\infty$ holds whatever $n \in \partial B_1$, see my work [282] via a two-scale expansion for the Poisson kernel. Quantitative rates of convergence, though, are very sensitive to the Diophantine properties of the normal $n$. Let $\kappa > \frac{1}{d-1}$ be fixed. We say that $n \in \partial B_1$ is 'Diophantine' or 'non-resonant' with constant $A > 0$ if

$$|(I_d - n \otimes n)\xi| \ge A|\xi|^{-\kappa}, \quad \text{for all } \xi \in \mathbb{Z}^d \setminus \{0\}, \qquad (3.6)$$

where $(I_d - n \otimes n)\xi$ denotes the projection of $\xi$ on the hyperplane $n^\perp$.

Let $M$ be an orthogonal matrix such that $Me_d = n$, and $N$ be the matrix of the $d-1$ first columns of $M$. The matrices $M$ and $N$ essentially give us a convenient coordinate system to work in. They of course depend on $n \in \partial B_1$ and can be chosen in such a way as to be locally smooth functions of $n$ in a neighborhood of any fixed $\bar{n} \in \partial B_1$. Thus, condition (3.6) is equivalent to

$$|N^T \xi| \ge A |\xi|^{-\kappa}, \quad \text{for all } \xi \in \mathbb{Z}^d \setminus \{0\}. \qquad (3.7)$$

A key idea to study the system (3.5) is to take advantage of the quasiperiodic structure of the system along the boundary. One lifts the problem to a higher-dimensional cylinder $\mathbb{T}^d \times (a, \infty)$, where $\Upsilon = \Upsilon(\theta, t)$ solves a sub-elliptic problem

$$\begin{cases} - \left( \frac{N^T \nabla_{\theta}}{\partial_t} \right) \cdot \left( M^T \mathbf{a}(\theta + tn) M \left( \frac{N^T \nabla_{\theta}}{\partial_t} \right) \Upsilon \right) = 0, & \theta \in \mathbb{T}^d, t > a, \\ \Upsilon = g(x, \theta + an), & \theta \in \mathbb{T}^d, t = a. \end{cases} \qquad (3.8)$$

Notice that $\mathcal{U} = \Upsilon(x - (x \cdot n)n, x \cdot n)$. This enables to recover compactness properties in the horizontal direction. The price one pays is the degeneracy of the elliptic system (3.8). Under the non-resonance assumption (3.6) (see also (3.7)), one can show a higher-order Poincaré inequality: the degenerate derivative $N^T \nabla_{\theta} \Upsilon$ controls $\Upsilon - \int_{\mathbb{T}^d} \Upsilon$, up to a finite loss of derivatives, see [155, Subsection 2.4]. Doing so it is possible to establish Saint-Venant type estimates for any $\alpha \in (\frac{1}{2}, 1)$, i.e. a differential inequality of the type

$$K(T) \le C_p (-K'(T))^\alpha$$

for the quantity

$$K(T) := \int_{\mathbb{T}^d} \int_T^\infty |N^T \nabla_{\theta} \Upsilon|^2 + |\partial_t \Upsilon|^2 d\theta dt,$$

which implies algebraic decay of $K(T)$ for $T \to \infty$. Hence, one obtains [156, Proposition 2.6], which roughly states the existence of a boundary layer tail $\mathcal{U}^\infty$ such that $\mathcal{U}$ and any derivatives converge to $\mathcal{U}^\infty$ faster than any polynomial of $Ay \cdot n$, where $A > 0$ is a constant such that (3.6).

Let us remark that the non-resonance condition was previously used to compensate for the degeneracy of $N^T \nabla_{\theta} \cdot$ in order to construct bounded correctors in the homogenization of quasiperiodic structures by Kozlov [222]. Moreover, the non-resonance condition enables to quantify ergodicity; see the quantitative ergodic theorem in our work [23, Proposition 2.1].
For further insights related to quantification of ergodicity for almost-periodic structures we refer to [20]. Furthermore, as is expected, the convergence in (3.5) can be arbitrarily slow for resonant directions, i.e. $n \in \partial B_1$ such that (3.6) does not hold. An explicit example is given in [282]; see also [12] for operators with oscillating coefficients. We finally emphasize that the existence of a boundary layer tail requires certain ergodicity properties, which here are implied by the quasiperiodicity of the boundary data along $y \cdot n = a$. Of course, in the case of a lack of ergodicity, no convergence at space infinity takes place, see for example [154, Proposition 11].

We conclude this overview of the literature, by a discussion of the homogenization of $(BL^\varepsilon)$ in the case of convex polygonal domains

$$ \Omega = \bigcap_{k=1}^{M} \{x, x \cdot n^k > a^k\} \subseteq \mathbb{R}^2. $$

If all normals $n^k$ are rational, i.e. $n^k \in \mathbb{R}Z^d \cap \partial B_1$, homogenization results were achieved by Moskow and Vogelius [266] and Allaire and Amar [15]. The work of Gérard-Varet and Masmoudi [155] handled the convergence in the case when $n^k \in \partial B_1$ satisfies the non-resonance condition (3.6). Let $\bar{V}$ be the homogenized limit in the two-scale expansion (3.1) and $g(x, y) = -\chi(y) \cdot \nabla \bar{V}$. For every edge $k \in \{1, \dots, M\}$, we call $U^{\infty,k}$ the boundary layer tail. Then, see [283, Theorem 3.3], $U^\varepsilon$ solution to $(BL^\varepsilon)$ converges to

$$ \left\{ \begin{array}{ll} -\nabla \cdot \bar{a} \nabla \bar{U} = 0, & x \in \Omega \\ \bar{U} = -U^{\infty,k} \cdot \nabla \bar{V}, & x \in \partial \Omega \cap \{x \cdot n^k = a^k\}, \quad \text{for all } 1 \le k \le M, \end{array} \right. \qquad (3.9) $$

with a rate $O(\varepsilon^{1/2})$ in $L^2(\Omega)$ on condition that $\bar{V}$ has enough regularity. Notice that the works [266, 283] are concerned with eigenvalue expansions for elliptic operators with highly oscillating coefficients, where the boundary layers determine the first-order corrections to the spectrum.

### 3.1.2 Boundary layers in convex domains

The first asymptotic convergence result for the homogenization of the system $(BL^\varepsilon)$ in general uniformly convex domains was obtained by Gérard-Varet and Masmoudi [156]. They proved the existence of an homogenized boundary condition

$$ \bar{g} \in L^\infty(\partial \Omega) $$

such that, for each $\delta > 0$ and $q \in [2, \infty)$,

$$ \|U^{\varepsilon} - \bar{U}\|_{L^q(\Omega)}^{q} \le C \varepsilon^{\frac{2(d-1)}{3d+5} - \delta}, \qquad (3.10) $$

where the constant $C(\delta, d, L, \lambda, \Omega, g, \mathbf{a}) \in (0, \infty)$ and $\bar{U}$ is the solution of the homogenized Dirichlet problem

$$ \left\{ \begin{array}{ll} -\nabla \cdot \bar{a} \nabla \bar{U} = 0 & \text{in } \Omega, \\ \bar{U}(x) = \bar{g}(x) & \text{on } \partial \Omega, \end{array} \right. \qquad (3.11) $$

and $\bar{a}$ is the usual homogenized tensor; see the similarity with the system (3.9). Notice that in [156], the estimate (3.10) is stated only for $q = 2$, but the statement for general $q$
can be recovered by interpolation since $L^\infty$ bounds are available for both $U^\epsilon$ and $\bar{U}$. Besides giving the quantitative rate in (3.10), this result was the even first qualitative proof of homogenization of $(BL^\epsilon)$.

It is natural to approximate $\partial\Omega$ locally by hyperplanes and thus the boundary layer by solutions of Dirichlet problems in flat tilted half-spaces. As emphasized above in Subsection 3.1.1, these hyperplanes destroy the periodic structure of the problem. The geometry of the domain $\Omega$ thus enters in a nontrivial way. The local behavior of the boundary layer depends on whether or not the angle of the normal vector to $\partial\Omega$ is non-resonant with the periodic structure of $g(x, \cdot)$ and $\mathbf{a}$ i.e the lattice $\mathbb{Z}^d$. In addition to that difficulty, already present for polygonal domains, there is a further issue in smooth domains related to pasting the different half-spaces together. This is complicated due to the strength of singularities in the boundary layer and to the difficulty in obtaining any regularity of the homogenized boundary condition $\bar{g}$.

The lack of a periodic structure means the problem requires a quantitative approach as opposed to the softer arguments based on compactness that are more commonly used in periodic homogenization. Such a strategy was pursued in [156], based on gluing together the solutions of half-space problems with boundary hyperplanes having Diophantine (non-resonant) slopes, and it led to the estimate (3.10). As pointed out by the authors of [156], the exponent in (3.10) is not optimal and was obtained by balancing two sources of error. Roughly, if one approximates $\partial\Omega$ by too many hyperplanes, then the constant in the Diophantine condition for some of the hyperplanes is small, leading to a worse estimate. If one approximates with too few planes, the error in the local approximation caused by the difference between the local hyperplane and $\partial\Omega$ becomes large. In [156], the authors approximate $\Omega$ by polygons having edges of comparable size $O(\epsilon^\alpha)$ for some $\alpha \in (0, 1)$, which is a source of non-optimality of the rate (3.10). This leaves room for improvement that we exploited in collaboration with Armstrong, Kuusi and Mourrat, see Theorem 3.1 below.

Given the role of the problem $(BL^\epsilon)$ in quantifying asymptotic expansions in periodic homogenization, obtaining the optimal convergence rate of $\|U^\epsilon - \bar{U}\|_{L^p(\Omega)}$ to zero is of fundamental importance. To make a guess for how far the upper bound for the rate in (3.10) is from being optimal, one can compare it to the known rate in the case that $\mathbf{a}$ is constant-coefficient (i.e., $\mathbf{a} = \bar{\mathbf{a}}$). In the latter case, the recent work of Aleksanyan, Shahgholian and Sjölin [14] gives, for every $q \in [1, \infty)$,

$$ \|U^{\epsilon} - \bar{U}\|_{L^q(\Omega)}^q \le C \cdot \begin{cases} \epsilon^{\frac{1}{2}} & \text{in } d=2, \\ \epsilon |\log \epsilon| & \text{in } d=3, \\ \epsilon & \text{in } d \ge 4. \end{cases} \qquad (3.12) $$

One should not expect a convergence rate better than $O(\epsilon^{\frac{1}{q}})$ for $\|U^\epsilon - \bar{U}\|_{L^q(\Omega)}$. Indeed, observe that the difference in the boundary conditions is $O(1)$ and that we should expect this difference to persist at least in an $O(\epsilon)$-thick neighborhood of $\partial\Omega$. Thus the solutions will be apart by at least $O(1)$ in a set of measure at least $O(\epsilon)$, and this already contributes $O(\epsilon^{\frac{1}{q}})$ to the $L^q$ norm of the difference, which is observed in [14, Theorem 1.6] and (3.12).
## 3.2 Main result

Our main result is the following quantitative homogenization result for the elliptic system with highly oscillating Dirichlet data (BL$^{\epsilon}$). There are two main points in the theorem: (i) a statement about the regularity of the homogenized boundary data, (ii) a convergence rate.

**Theorem 3.1** (nearly optimal quantitative estimates; [23, Theorem 1], in collaboration with Armstrong, Kuusi and Mourrat). Assume that $\Omega \subseteq \mathbb{R}^d$ is a smooth bounded uniformly convex domain, that $\mathbf{a}$ is uniformly elliptic and that $\mathbf{a}$ and $g$ are smooth and $\mathbb{Z}^d$-periodic (see assumptions (3.1), (3.2) and (3.3)). Let $\bar{\mathbf{a}}$ denote the homogenized coefficients associated to $\mathbf{a}$ obtained in periodic homogenization. Then,

(1) there exists a function $\bar{g} \in L^{\infty}(\partial\Omega)$ satisfying

$$ egin{cases} \bar{g} \in W^{s,1}(\partial\Omega) & \text{for all } s < \frac{2}{3} \text{ in } d=2, \\ \nabla \bar{g} \in L^{\frac{2(d-1)}{3},\infty}(\partial\Omega) & \text{in } d>2, \end{cases} $$

(2) for every $q \in [2, \infty)$ and $\delta > 0$, for every $\varepsilon \in (0, 1]$, the solutions $U^{\varepsilon}$ and $\bar{U}$ of the problems (BL$^{\varepsilon}$) and (3.11) satisfy the estimate

$$ \|U^{\varepsilon} - \bar{U}\|_{L^q(\Omega)}^{q} \le C \cdot \begin{cases} \varepsilon^{\frac{1}{3}-\delta} & \text{in } d=2, \\ \varepsilon^{\frac{2}{3}-\delta} & \text{in } d=3, \\ \varepsilon^{1-\delta} & \text{in } d \ge 4, \end{cases} \quad (3.13) $$

where $C(q, \delta, d, \lambda, \mathbf{a}, g, \Omega) \in (0, \infty)$ is independent of $\varepsilon$.

In addition we obtain an explicit formula for the homogenized boundary data $\bar{g}$, see [23]. The formula for $\bar{g}(x)$, $x \in \partial\Omega$ fixed, is an average of $g$ agains a kernel $\omega(x, \cdot)$ that depends on the normal $n(x)$:

$$ \bar{g}(x) = \int_{\mathbb{T}^d} g(x, y)\omega(x, y) dy. \quad (3.14) $$

The kernel $\omega(x, \cdot)$ comes from the two-scale expansion of the Poisson kernel around $x$, see [23, Corollary 5.2]. It involves half-space boundary layer correctors around the point $x$.

**Remark 3.2.** The fundamental role of the uniform convexity assumption is to ensure that there are enough good non-resonant directions. We introduce the function $A = A(n) \in [0, 1]$ defined for $n \in \partial B_1$ in the following way

$$ A(n) := \sup \{A \ge 0 : \text{ A satisfies (3.6)}\}. \quad (3.15) $$

In a bounded uniformly convex domain $\Omega$, we have

$$ A^{-1}(n(\cdot)) \in L^{d-1, \infty}(\partial \Omega). \quad (3.16) $$

The assumption of uniform convexity can be relaxed, see the paper of Zhuge [353] building upon our analysis. In a domain of finite type $k$, $A^{-1}(n(\cdot)) \in L^{\frac{1}{k-1}, \infty}(\partial\Omega)$. The error estimates are worse as expected.
*Remark 3.3* (about low dimensions, $d = 2, 3$). The reason that the rate is worse in low dimensions is because in some places near points of $\partial\Omega$ with good Diophantine normals, the boundary layer where $|U^\epsilon - \bar{U}| \gtrsim 1$ will be $O(\epsilon)$ thick, but in other places near points with rational normals with small denominator relative to $\epsilon^{-1/2}$, the boundary layer will actually be worse, up to $O(\epsilon^{1/2})$ thick. In small dimensions, i.e. in $d = 2$ and with $d = 3$ being critical, the ‘bad’ points actually take a relatively large proportion of the surface area of the boundary (see the integrability of $A^{-1}(n(\cdot))$ given by (3.16)), leading to a worse error.

*Remark 3.4* (on the smoothness assumption). We do not expect it to be possible to eliminate the small loss of exponent represented by $\delta > 0$ without upgrading the qualitative regularity assumption (3.2) on the smoothness of $a$ and $g$ to a quantitative one (for example, that these functions are analytic). Note that this regularity assumption plays an important role in the proof of Theorem 3.1 and is not a mere technical assumption or one used to control the small scales of the solutions. Rather, it is used to obtain control over the large-scale behavior of the solutions via the quasiperiodic structure of the problem since it gives us a quantitative version of the ergodic theorem, see [23, Proposition 2.1]. In other words, the norms of high derivatives of $a$ and $g$ control the ergodicity of the problem and thus the rate of homogenization.

### Novelty of our result

Our result improves the best known results at that time in two directions:

(1) The rates of convergence that we obtain (3.13) are better in any dimension than the rates (3.10) obtained by Gérard-Varet and Masmoudi in [156]. In dimensions $d \ge 4$, we obtain the optimal convergence rate up to an arbitrarily small loss of exponent, since it agrees with (3.12). The difference in small dimensions between our rate and (3.12) is due to an error which arises only in the case of operators with oscillating coefficients: the largest source of error comes from the possible irregularity of the homogenized boundary condition $\bar{g}$. Reducing this source of error requires to improve the regularity of $\bar{g}$; see below the paragraph ‘Further developments’.

(2) The statement asserting that $\nabla \bar{g} \in L^{\frac{2(d-1)}{3}, \infty}$ in $d > 2$ and $\bar{g} \in W^{\frac{2}{3}-, 1}$ in $d = 2$ is an improvement upon the one proved in [156, consequence of Corollary 2.9], where it was shown that $\nabla \bar{g} \in L^{\frac{d-1}{2}, \infty}$ in $d > 2$ and $\bar{g} \in W^{\frac{1}{2}-, 1}$ in $d = 2$. The limitation of this regularity comes from the lack of integrability of the inverse of the Diophantine constant, $A^{-1}(n(\cdot)) \in L^{d-1, \infty}$, see (3.16).

As we already mentioned and comment more on below, progress on the regularity of $\bar{g}$ enables us to improve the error estimates.

### Further developments

The work of Gérard-Varet and Masmoudi [156] and our work with Armstrong, Kuusi and Mourrat [23] were followed by an intense research activity, that lead to significant progress in several directions:

(1) **Optimal estimates in low dimensions**

Several months after our paper [23] first appeared on the arXiv, Zhongwei Shen kindly pointed out to us that our method leads to optimal estimates for the boundary layer in dimensions $d = 2, 3$, up to an arbitrarily small loss of exponent. Indeed, Shen
and Zhuge [316] were able to upgrade the regularity statement for the homogenized boundary data in Theorem 3.1, reaching $\nabla \bar{g} \in L^q$ for any $q < d-1$ in dimension $d \ge 3$, and $\bar{g} \in W^{s,1}$ for any $s < 1$ in dimension $d = 2$. Their proof of the regularity of the homogenized boundary data follows ours, with a new ingredient, namely a weighted estimate for the boundary layer. Using this improvement of regularity for $\bar{g}$ and then following our argument for estimating boundary layers leads to the following improvement of the estimates of Theorem 3.1 in $d = 2, 3$, which is also proved in [316]: for every $q \in [2, \infty)$ and $\delta > 0$, there is a constant $C(q, \delta, d, \lambda, \mathbf{a}, g, \Omega) \in (0, \infty)$ such that, for every $\varepsilon \in (0, 1]$, the solutions $U^\varepsilon$ and $\bar{U}$ of the problems (BL$^\varepsilon$) and (3.11) satisfy the estimate

$$ \|U^\varepsilon - \bar{U}\|_{L^q(\Omega)}^q \le C \cdot \begin{cases} \varepsilon^{\frac{1}{2}-\delta} & \text{in } d=2, \\ \varepsilon^{1-\delta} & \text{in } d=3. \end{cases} \quad (3.17) $$

This is optimal since it agrees with (3.12), up to an arbitrarily small loss of exponent.

## (2) Regularity of the homogenized boundary data

As mentioned above, in the paper [316] Shen and Zhuge were able to upgrade the regularity of the homogenized boundary data by using a weighted estimate near the boundary for the boundary layers. In the subsequent paper [317], Shen and Zhuge manage a further step forward. They prove that $\bar{g} \in W^{1,p}(\partial\Omega)$ for any $p \in [1, \infty)$. This follows from combining: (i) bounds coming from the analysis of the half-space boundary layer problem in the higher-dimensional torus $\mathbb{T}^d \times (0, \infty)$, which involve the Diophantine constant $A$, see Subsection 3.1.1 above, (ii) bounds coming from the Poisson kernel in the ‘physical’ half-space and (iii) a formula [317, equation (2.18)] linking the higher-dimensional torus boundary layer to the ‘physical’ space boundary layer. With these simple ideas, they succeed in circumventing the lack of integrability of the inverse of the Diophantine constant $A^{-1}(n(\cdot))$. Along a different line of research, let us also mention the work of Feldman and Zhang [132]. They also manage to prove Hölder continuity of the homogenized boundary data using an intermediate scale homogenization problem related to the variation of the boundary layer tail in the vicinity of a given rational direction. We summarize progress on the regularity of the homogenized boundary data in Figure 3.1.

## (3) Relaxation of the convexity assumption

In the work [353], Zhuge was able to extend the quantitative homogenization of the system (BL$^\varepsilon$) to domains that are not convex and are of finite type, which is more or less the most general framework in which one can carry out a quantitative analysis. Indeed, it is understood that in domains that have a flat part or infinite vanishing of derivatives at some point of the boundary, no algebraic rate of convergence can be expected, although some qualitative convergence may take place in certain conditions, see [148, 218]. The absence of an algebraic rate follows from results about convergence to the boundary layer tail in the half-space, see [282]. The qualitative convergence was proved in [218] by Kim, Lee and Shahgholian under the assumption that the domain satisfies an Irrational Direction Dense Condition, see [218, Theorem 1.5]. It is noted, see [218, Example 4.10], that the homogenized boundary data may be discontinuous if the boundary contains a flat part with a rational direction.

## (4) Other types of oscillating boundary conditions

Progress on the quantitative homogenization of problems with highly oscillating Dirichlet data also opened the way to investigations of other types of oscillating boundary
<table><caption>Figure 3.1 – Regularity of the homogenized boundary data</caption><tr><td>Gérard-Varet and Masmoudi (2012) [156]</td><td>g&#773; &isin; L<sup>&infin;</sup>(&part;&Omega;) &cap; L<sup><sup>d&minus;1</sup>&frasl;<sub>2</sub>,&infin;</sup>(&part;&Omega;)<br/>for d &gt; 2</td></tr><tr><td>Armstrong, Kuusi, Mourrat and Prange (2017) [23]</td><td>&nabla;g&#773; &isin; L<sup><sup>2(d&minus;1)</sup>&frasl;<sub>3</sub>,&infin;</sup>(&part;&Omega;) for d &gt; 2</td></tr><tr><td>Feldman and Zhang (2019) [132]<br/>via intermediate scale boundary layer problem</td><td>g&#773; &isin; C<sup>0, <sup>1</sup>&frasl;<sub>d</sub></sup>(&part;&Omega;)</td></tr><tr><td>Shen and Zhuge (2020) [317]<br/>via weighted estimates</td><td>&nabla;g&#773; &isin; L<sup>&infin;</sup>(&part;&Omega;)</td></tr></table>

conditions. Shen and Zhuge studied the Neumann problem in [316]. Geng and Zhuge studied the Robin problem $n(x) \cdot a(x/\epsilon)\nabla U^\epsilon + b(x/\epsilon)U^\epsilon = h$ in [148]. Notice that for Robin boundary conditions, the rates are obtained using in particular a duality argument from the Neumann problem.

(5) Eigenvalue expansions

Progress on the rates of convergence for the homogenization of the boundary layer problem opens the way to improved error estimates for the first-order expansions of eigenvalues and eigenvectors of the elliptic operator $-\nabla \cdot a(x/\epsilon)\nabla U^\epsilon$ in a bounded domain $\Omega$. Such an analysis was done in my paper [283] for polygonal domains building upon [155], and by Zhuge in [354] for smooth uniformly convex domains building upon [23, 316]; for an applied view on wave propagation in bounded periodic media, see [108].

(6) Random structures

Half-space boundary layer correctors are constructed by Fischer and Raithel [136, Theorem 1] for random stationary ergodic structures. In this setting contrary to periodic structures, the position of the boundary of the half-space does not matter. The homogenization result for the system (BL$^\epsilon$) in bounded domains with a random structure was announced at the '9th GAMM-Workshop on Analysis of Partial Differential Equations' at the University of Freiburg in October 2021.

(7) Interfaces and corners

The homogenization of elliptic systems with oscillating coefficients with different structures in two half-spaces separated with a flat interface is a very similar problem to the analysis of the half-space boundary layer problem (3.5). Progress on this problem was achieved by Josien [203] for periodic structures with commensurable periods and by Josien and Raithel [204] for general periodic or random structures. Such an analysis was further extended to the case of a domain with a corner by Josien, Raithel and Schäffner [205].

## 3.3 New ideas and strategy for the proofs

We first underline the novelty of our approach, before focusing on two specific aspects: the non uniform size of the boundary layer which is a discovery of [23] and the continuity estimates for the homogenized boundary data.
### 3.3.1 High level strategy

The proof of Theorem 3.1 blends techniques from previous works on the problem [155, 156, 13, 14] with some original estimates and then combines them using a new strategy. Like the approach of [156], we cut the boundary of $∂Ω$ into pieces and approximate each piece by a hyperplane. However, rather than gluing approximations of the solution together, we approximate, for a fixed $x_0$, the contribution of each piece of the boundary in the Poisson formula

$$U^{\epsilon}(x_0) = \int_{\partial \Omega} P^{\epsilon}(x_0, x) g(x, \frac{x}{\epsilon}) dx, \qquad (3.18)$$

where $P^{\epsilon}$ is the Poisson kernel for the heterogeneous operator $-\nabla \cdot \mathbf{a}(\cdot, \frac{\cdot}{\epsilon})\nabla \cdot$.

The first step in our argument is to replace the Poisson kernel $P^{\epsilon}(x_0, x)$ by its two scale expansion, using a result of Kenig, Lin and Shen [216] (based on the classical regularity theory of Avellaneda and Lin [29, 30]), which states that

$$P^{\epsilon}(x_0, x) = \bar{P}(x_0, x) \omega^{\epsilon}(x) + \text{small error,}$$

where $\bar{P}(x_0, x)$ is the Poisson kernel for the homogenized operator $-\nabla \cdot \bar{\mathbf{a}}\nabla \cdot$ and $\omega^{\epsilon}(x)$ is a highly oscillating function which is given explicitly in [216] and which depends mostly on the coefficients in an $O(\epsilon)$-sized neighborhood of the point $x \in \partial\Omega$. We then show that this function $\omega^{\epsilon}(x)$ can be approximated by the restriction of a smooth, $\mathbb{Z}^d$-periodic function on $\mathbb{R}^d$ which depends only on the outer unit normal to $\partial\Omega$ at $x$:

$$\omega^{\epsilon}(x) = \omega(x, \frac{x}{\epsilon}) + \text{small error,} \qquad (3.19)$$

for a smooth $\mathbb{Z}^d$-periodic function $\omega(x, \cdot) \in C^{\infty}(\mathbb{R}^d)$. This is true because the boundary of $\partial\Omega$ is locally close to a hyperplane which is then invariant under $\mathbb{Z}^d$-translations. To bound the error in this approximation we rely in a crucial way on the $C^{1,1-}$ regularity theory of Avellaneda and Lin [29, 30] up to the boundary for periodic homogenization.

We can therefore approximate the Poisson formula (3.18) by

$$u^{\epsilon}(x_0) = \int_{\partial \Omega} \bar{P}(x_0, x) \omega(x, \frac{x}{\epsilon}) g(x, \frac{x}{\epsilon}) dx + \text{small error.} \qquad (3.20)$$

Finally, we cut up the boundary of $∂Ω$ into small pieces which are typically of size $O(\epsilon^{1-})$ but sometimes as large as $O(\epsilon^{\frac{1}{2}-})$, depending on whether the local outer unit normal to $∂Ω$ resonates with the lattice structure of $\mathbb{Z}^d$ on scales smaller than $O(\epsilon^{-1})$. This chopping has to be done in a careful way, which we handle by performing a Calderón-Zygmund-type cube decomposition; see Subsection 3.3.2. In each piece, we freeze the macroscopic variable $x = \bar{x}$ on both $\omega$ and $g$ and approximate the boundary by a piece of a hyperplane, making another small error. The integral on the right of (3.20) is then replaced by a sum of integrals, each of which is a slowly varying smooth function $\bar{P}(x_0, \cdot)$ times the restriction of a smooth, $\epsilon\mathbb{Z}^d$-periodic function $\omega(\bar{x}, \cdot, \frac{\cdot}{\epsilon})g(\bar{x}, \cdot, \frac{\cdot}{\epsilon})$ to a hyperplane. This is precisely the situation in which an appropriate quantitative form of the ergodic theorem for quasiperiodic functions allows us to compute the integral of each piece, up to a tiny error, which turns out to be close to the integral of $\bar{P}(x_0, \cdot)$ times $\int_{\mathbb{T}^d} g(\bar{x}, y)\omega(\bar{x}, y) dy$, the mean of the local periodic function. Therefore we deduce that

$$u^{\epsilon}(x_0) = \int_{\partial \Omega} \bar{P}(x_0, x) \left( \int_{\mathbb{T}^d} g(x, y) \omega(x, y) dy \right) dx + \text{small error.} \qquad (3.21)$$
The right side is now $\bar{u}(x_0)$ plus errors, since now we can see that the homogenized boundary condition should be defined by (3.14).

There is an important subtlety in the final step, since the function $\bar{g}$ is not known to be very regular. As a consequence, we have to be careful in estimating the error made in approximating the homogenized Poisson formula with the sum of the integrals over the flat pieces. This is resolved by showing that $\bar{g}$ is continuous at every $x \in \partial\Omega$ with Diophantine normal $n(x)$, with a quantitative bound for the modulus which leads to the conclusion that $\nabla \bar{g} \in L^{\frac{2(d-1)}{3}-}$. This estimate is a refinement of those of [156] and also uses ideas from [282]; see the discussion in Subsection 3.3.3.

We conclude this section by remarking that, while many of the arguments in the proof of Theorem 3.1 are rather specific to the problem, the high-level strategy based on two-scale expansion of the Poisson kernel, a suitable regularity theory (like that of Avellaneda and Lin) and the careful selection of approximating half-spaces (done here using a Calderón-Zygmund cube decomposition of the boundary based on the local Diophantine quality) is quite flexible and useful in other situations; see the paragraph ‘Further developments’ in Section 3.2.

### 3.3.2 The size of the boundary layer is not uniform

<figure><img src="image_6.png" /><figcaption>Figure 3.2 – Non-uniform boundary layer size; courtesy of Antti Hannukainen (Aalto University, Finland)</figcaption></figure>

The main discovery of our paper [23] is that the boundary layer, i.e. the zone where $U^\epsilon$ and $\bar{U}$ are $O(1)$ apart has a variable size that depends on the geometry of the boundary $\partial\Omega$ and the resonance/non-resonance with the lattice $\mathbb{Z}^d$ of the underlying periodic structure. The function $A(n(\cdot))$ defined by (3.15) determines the local speed of convergence to the boundary layer tail, see the ergodic theorem [23, Proposition 2.1]. Roughly speaking the boundary layer will be thick where $A(n(\cdot))$ is small and thin where $A(n(\cdot))$ is close to 1. It turns out that: (i) near rational directions $\frac{\xi}{|\xi|}$ with $\gcd(\xi) = 1$ and $|\xi|$ small, the irrational non-resonant directions $n$ have small Diophantine constant $A(n)$, while (ii) near rational directions $\frac{\xi}{|\xi|}$ with $\gcd(\xi) = 1$ and $|\xi|$ large, the irrational non-resonant directions $n$ have
large Diophantine constant $A(n)$. Hence, close to the rational directions $\frac{\xi}{|\xi|}$ with $|\xi|$ small, the boundary layer will be locally thick. In the vicinity of rational directions $\frac{\xi}{|\xi|}$ with $|\xi|$ large, the boundary layer will be locally thin. This can be seen on Figure 3.2, which is an unpublished simulation done by Antti Hannukainen (Aalto University, Finland) of the Laplace equation in a disk with oscillating Dirichlet boundary data.

These observations are fundamental when designing the chopping of the boundary. Indeed, one needs to balance the error between approximating by too many tangent hyperplanes, which results in capturing resonant normals, and approximating by too few tangent hyperplanes, which results in a large local error.

<figure><img src="image_7.png" /><figcaption>Figure 3.3 – Stopping time argument for the decomposition of the boundary layer</figcaption></figure>

We perform a Calderón-Zygmund-type decomposition of the domain near the boundary [23, Proposition 3.1] which, when applied to the Diophantine constant of the normal to the boundary, will help us construct the approximation of the boundary layer. The decomposition is proved by a stopping time argument; see Figure 3.3. The abstract result is applied to the function

$$F(x) := \varepsilon^{1-\delta} A^{-1}(n(x)), \quad x \in \partial\Omega,$$

where $A(n(\cdot))$ is defined by (3.15) and yields $P_\varepsilon$ a the collection of triadic cubes; see Figure 3.4. The boundary layer is the union of the cubes in the decomposition. The decomposition has the following properties:
(i) for each cube $\square \in \mathcal{P}_\varepsilon$, there exists $\bar{x}(\square) \in 3\square \cap \partial\Omega$ such that

$$A(n(\bar{x}(\square))) \geq \frac{\varepsilon^{1-\delta}}{\text{size}(\square)}, \qquad (3.22)$$

which means that each cube has at least a 'good' non-resonant direction;

(ii) we have an estimate for the number of cubes bigger than a certain threshold

$$\#\{\square \in \mathcal{P}_\varepsilon : \text{size}(\square) \geq 3^n\} \leq C3^{-2n(d-1)}\varepsilon^{(1-\delta)(d-1)}; \qquad (3.23)$$

notice that (3.23) implies in particular that the largest cube in $P_\varepsilon$ has size at most $C\varepsilon^{\frac{1-\delta}{2}}$; since $F$ is bounded below by $\varepsilon^{1-\delta}$, we have

$$\text{for all } \square \in \mathcal{P}_\varepsilon, \quad c\varepsilon^{1-\delta} \leq \text{size}(\square) \leq C\varepsilon^{\frac{1-\delta}{2}}. \qquad (3.24)$$

<figure><img src="image_8.png" /><figcaption>Figure 3.4 – Calderón-Zygmund decomposition of the boundary layer and good non-resonant direction</figcaption></figure>

### 3.3.3 Regularity estimates for the homogenized boundary data

We explain here how to investigate the variation of $U(\cdot; a, n(x))$ the half-space boundary layer corrector solving (3.5) in terms of the normal $n(x)$ near a non-resonant normal. This analysis results in 'continuity' estimates with respect to $x$ for the kernel $\omega(x, \cdot)$ defined
in (3.19), see [23, Proposition 5.4], and as a consequence for the boundary layer tail and for the homogenized boundary data $\bar{g}$ defined by (3.14), see [23, Proposition 6.3].

Gérard-Varet and Masmoudi were able to prove a Lipschitz estimate for the boundary layer tails for two directions $n_1$ and $n_2$, see [156, Corollary 2.9]. The Lipschitz constant is proportional to $\max(A(n_1)^{-2}, A(n_2)^{-2})$, which yields an estimate of the gradient in $n$ like $A^{-2}(n(\cdot)) \in L^{\frac{d-1}{2}}(\partial\Omega)$ for $d > 2$, see Figure 3.1.

The method developed in [23] relies on ideas of [156]. A central idea is to compare $U(\cdot; a, n_1)$ and $U(\cdot; a, n_2)$ for $|n_1 - n_2| \ll 1$ at the level of the lifted boundary layers $\Upsilon_1$ and $\Upsilon_2$ solving ad hoc variants of (3.8). The advantage of this is that the boundary data for the difference vanishes. We only assume that $n_2$ satisfies the Diophantine condition (3.6). The source term in the system for the difference $\Upsilon_1 - \Upsilon_2$ is expressed in terms of derivatives of $\Upsilon_2$ only. The estimates of [156, Proposition 2.6], see also Subsection 3.1.1 above, then allow to estimate this source term. Thus,

$$ \int_{\mathbb{T}^d} \int_T |N^T \nabla_\theta (\Upsilon_1 - \Upsilon_2)|^2 + |\partial_t (\Upsilon_1 - \Upsilon_2)|^2 d\theta dt \le \frac{C|n_1 - n_2|^2}{A^3} \left(1 + \frac{|n_1 - n_2|^2}{A^2}\right), $$

as well as for derivatives of any order. These estimates turn into $\bar{g} \in L^{\frac{2(d-1)}{3}, \infty}$ for $d > 2$, see Figure 3.1.

*Remark 3.5 (On the improvement by Shen and Zhuge).* Let us comment on the upgrade of the regularity of $\nabla \bar{g}$ by Shen and Zhuge in [316]. Using a weighted estimate, they are able to refine the bounds on $\Upsilon_1 - \Upsilon_2$ in a layer close the boundary in the following way: for all $\sigma \in (0, 1)$, there exists a constant $C(d, L, \lambda, \mathbf{a}, \mathcal{V}_0, \sigma) \in (0, \infty)$ such that

$$ \int_{\mathbb{T}^d} |N^T \nabla_\theta (\Upsilon_1(\theta, 0) - \Upsilon_2(\theta, 0))| d\theta \le \frac{C|n_1 - n_2|}{A^{1+\sigma}} \left(1 + \frac{|n_1 - n_2|}{A}\right). $$

Consequently, they can prove (see [316, Theorem 6.1]) that for any $\sigma \in (0, 1)$, there exists a constant $C(d, L, \lambda, \mathbf{a}, g, \sigma) < \infty$ such that for any $x_1, x_2 \in \partial\Omega$, if $n_2$ Diophantine with constant $A$ and $|n_1 - n_2| \le \nu_0$, then

$$ |\bar{g}(x_1) - \bar{g}(x_2)| \le \frac{C|n(x_1) - n(x_2)|}{A^{1+\sigma}} \left(1 + \frac{|n(x_1) - n(x_2)|}{A}\right). $$

Following our proof of Proposition [23, Proposition 6.3], this yields the improved regularity $\nabla \bar{g} \in L^q(\partial\Omega)$ for any $q < d - 1$ in dimension $d \ge 3$, and $\bar{g} \in W^{s,1}(\partial\Omega)$ for any $s < 1$ in dimension $d = 2$. We emphasize that these results were further strengthened by Shen and Zhuge in [317], see paragraph ‘Further developments’ in Section 3.2.
Part II

**Regularity and concentration for
unsteady fluids**
<figure><img src="image_9.png" /></figure>
# Chapter 4

# Geometric concentration

This chapter relies mainly on the papers:

- [36], with Tobias Barker, *Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions*, *Arch. Ration. Mech. Anal.* (2020).

- [247], with Yasunori Maekawa and Hideyuki Miura, *Local energy weak solutions for the Navier-Stokes equations in the half-space*, *Comm. Math. Phys.* (2019).

- [249], with Yasunori Maekawa and Hideyuki Miura, *Estimates for the Navier-Stokes equations in the half-space for non localized data*, *Analysis & PDE* (2020).

## Contents

<table><tbody><tr><td><strong>4.1 Context: state of the art and obstacles</strong></td><td><strong>67</strong></td></tr><tr><td>4.1.1 Global regularity in the half-space: some positive results</td><td>67</td></tr><tr><td>4.1.2 Global regularity in the half-space: some negative results</td><td>68</td></tr><tr><td>4.1.3 Local boundary regularity: some positive results</td><td>70</td></tr><tr><td>4.1.4 Local boundary regularity: some negative results</td><td>71</td></tr><tr><td>4.1.5 Geometric regularity criteria</td><td>71</td></tr><tr><td>4.1.6 Regularity in the half-space: some open problems</td><td>72</td></tr><tr><td><strong>4.2 Main results</strong></td><td><strong>73</strong></td></tr><tr><td>4.2.1 Fractional pressure estimates in the half-space</td><td>73</td></tr><tr><td>Novelty of our result</td><td>74</td></tr><tr><td>4.2.2 Unification of Type I blow-up notions in the half-space</td><td>74</td></tr><tr><td>Novelty of our result</td><td>75</td></tr><tr><td>4.2.3 Localized vorticity alignment with no slip boundary condition</td><td>76</td></tr><tr><td>Novelty of our result</td><td>77</td></tr><tr><td><strong>4.3 New ideas and strategy for the proofs</strong></td><td><strong>78</strong></td></tr><tr><td>4.3.1 Resolvent pressure formulas in the half-space and pressure estimates</td><td>78</td></tr><tr><td>4.3.2 Boundedness of scale-invariant energies in Type I blow-up scenarios</td><td>80</td></tr><tr><td>4.3.3 A new strategy for regularity under vorticity alignment in a scale-invariant regime</td><td>82</td></tr><tr><td>A strategy based on a Liouville theorem</td><td>82</td></tr><tr><td>Our new strategy based on persistence of singularities</td><td>83</td></tr></tbody></table>
This chapter is devoted to nonlinearity depletion mechanisms under local coherence of the vorticity direction. This line of research was pioneered by Constantin and Fefferman in the early 1990s [107]. There is a clear physical motivation behind these studies. Concentrated vortices, in particular vortex filaments that have a constant vorticity direction, play a crucial role in the self-organization and spatial intermittency of turbulent fluids: see [223, 224] for experimental, [343, 344] for numerical and [264, 284, 285] for theoretical evidence. Concentrated vortices were also used to model the turbulent three-dimensional cascade of energy toward the small scales, see [244, 164, 274], in which a time-dependent linear strain, as in the Burgers vortex or Lundgren's spiral, represents the stretching of other vortices. Mathematically, models for concentrated vortices, notably the Burgers vortex, were analyzed for instance in works by Gallay and Wayne [145] for low Reynolds numbers and Gallay and Maekawa [143] for high Reynolds numbers.

In this chapter, we investigate vorticity depletion mechanisms in the whole-space and in the half-space. The combination of the incompressibility condition and the no-slip boundary condition for the velocity create strong nonlocal effects. This creates important difficulties and explains why certain results known in the whole-space are open questions for the half-space. We list such problems in Subsection 4.1.6 below. Some difficulties of the regularity in the half-space can be linked to the following facts:

(1) In the (flat) half-space one can freely differentiate in the direction tangential to the boundary, but not in the vertical direction. Therefore, to estimate the vertical derivatives $\partial_t^2 U$ one uses the equation and needs information about $\partial_t U$ and $\nabla P$, which gives more interactions than in the whole-space. This can be seen in the example (4.2) of a parasitic flow in the half-space. Such flows are basic flows for finding counter-examples to the local regularity theory, see Subsection 4.1.4.

(2) In addition to the Helmholtz pressure, in the half-space there is also a non trivial harmonic pressure that is driven by the flow on the boundary, see for instance (4.5). This makes the pressure depend in a very nonlocal way on the pressure. We investigate this nonlocality in Subsection 4.3.1 and Subsection 5.3.2.

(3) The vorticity satisfies a nonlocal nonlinear boundary condition as shown by Maekawa [246]: on the half-plane

$$ \partial_2 \omega + (-\partial_1^2)^{\frac{1}{2}} \omega = N(\omega, \omega) \qquad (4.1) $$

with $N$ a bilinear operator involving the Biot-Savart kernel; similar formulas hold in higher dimensions. This makes the use of the vorticity unwieldy in the half-space, while in the whole space the vorticity is a very convenient tool, see for instance Subsection 4.1.5 below. Note that in the case of perfect slip for the velocity, the vorticity satisfies a homogeneous Dirichlet boundary condition, and hence the vorticity can be used successfully as in the whole-space. The complexity of the boundary condition (4.1) reflects the intricate vorticity generation mechanisms near the boundary that are understood, up to now, to a very limited extent.

In Section 4.1, we review some known results and obstacles for solutions in the half-space. Our fundamental contribution to the analysis of half-space solution is the derivation of pressure formulas that enable to circumvent the lack of certain estimates in the half-space. Results in these directions play a key role in this chapter and the next one. We refer in particular to the fractional pressure estimates stated in Theorem 4.1 and to their proof in Subsection 4.3.1, and to the local pressure estimates in Subsection 5.3.2 in the
next chapter. The implications of these results are important, see: in this chapter Theorem 4.2 about the unification of notions of Type I blow-up in the half-space, which enables us to clear a significant hurdle, Theorem 4.3 about geometric concentration in the half-space, which is an entirely new result even in the whole-space; and in the next chapter Theorem 5.10 about the global-in-time existence of local energy solutions, which answers an open question asked in [40], Theorem 5.11 about local-in-space short-time smoothing in the half-space and Theorem 5.13 about norm concentration in the half-space.

# 4.1 Context: state of the art and obstacles

## 4.1.1 Global regularity in the half-space: some positive results

Several explicit representation formulas are derived for the unsteady Stokes system in the half-space $\mathbb{R}_+^3$. Golovkin [168] derived the formula for the Poisson kernel of the unsteady homogeneous Stokes system with a Dirichlet boundary condition. Using this Golovkin kernel and the Green kernel for the Stokes system, Solonnikov derived in a series of works [320, 321, 322], see also [323, 324, 252], the representation formulas for the unsteady nonhomogeneous Stokes system with a Dirichlet boundary condition. Note that these formulas are for divergence-free source terms, and hence dubbed ‘restricted Green Stokes kernel’. A major drawback of this assumption is that it requires a tedious computation of the Helmholtz-Leray projection in the half-space when one uses the kernel to solve the Navier-Stokes equations, because the nonlinear term $\nabla \cdot (U \otimes U)$ is not divergence-free in general. The restriction on the source term was recently removed by Kang, Lai, Lai and Tsai [209], where bounds are proved for the ‘unrestricted Green Stokes kernel’.

Other approaches include the one of Ukai [340] that derives an explicit solution formula, in terms of Riesz transforms and the fundamental solutions of the heat and Laplace equations, in the case of the unsteady homogeneous Stokes system with a Dirichlet boundary condition. The formula was extended to the nonhomogeneous case by Cannone, Planchon and Schonbek [79] and Danchin and Zhang [117]. These formulas work well in the case of data in nonendpoint Lebesgue spaces, but not for $L^1$ or $L^\infty$ due to the unboundedness of the Riesz transform there.

The paper by Desch, Hieber and Prüss [124] pioneers a novel approach that avoids the use of the Helmholtz-Leray projection to compute the Stokes semigroup. The authors prove the analyticity of the Stokes semigroup in $L_\sigma^\infty(\mathbb{R}_+^d)$. Their approach is based on the study of the Stokes resolvent problem. In order to circumvent the use of the Helmholtz-Leray projection, one of the key ideas is to decompose the resolvent operator into a part corresponding to the Dirichlet-Laplace part and another part associated with the nonlocal pressure term. Our work [249] with Maekawa and Miura, that underlies our work [247] and the pressure formulas in Subsection 4.3.1 and Subsection 5.3.2, uses the same idea to handle non-decaying data that is $L_{uloc}^p$ for $p \in (1, \infty]$.

On a tangential aspect, let us mention the works of Abe and Giga [1, 2] that proves the analyticity of the Stokes semigroup in $L^\infty$ on admissible domains, such as for instance the half-space or a bounded $C^3$ domain. The authors rely on a compactness proof rather than on explicit formulas. A weighted (that degenerates near the boundary) scale-invariant estimate for the pressure is used to get a Liouville theorem for the limit blown-up problem; see also [9, Appendix C].

As is expected, the bounds obtained in these works for the potentials make it possible
to prove space-time smoothness of solutions to the Stokes system [300, Theorem 2.1] and of bounded mild solutions to the Navier-Stokes equations in the half-space, see for instance [302, 39]. Moreover Solonnikov proved maximal regularity estimates for the Stokes system in the half-space, for isotropic [321] and anisotropic [323] space-time Lebesgue spaces. The pressure is estimated in these works. Let us also mention the maximal regularity result of Sohr and von Wahl [319]. These estimates though are not global in time in the sense that the constant depends on the time interval. This point was improved by Giga and Sohr in [163], where the estimates are global. The pressure estimates were however not given in [163]. The Schauder-type estimates, i.e. maximal regularity in space-time parabolic Hölder spaces, were proved by Solonnikov in [325].

## 4.1.2 Global regularity in the half-space: some negative results

**On the endpoints $L^1$ and $L^\infty$** The Stokes semigroup on the half-space fails to be analytic on $L^1(\mathbb{R}_+^d)$. Indeed, Desch, Hieber and Prüss [124, Theorem 5.1] proved that there exists data in $L^1(\mathbb{R}_+^d)$ such that the solution to the Stokes resolvent problem does not belong to $L^1(\mathbb{R}_+^d)$. Han [181] showed that a correction term involving the net force exerted by the fluid on the boundary must be subtracted to recover the $L^1$ summability. Note though that the expected semigroup estimate holds for the gradient of the velocity, which belongs to $L^1$ for initial data in $L^1$ [161, Theorem 0.1] and [124, Proposition 5.3]; see also [251].

On the other end of the Lebesgue spaces scale, certain mild decay conditions have to be imposed in order to ensure a representation formula for the pressure. Indeed there are examples of non-decaying flows, solutions to the Stokes system in the half-space, that are sometimes dubbed ‘parasitic solutions’ because they are driven by the pressure. For instance in $\mathbb{R}_+^3$,

$$ \mathcal{U}(x,t) := (V_1(x_3, t), V_2(x_3, t), 0) \quad \text{and} \quad \mathcal{P}(x,t) := -f(t) \cdot x', \quad (4.2) $$

where $f \in C_0^\infty((0, \infty); \mathbb{R}^2)$ and $V = V(x_3, t)$ solves the heat equation $\partial_t V - \partial_3^2 V = f$ with $V(0, t) = 0$. These Poiseuille-type flows generalize the ‘Serrin-type’ flows known in the whole-space [311]. Such flows also give counter-examples to the local regularity theory, see Subsection 4.1.4 below. Notice that (4.2) also solves the Navier-Stokes equations in the half-space with no-slip boundary condition. A Liouville theorem for infinite energy solutions of the Stokes system in the half-space was worked out by Jia, Seregin and Šverák [194, 195]. Their result holds for weak solutions in $L^\infty(\mathbb{R}_+^3 \times (-\infty, 0))$, so-called bounded ancient solutions. The proof in [194] is based on the use of the Fourier transform in the tangential variable, while the proof in [195] uses a duality argument and the kernel bounds by Solonnikov [320, 321, 322, 324]. Notice that in collaboration with Maekawa and Miura [249] we extended these results under the weaker integrability assumption that the velocity belongs to $L^\infty(0, T; L_{uloc,\sigma}^1(\mathbb{R}_+^3))$. For further discussions of Liouville theorems for the Stokes system in the half-space, we refer to [1, Section 4], [159, Section 4].

**On difficulties related to the pressure** There is a fundamental difference between the pressure in the half-space and the pressure in the whole-space. In the whole-space, if parasitic solutions are excluded for instance thanks to some mild decay assumption at space infinity, the pressure is equal to the Helmholtz pressure which comes from the Helmholtz-Leray decomposition of the source term; see [135, 69] for related discussions. In the half-space, the non-locality of the pressure is much stronger due to the interaction of the incompressibility condition and the no-slip boundary conditions. For the Stokes system in the
half-space

$$ \partial_t U - \Delta U + \nabla P = \nabla \cdot F, \quad \nabla \cdot U = 0 \quad \text{in } \mathbb{R}_+^3 \times (0, T) \qquad (4.3) $$

with the no-slip boundary condition

$$ U(\cdot, t) = 0 \quad \text{on} \quad \partial \mathbb{R}_+^3 \qquad (4.4) $$

and the initial data $U(\cdot, 0) = 0$, one can first carry out the Helmholtz-Leray decomposition of the source term $\nabla \cdot F = \nabla P_{Helm}^F + \mathbb{P}(\nabla \cdot F)$, where $P_{Helm}^F$ is the Helmholtz pressure and $\mathbb{P}$ is the Leray projector on divergence-free fields. Then taking formally the divergence of (4.3), one ends up with the following elliptic system

$$ \left\{ \begin{array}{ll} -\Delta P_{harm}^F(\cdot, t) = 0 & \text{in } \mathbb{R}_+^3, \\ \partial_3 P_{harm}^F(\cdot, t) = \gamma|_{x_3=0} \Delta U_3(\cdot, t) & \text{on } \partial \mathbb{R}_+^3, \end{array} \right. \qquad (4.5) $$

for a pressure $P_{harm}^F(\cdot, t)$ that is called the ‘harmonic’ part of the pressure, since it is harmonic. Note that in the whole-space the harmonic pressure is harmonic in the whole of $\mathbb{R}^3$, hence constant if the pressure is assumed to grow sub-linearly for instance.

Koch and Solonnikov [220, Theorem 1.3] showed that there are examples of Stokes systems in the half-space with divergence-form source term $\nabla \cdot F$ for which the harmonic pressure is not integrable in time. Hence in short, $P \neq U \otimes U$ for the half-space, contrary to the whole-space. Indeed the example constructed in [220] is of the form $P = P_{Helm} + \partial_t Q$ with:

(i) the following integrability for the Helmholtz pressure $P_{Helm} \in L_{t,x}^q$, which is expected when the source term $F \in L_{t,x}^q$,

(ii) and $Q \in L_t^q W_x^{2,q}$, which means that the harmonic pressure $\partial_t Q$ is not integrable in time.

This fact is a source of considerable difficulties for our proof in Subsection 4.3.2 of Theorem 4.2. We go around this obstacle thanks to the fractional pressure estimates stated in Theorem 4.1. These two theorems are among the main contributions of our paper [36].

Tolksdorf [335] points to another difficulty can be seen at the level of the resolvent problem. Assuming that an estimate of the following type holds for the pressure of the resolvent problem

$$ \|P\|_{L^2(\Omega)} \leq C(\lambda) \|f\|_{L^2(\Omega)}, \qquad (4.6) $$

the question is to find an explicit quantitative dependence of $C(\lambda)$ in terms of the resolvent parameter $\lambda$ for various bounded or unbounded domains $\Omega \subseteq \mathbb{R}^3$. Via Dunford's formula such estimates would imply short-time estimates for the pressure of the evolution problem. In the whole-space or the half-space, if an estimate such as (4.6) was true (which is unclear), scaling arguments would imply that $C(\lambda) \simeq |\lambda|^{-\frac{1}{2}}$. Tolksdorf [335, Proposition 3.1] shows that on bounded $C^4$ domains, $C(\lambda) \simeq |\lambda|^{-\frac{1}{2}}$ scales as is expected in view of scaling for homogeneous Neumann-type boundary conditions. Somewhat surprisingly, Tolksdorf [335, Proposition 3.3] shows that for no-slip boundary conditions on bounded Lipschitz domains, the following estimate is optimal: for any $\alpha \in [0, \frac{1}{4})$, $C_\alpha(\lambda) \lesssim_\alpha |\lambda|^{-\alpha}$ for $\lambda$ large; see also [273]. Hence the scaling is broken for homogeneous Dirichlet boundary conditions. We further discuss this topic in Subsection 5.3.2 in connection with the pressure estimates of our work with Maekawa and Miura [247] for uniformly locally integrable data.
### 4.1.3 Local boundary regularity: some positive results

Despite the difficulties related to the strong nonlocality of the Navier-Stokes equations in the half-space, see Subsection 4.1.2 above and Subsection 4.1.4 below, many regularity results can be localized. For a general survey of the local regularity in the half-space, we refer to the paper by Seregin and Shilkin [300].

First, it is possible to localize maximal regularity results in the half-space. This was done in several steps by Seregin in the 2000s. The most accomplished result is in the paper [308]. There Seregin is able to remove extra integrability assumptions that were needed in previous works, such as [306, 305].

Second, we define the following scale-invariant quantities: for $r \in (0, \infty)$,

$$A(U, r) := \sup_{-r^2 < s < 0} \frac{1}{r} \int_{B_+(r)} |U(y, s)|^2 dy, \qquad (4.7)$$

$$C(U, r) := \frac{1}{r^2} \int_{-r^2}^{0} \int_{B_+(r)} |U|^3 dyds, \qquad (4.8)$$

$$E(U, r) := \frac{1}{r} \int_{-r^2}^{0} \int_{B_+(r)} |\nabla U|^2 dyds, \qquad (4.9)$$

$$D_{\frac{3}{2}}(P, r) := \frac{1}{r^2} \int_{-r^2}^{0} \int_{B_+(r)} |P - (P(\cdot, s))_{B_+(r)}|^{\frac{3}{2}} dyds. \qquad (4.10)$$

It was proved by Mikhaylov in [258] that if $U$ is a suitable solution in $Q_+(1)$, boundedness of one of the quantities $\sup_{r \in (0,1)} A(U, r)$, $\sup_{r \in (0,1)} C(U, r)$ or $\sup_{r \in (0,1)} E(U, r)$ implies boundedness of all quantities $\sup_{r \in (0,1)} A(U, r)$, $\sup_{r \in (0,1)} C(U, r)$, $\sup_{r \in (0,1)} E(U, r)$ and $\sup_{r \in (0,1)} D_{\frac{3}{2}}(P, r)$.

Third, we have partial regularity results. Indeed, Seregin [306, Theorem 2.3] proved that for a suitable solution $U$ in $Q_+(1)$ smallness of one of the quantities $\sup_{r \in (0,1)} A(U, r)$, $\sup_{r \in (0,1)} C(U, r)$ or $\sup_{r \in (0,1)} E(U, r)$ implies parabolic Hölder regularity in a neighborhood of $(0, 0)$; see also variations in [180] and [259]. Seregin, Shilkin and Solonnikov [309, Theorem 1.1] prove a one-scale $\varepsilon$-regularity criteria in the half-space with a compactness argument à la Lin [243]. In [306] and [309] the definition of suitability includes bounds that can be removed using maximal regularity estimates that are obtained in [308]. In [295], Seregin shows a one-scale regularity criteria based on the sole smallness of the velocity. It is reminiscent of Wolf's criteria [347] in the whole-space, see also the recent paper by Kwon [225].

Fourth, there are local Ladyženskaja-Prodi-Serrin type criteria under boundedness of non borderline critical norms. This is proved by Takahashi [329], see also Seregin's result [307], for finite energy weak solutions and extra integrability assumptions on the velocity gradient and the pressure, that can be removed using the maximal regularity result of [308]. For suitable solutions, the regularity follows from the $\varepsilon$-regularity criteria by Gustafson, Kang and Tsai [180].

Finally, we have versions of the Escauriaza-Seregin-Šverák [129] result. Regularity under $L_t^\infty L_x^3$ local control of a finite-energy weak solution is obtained by Seregin in [293] near
a flat boundary and by Mikhailov and Shilkin in [257] near a curved boundary. Seregin's result [294] about the blow-up of the $L^3$ norm near a potential singularity was generalized to the half-space by Barker and Seregin [40] near a flat boundary. Note that the papers [294] and [40] are global results. A local version of these results near a curved boundary was obtained by Albritton and Barker in [9].

### 4.1.4 Local boundary regularity: some negative results

Contrary to local solutions in the whole-space, there are obstructions to spatial smoothing in the half-space for suitable solutions: in the local setting, there are bounded flows with unbounded derivatives. This was demonstrated in two bodies of works:

#### (i) Finite-energy

Using a compactly supported Hölder continuous flux term on the boundary, Kang [208] constructed an example of a bounded Stokes flow $U$ in the half-space such that $\nabla U$ is unbounded away from the support of the flux. Kang, Lai, Lai and Tsai [210] recently showed that Kang's example has global finite-energy. They also find a flow with similar properties solving the Navier-Stokes equations. In the same vein, the optimality of the local maximal regularity estimates of Seregin [306, 305, 308] is investigated in the work by Chang and Kang [91].

#### (ii) Shear flow

Using the shear flow solution (4.2) for a well-chosen singular time-dependent source term $f$, Seregin and Šverák show the existence of a bounded flow with unbounded first-order derivatives. We emphasize that this flow is not decaying at space infinity, contrary to the flows constructed in [208, 210].

Finally, let us remark that Chang and Kang [90] proved the failure of Caccioppoli-type inequalities for suitable solutions of the Navier-Stokes equations in the half-space. Such inequalities, that do not involve the pressure, are known for suitable solutions in the whole-space [200, 348].

### 4.1.5 Geometric regularity criteria

For a general overview of geometric regularity criteria for the Navier-Stokes equations, we refer to the recent survey paper of Miller [261]. These criteria encapsulate some geometric information about the flow, either in the fact that certain quantities, such as the helicity, are small, or that the direction of the velocity or the vorticity is constant or varies smoothly... The observed structure of three-dimensional turbulent flows indicates that there is a high level of spatial intermittency, or sparseness, and anisotropy for the vorticity, with high vorticity regions organized as filaments; see for instance [130].

In [107], Constantin and Fefferman provided a geometric regularity criteria for the vorticity $\omega = \nabla \times U$ of solutions on the whole-space, which remarkably does not depend on scale-invariant quantities. Specifically, they showed that if

$$ \begin{gathered} |\sin(\angle(\omega(x+y,t), \omega(x,t)))| \le C|y| \\ \text{for } (x,t) \in \Omega_d := \{(y,s) \in \mathbb{R}^3 \times (0,T) : |\Omega(y,s)| > d\} \quad (4.11) \end{gathered} $$

(here $\angle(a, b)$ denotes the angle between the vectors $a$ and $b$), then $U$ is smooth on $\mathbb{R}^3 \times (0, T]$. Notice that since the angle $\angle(\omega(x+y,t), \omega(x,t))$ appears in (4.11), antiparallel
vortex alignment is covered by their result. The proof uses energy estimates for the vorticity equation

$$ \partial_t \omega - \Delta \omega + U \cdot \nabla \omega - \omega \cdot \nabla U = 0 \qquad (4.12) $$

to get

$$ \|\omega(\cdot, t)\|_{L_2(\mathbb{R}^3)}^2 + 2 \int_0^t \int_{\mathbb{R}^3} |\nabla w|^2 dxdt \le 2 \int_0^t \int_{\mathbb{R}^3} (\omega \cdot \nabla U) \cdot \omega dxdt \qquad (4.13) $$

Then Constantin and Fefferman proceed with a careful analysis of the stretching term, i.e. the right hand side of (4.13). In particular, the Biot-Savart law, integration by parts and linear algebra identities are used to show that the most singular contribution of the stretching term can be controlled by

$$ \int_0^t \iint_{\Omega_d \times \Omega_d} \frac{|\omega(x, s)|^2 |\omega(y, s)| |\sin(\angle(\omega(y, s), \omega(x, s)))|}{|x - y|^3} dydxds. \qquad (4.14) $$

Crucially (4.11) depletes the singularity in the integral of this stretching term. For certain extensions of this geometric regularity criterion, we refer (non-exhaustively) to [48, 172, 351, 174, 171, 241].

In the paper by Giga and Miura [162], a new strategy for regularity under a continuous alignment condition for the vorticity was introduced for the Navier-Stokes in the whole-space with a scale-invariant a priori control (a Type I condition). The Type I assumption gives compactness and a Liouville theorem for a two-dimensional flow leads to the conclusion. This scheme was adapted successfully to the Navier-Stokes equations in the half-space by Giga, Hsu and Maekawa [159]. In this case, the proof of the Liouville theorem is convoluted due to the intricate boundary condition (4.1) for the vorticity. One of our contribution, see Theorem 4.3 and Subsection 4.3.3 below, is to develop an approach that bypasses the use of Liouville theorems.

### 4.1.6 Regularity in the half-space: some open problems

The issue of vorticity creation at the boundary means that certain other situations that are understood for the Navier-Stokes equations in the whole-space, remain open, as far as we know, for the case of the fluid occupying the half-space with no-slip boundary condition.

**On backward self-similar singularities in the half-space** It is not known whether there exists backward self-similar solutions

$$ U(x, t) = \frac{1}{\sqrt{T-t}} U\left(\frac{x}{\sqrt{T-t}}\right), \quad (x, t) \in \mathbb{R}_+^3 \times (0, T) $$

with finite local energy and no-slip boundary condition. For the whole-space such solutions were shown not to exist in [338]. Notice that the case when the profile U has more decay i.e. belongs to $L^3(\mathbb{R}_+^3)$ is ruled out by the blow-up result of [293].

**On axisymmetric flows in the half-space** The smoothness of solutions to the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, \infty)$ with no-slip boundary condition that are axisymmetric
without swirl is not known. Such solutions have the following form in cylindrical coordinates

$$U(r, z, t) = U_r(r, z)\vec{e}_r + U_z(r, z)\vec{e}_z,$$

where $\vec{e}_r = (\cos(\theta), \sin(\theta), 0)$, $\theta \in (-\pi, \pi)$ and $\vec{e}_z = (0, 0, 1)$. For the whole-space, such solutions were shown to be smooth by Ladyženskaja [228]. For the half-space Kang proved in [207] that such solutions blow-up at maximum one point located at the intersection of the axis of symmetry and of the boundary.

Moreover, the regularity of axisymmetric flows in the half-space but under a Type I assumption is still open, as mentioned in [96]. We recall that such results are known for the whole-space since the works of Chen, Strain, Tsai and Yau [94, 93] on the one hand, and of Koch, Nadirashvili, Seregin and Šverák [219] on the other hand.

### On vorticity depletion mechanisms in the half-space
For the case of the Navier-Stokes equations in $\mathbb{R}_+^3$ with no-slip boundary conditions, vorticity is generated at the boundary. Specifically, $\omega$ has a non-zero trace on $\partial\mathbb{R}_+^3$ and satisfies a nonlocal boundary condition as showed by Maekawa [246], see (4.1) (written in two space dimensions). This provides an obstacle to applying energy methods to the vorticity equation (4.12). Consequently, the following statement analogous to the result of Constantin and Fefferman for the whole-space [107] remains open for the half-space

(Q): For finite energy solutions of the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, T)$, with no-slip boundary condition and divergence-free initial data $U_0 \in C_0^\infty(\mathbb{R}_+^3)$, does the Lipschitz continuity of the vorticity direction (4.11) imply that $U$ is smooth on $\mathbb{R}_+^3 \times (0, T]$?

Note that with perfect slip boundary conditions

$$U_3 = \partial_3 U_2 = \partial_3 U_1 \quad \text{on} \quad \partial\mathbb{R}_+^3, \quad U = (U_1, U_2, U_3), \qquad (4.15)$$

vorticity is not created at the boundary and the geometric regularity condition is known to hold [111]. Li [242] was able to generalize this result for Navier-slip boundary conditions.

Theorem 4.3 below obtained in collaboration with Barker [36] investigates regularity under local vorticity alignment under a Type I assumption. The criticality enables to handle the strong nonlocal effects in the half-space due to the pressure or to the non-zero vorticity on the boundary.

## 4.2 Main results

### 4.2.1 Fractional pressure estimates in the half-space

The result below is a reformulation of global pressure estimates in the half-space that first appeared in the work of Chang and Kang [89]. These estimates are stated for the Stokes system in the half-space (4.3) with the no-slip boundary condition (4.4). We dub these estimates ‘fractional pressure estimates’ since they involve fractional derivatives of the source term $F$.

**Theorem 4.1** (fractional pressure estimates; [36, Proposition 7], in collaboration with Barker). Let $q \in [1, \infty]$, $p \in (1, \infty)$, $F = (F_{\alpha\beta})_{1\le\alpha,\beta\le3} \in L^q(0, T; W_0^{1,p}(\mathbb{R}_+^3))$. We decompose the pressure of the system (4.3) with the no-slip boundary condition (4.4) and the zero
initial data $U(\cdot, 0) = 0$ into $P = P_{Helm}^F + P_{harm}^F$, where $P_{Helm}^F$ is the Helmholtz part of the pressure and $P_{harm}^F$ is the harmonic part of the pressure.
Then the following estimates hold: for all $F \in L^q(0, T; W_0^{1,p}(\mathbb{R}_+^3))$, for all $\kappa \in (\frac{1}{p}, 1]$,

$$ \begin{aligned} \|P_{Helm}^F\|_{L^q(0,T;L^p(\mathbb{R}_+^3))} &\le C\|F\|_{L^q(0,T;L^p(\mathbb{R}_+^3))}, \\ \|P_{harm}^F\|_{L^q(0,T;L_{x_3}^\infty(0,\infty;L_{x'}^p(\mathbb{R}^2)))} &\le C T^{\frac{1}{2}(\kappa-\frac{1}{p})} \|F\|_{L^p(\mathbb{R}_+^3)}^{1-\kappa} \|\nabla F\|_{L^p(\mathbb{R}_+^3)}^{\kappa} \|F\|_{L^q(0,T)}^{1-\kappa}, \end{aligned} $$

with a constant $C(\kappa, p) \in (0, \infty)$.

## Novelty of our result

This result is not new. A similar estimate is contained in the work of Chang and Kang [89, Theorem 1.2, (1.14)]. Notice that we gain boundedness in the vertical direction for $p_{harm}^F$, but we are not able to recover the $L_{x_3}^p$ integrability as in Chang and Kang. Notice that due to the example of Koch and Solonnikov [220] it is not possible to take $\kappa = 0$; see Subsection 4.1.2.

Our point in [36] is to revisit the proof of the estimate of Chang and Kang. We provide a completely new and elementary proof of these estimates, which takes advantage of new pressure formulas for the half-space discovered in [249, 247]. In particular, we avoid the use of space-time fractional Sobolev norms. A key point in the proof is that $F$ vanishes on the boundary of the half-space, which is not a restriction in view of applications to the Navier-Stokes system where $F = -U \otimes U$. Indeed, the proof relies on the use of the one-dimensional Hardy inequality in the vertical direction, see Subsection 4.3.1 below.

We also provide a completely new and elementary proof of these estimates based on formulas for the harmonic pressure that I obtained in collaboration with Maekawa and Miura [247].

## 4.2.2 Unification of Type I blow-up notions in the half-space

The following result is a local result for solutions of the Navier-Stokes equations near the boundary of the half-space. It enables to infer the boundedness of scale-invariant energies under an 'ODE blow-up rate' notion of Type I singularities, which is the most classical notion of Type I singularity corresponding to the ODE blow-up rate in semilinear heat equations (see for instance Giga and Kohn [160]).

**Theorem 4.2** (boundedness of scale-invariant energies; [36, Theorem 2], in collaboration with Barker). Let $M \in (0, \infty)$. Suppose that $(U, P)$ is a suitable solution of the Navier-Stokes equations in $Q_+(1)$ (see [308, Definition 1.3] and [300, Definition 1.2] for a definition of suitable solutions in the half-space) that satisfies the 'ODE blow-up rate' Type I bound

$$ |U(x, t)| \le \frac{M}{\sqrt{-t}} \quad \text{in } Q_+(1). \qquad (4.16) $$

Then,

$$ \sup_{0<r<1} \{A(U, r) + E(U, r) + D_{\frac{3}{2}}(P, r)\} \le \mathcal{M}(M, A(U, 1), E(U, 1), D_{\frac{3}{2}}(P, 1)), \quad (4.17) $$

where $A$ (resp. $E$, $D$) are defined by (4.7) (resp. (4.9), (4.10)).
The importance of Theorem 4.2 lies in the fact that it links the natural notion of Type I blow-up (4.16) to a workable notion of Type I for solutions to the Navier-Stokes equations in the half-space with no-slip boundary condition. Indeed it turns out that the scale-invariant condition (4.17) is exactly what is needed to prove a number of results. This can be seen in particular in three situations:

(1) The generalized Type I bound (4.17) ensures that a mild bounded ancient solution originating from $(U, P)$ has some sort of decay at space infinity.

(2) In the paper [35] on concentration phenomena for Type I blow-up solutions of the Navier-Stokes equations, the boundedness of scale-invariant energies was used to provide a control in $L_{uloc}^2(\mathbb{R}^3)$ of a rescaled solution, see Chapter 5.

(3) The scale-invariant bound (4.17) ensures that solutions rescaled according to the natural scaling of the Navier-Stokes equations satisfy the uniform bound in the energy norm required for applying the persistence of singularities. This fact is crucial for our new strategy to prove regularity under vorticity alignment, see Theorem 4.3 below and the description of the strategy in Subsection 4.3.3.

### Novelty of our result

The implication ‘ODE blow-up rate’ Type I implies boundedness of scale-invariant energies is well-known in the whole-space. It is known that for a suitable weak solution $(U, P)$ of the Navier-Stokes equations in $Q(1)$, the Type I condition

$$ |U(x, t)| \le \frac{M}{\sqrt{-t}} \quad \text{in } Q(1) \qquad (4.18) $$

implies

$$ \sup_{0<r<1} \{A(U,r) + E(U,r) + D_{\frac{3}{2}}(P,r)\} < \infty. \qquad (4.19) $$

This was established by Seregin and Zajaczkowski in [310]; see also Seregin and Šverák [303] (statement and proof). It is discussed in [8] that the definition of Type I singularities given by (4.19) is very natural and includes most other popular notions of Type I used in the literature. The proofs in the whole-space crucially use the fact that $P \simeq U \otimes U$. For more on the whole-space we refer to Remark 5.6 and to the sketch of the proof in Subsection 4.3.2 below.

For the case of a suitable solution $(U, P)$ in the half-space with no-slip boundary condition, it was shown by Mikhaylov [258] and by Seregin [296] that the definition of Type I singularity (4.17) is implied by $|U(x, t)| \le \frac{M}{|x|}$; see also a similar result of Chernobay [96] that applies to axisymmetric Type I solutions. However, it was previously not understood if, for solutions in the half-space with no-slip boundary conditions, (4.17) was consistent with the ‘ODE blow-up rate’ notion of Type I singularities (4.16). Our Theorem 4.2 fills this gap and demonstrates that (4.17) is a reasonable notion of Type I singularity for suitable solutions in the half-space with no-slip boundary condition.

The fractional pressure estimates of Theorem 4.1 above are the main tools enabling our breakthrough as explained below in Subsection 4.3.2. Indeed, these estimates for the pressure enable to circumvent the fact that in the half-space $P \not\simeq U \otimes U$ (see [220] and Subsection 4.1.2 above).
### 4.2.3 Localized vorticity alignment with no slip boundary condition

The following result shows that for a potential Type I singularity the direction of the vorticity is very rough in the vicinity of the blow-up point.

<figure><img src="image_10.png" /><figcaption>Figure 4.1 – Continuous alignment on concentrating sets</figcaption></figure>

**Theorem 4.3** (geometric concentration in the half-space; [36, Theorem 1], in collaboration with Barker). Let $M \in (0, \infty)$. Suppose $U$ is a mild solution to the Navier-Stokes equations in $\mathbb{R}_+^3 \times (0, T)$, with the no-slip boundary condition (4.4) and initial data $U_0 \in C_{0,\sigma}^\infty(\mathbb{R}_+^3)$. Furthermore, suppose that for $(x, t) \in \mathbb{R}_+^3 \times (0, T)$:

$$|U(x, t)| \le \frac{M}{\sqrt{T-t}}. \qquad (4.20)$$

Let $t_n \uparrow T$, $\bar{x} \in \overline{\mathbb{R}}_+^3$, $d > 0$ and $\delta > 0$. Define $\omega = \nabla \times U$, $\xi := \frac{\omega}{|\omega|}$, the set of high vorticity,

$$\Omega_d := \{(x, t) \in \mathbb{R}_+^3 \times (0, T) : |\omega(x, t)| > d\},$$

the cone

$$C_{\delta, \bar{x}} := \bigcup_{t \in (T-1, T)} \{x \in \mathbb{R}_+^3 : |x - \bar{x}| < \delta \sqrt{T-t}\} \qquad (4.21)$$

and the time-sliced cone

$$C_{t_n, \delta, \bar{x}} := \bigcup_n \{(x, t_n) : x \in \mathbb{R}_+^3 \text{ and } |x - \bar{x}| < \delta \sqrt{T - t_n}\}. \qquad (4.22)$$

Let $\eta: \mathbb{R} \to \mathbb{R}$ be a continuous function with $\eta(0) = 0$.

Under the above assumptions, there exists $\delta(M, U_0) > 0$ such that the following holds true: on the one hand

$$\sup_n \sup_{(x,t) \in C_{t_n, \delta, \bar{x}}} |\omega(x,t)| < \infty \Rightarrow (x_0, T) \text{ is a regular point of } U, \qquad (4.23)$$