# Bounded Solutions in Incompressible Hydrodynamics

Dimitri Cobb

► To cite this version:

Dimitri Cobb. Bounded Solutions in Incompressible Hydrodynamics. 2021. [hal-03223282](https://hal.science/hal-03223282)

HAL Id: hal-03223282

https://hal.science/hal-03223282

Preprint submitted on 10 May 2021

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# BOUNDED SOLUTIONS IN INCOMPRESSIBLE HYDRODYNAMICS

Dimitri Cobb

UNIVERSITÉ DE LYON, UNIVERSITÉ CLAUDE BERNARD LYON 1

Institut Camille Jordan - UMR 5208

43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, FRANCE

cobb@math.univ-lyon1.fr,

May 10, 2021

## Abstract

In this article, we study bounded solutions of Euler-type equations on $\mathbb{R}^d$ which have no integrability at $|x| \to +\infty$. As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at $|x| \to +\infty$. We find this condition to be necessary and sufficient. We deduce a full well-posedness result for the Euler equations in the space $C_T^0(B_{\infty,1}^1)$. We also apply the methods developed for the Euler system to other hydrodynamic models: the Navier-Stokes equations and magnetohydrodynamics.

**2010 Mathematics Subject Classification:** 35Q35 (primary); 35A02, 35B30, 35Q31, 35S30 (secondary).

**Keywords:** Leray projection, incompressible fluids, bounded solutions, Euler equations, far-field condition.

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Role of the pressure and Leray projection</td><td>2</td></tr><tr><td>1.2</td><td>Bounded solutions in fluid mechanics</td><td>3</td></tr><tr><td>1.3</td><td>Outline of the paper and summary of the results</td><td>4</td></tr><tr><td>2</td><td>Littlewood-Paley analysis</td><td>7</td></tr><tr><td>3</td><td>Euler equations</td><td>10</td></tr><tr><td>3.1</td><td>Bounded weak solutions</td><td>10</td></tr><tr><td>3.2</td><td>The projected problem</td><td>11</td></tr><tr><td>3.3</td><td>Main result</td><td>13</td></tr><tr><td>3.4</td><td>Equivalence of the two formulations: smooth in time solutions</td><td>13</td></tr><tr><td>3.5</td><td>Equivalence of the two formulations: low time regularity</td><td>17</td></tr><tr><td>3.6</td><td>A full well-posedness result</td><td>19</td></tr><tr><td>4</td><td>Navier-Stokes equations</td><td>19</td></tr><tr><td>5</td><td>Elsässer variables in ideal magnetohydrodynamics</td><td>22</td></tr><tr><td>5.1</td><td>Weak solutions</td><td>23</td></tr><tr><td>5.2</td><td>Equivalence of the formulations</td><td>24</td></tr><tr><td>5.3</td><td>Further reamrks on the ideal MHD system</td><td>25</td></tr><tr><td>A</td><td>Appendix: Fourier multipliers at high frequencies</td><td>26</td></tr></table>
# 1 Introduction

This article is concerned with bounded solutions to systems of PDEs describing incompressible fluid mechanics. The simplest of these equations is the incompressible (homogeneous) Euler system, which reads

$$ (1) \qquad \left\{ \begin{array}{l} \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = 0 \\ \operatorname{div}(u) = 0. \end{array} \right. $$

This system is set on the whole space $\mathbb{R}^d$, and its unknowns are the velocity field $u(t, x) \in \mathbb{R}^d$ and the pressure field $\pi(t, x) \in \mathbb{R}$. As we will see, the issues raised in this article are very specific to the non-compact nature of $\mathbb{R}^d$, so we will not be interested by flows defined on the torus $\mathbb{T}^d$.

## 1.1 Role of the pressure and Leray projection

In system (1), the pressure plays an apparently minor role, as it is entirely determined by the velocity field. Therefore, although (1) has two unknowns, only the velocity field $u$ really matters when solving the system. In fact, we may formally compute $\nabla \pi$ as a function of $u$ by taking the divergence of the first equation in (1) and solving the elliptic equation thus produced: we obtain

$$ (2) \qquad \nabla \pi = \nabla(-\Delta)^{-1} \sum_{j,k} \partial_j \partial_k (u_j u_k). $$

From the previous equation, we introduce the Leray projection operator

$$ (3) \qquad \mathbb{P} = \mathrm{Id} + \nabla(-\Delta)^{-1} \operatorname{div} $$

which may be seen as the $L^2$-orthogonal projector on the subspace of divergence-free functions. In other words, we may see the pressure term in $\nabla \pi$ in (1) as an orthogonal projection enforcing the divergence-free condition $\operatorname{div}(u) = 0$ at all times. Applying $\mathbb{P}$ to (1), we get a new equation

$$ (4) \qquad \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u) = 0, $$

which is (up to a commutator term) a transport equation.

The Leray projection operator (3) may be seen as a Fourier multiplier<fnref n="1" /> whose symbol $m(\xi)$ is a bounded homogeneous function of degree zero. As such, Calderón-Zygmund theory applies to show that it defines a bounded operator $\mathbb{P}: L^p \to L^p$, as long as $1 < p < +\infty$. However, the theory breaks down at the endpoint exponents $p = 1$ or $p = +\infty$.

In the case $p = +\infty$, the Leray projector is not clearly defined as a Fourier multiplier on the whole space $L^\infty$, or even on a dense subspace. This is due to the singularity of the symbol $m(\xi)$ at $\xi = 0$, point at which it is discontinuous. For instance, if $f = 1$ is a constant function, it is not possible to define $\mathbb{P}f$ by multiplication of the Fourier transform, since the product $m(\xi) \times \delta_0(\xi)$ is ill-defined.

Moreover, the nature of this example shows that all $L^\infty$-based spaces (such as $C_b^0$, $W^{1,\infty}$, $B_{\infty,1}^1$, etc.) will suffer from the same pathology, as they all contain constant functions.

<footnote n="1">¹If $\sigma(\xi)$ is a complex valued function on $\mathbb{R}^d$, the Fourier multiplier of symbol $\sigma(\xi)$, which we note $\sigma(D)$, is the operator defined by $\sigma(D)f = \mathcal{F}^{-1}[\sigma(\xi)\hat{f}(\xi)]$.</footnote>
**Remark 1.1.** The behavior of singular integral operators on $L^\infty$ is at the center of deep results. For instance, it can be shown, using Fefferman-Stein duality, that the Leray projector can be extended from the quotient $L^\infty/\mathbb{R}[x]$ to a bounded operator on the space BMO of functions of bounded mean oscillations (see [25], Chapters 3 and 4).

On the other hand, it is possible to find examples of $L^\infty$ function whose image by a singular integral operator is not bounded (see Section 3.2 in [4]).

## 1.2 Bounded solutions in fluid mechanics

The fact that the Leray projector is ill-defined on spaces of bounded functions is very unfortunate, as the endpoint spaces $p = +\infty$ often play a special role in fluid mechanics. In particular, the space $W^{1,\infty}$ of Lipschitz functions is the largest space in which one can hope to solve most equations of ideal fluid mechanics – equations (1) and (4) are nearly transport equations. So far, all well-posedness results have been proved for solutions in spaces embedded in $W^{1,\infty}$.

In addition, infinite energy solutions have also been the center of much attention in the past twenty years, and present their own interesting challenges (see [14] for an introduction to these questions), so that a precise theory of non-integrable bounded solutions seems quite desirable.

To circumvent the problem of defining the Leray projection on $L^\infty$, a trick that was used in [20] and [18] is to take advantage of the integrability properties of the kernel of the inverted Laplacian $(-\Delta)^{-1}$ to prove that, for all $j \in \{1, ..., d\}$, we have a bounded operator on non-homogeneous Besov spaces: for all $s \in \mathbb{R}$,

$$ \mathbb{P} \partial_j : B_{\infty,1}^s \to B_{\infty,1}^{s-1}, $$

is continuous so that all the terms in the projected equation (4) are well-defined<fnref n="2" /> if, for example, $u \in L_{\text{loc}}^2(L^\infty)$. One can then proceed to studying equation (4) without further trouble, proving (local) existence and uniqueness of a $C^0(B_{\infty,1}^1)$ solution.

The catch is that equation (4) is not equivalent to problem (1) for solutions that are solely bounded, regardless of their smoothness. Although all solutions of (4) solve (1), the converse is not true: consider, for example, the uniform Poiseuille-type flow (see [8], Section 4.3 for a detailed discussion)

$$ (5) \qquad u(t,x) = f(t)e_1 \quad \text{and} \quad \pi(t,x) = -f'(t)x_1, $$

where $f \in C^\infty(\mathbb{R})$. Then (5) solves the Euler problem (1), but is not a solution of (4). Note that this solution is indeed $C^\infty$ with respect to both time and space variables, so that the problem entirely lies with its lack of integrability at $|x| \to +\infty$.

Worse than that: if $f$ is compactly supported away from $t=0$, we see that (5) is a smooth nonzero solution of the Euler equations (1) with initial datum $u(0) = 0$ (again, see [8]). Therefore, while problem (4) is well-posed in $C^0(B_{\infty,1}^1)$, it is not the case for problem (1). Note that this issue is of a different nature than the $C^1 \cap L^2$ ill-posedness displayed in [11], as the authors of [11] rely on singular integral operators not mapping $C^0 \cap L^2$ to $L^\infty$, while our example (5) hinges on the lack of integrability (in the case of (5), $u(t) \notin L^2$).

This ill-posedness issue is not entirely surprising, as (1) has no “boundary condition” at $|x| \to +\infty$, whereas the presence of the inverted Laplace operator in (3) implicitly equips (4) with one such condition: $(-\Delta)^{-1}$ has its range in a space of functions with no harmonic part. A lack of “boundary condition” means that the fluid may be driven by an exterior pressure, whereas the dynamics of a physical system can be deterministic only if there is no arbitrary outside interference.

Such unpleasant behavior is to be expected when dealing simultaneously with constant functions, as in (5), and the Leray projection, as these functions are, at the same time, gradients and divergence-free functions. In order to rule out solutions of this type, it is customary to equip (1)

<footnote n="2">²See the second case in the proof of Proposition 8, p. 1161 in [20] and equations (2.50) to (2.53) in [18].</footnote>
with a (loosely stated) “far-field condition”, namely that the fluid is, far away, at rest in some given inertial reference frame: there is a fixed $V \in \mathbb{R}^d$ such that

$$u(t, x) \rightarrow V \quad \text{as} \quad |x| \rightarrow +\infty.$$

However, this approach remains incomplete for dealing with general bounded solutions. For example, periodic bounded flows, although they have an average velocity, can hardly be said to possess a limit at $|x| \rightarrow +\infty$.

The question we ask, in this article, is the following: under what condition do bounded solutions of the Euler problem (1) also solve the projected problem (4)?

In [8], we have proven that a loose integrability condition at $|x| \rightarrow +\infty$ is a sufficient answer to this question. However, this result is somewhat incomplete, since (4) possesses solutions that are not integrable at $|x| \rightarrow +\infty$, and these solutions must also solve (1) (take for example the constant flow $u(t, x) = Cst$). Similarly, another sufficient condition is given<fnref n="3" /> in [16] (point ii of Theorem 11.1, pp. 109-111), where the flow is required to satisfy the Morrey-type condition

$$(6) \qquad \forall t_1 < t_2, \quad \sup_{x \in \mathbb{R}^d} \frac{1}{\lambda^d} \int_{t_1}^{t_2} \int_{|x-y| \le \lambda} |u(t, y)|^2 dy dt \xrightarrow{\lambda \to +\infty} 0.$$

Here again, we note that this condition (although weaker than that of [8]) is not optimal, since any nonzero periodic solution of (4) defines a solution of the Euler equations that is not covered by (6).

As we will see below, it is possible to recover a necessary and sufficient condition that answers our question. To the best of our knowledge, it is the first time this question has received a full answer in the mathematical literature.

**Remark 1.2.** It should be noted that our problem is due to the fact that the solutions $(u, \pi)$ are defined on the non-compact space $\mathbb{R}^d$. For the Euler problem set on the torus $\mathbb{T}^d$, the issue completely dissapears. The pressure solves a Poisson equation, and hence is given by its Fourier coefficients

$$\forall k \in \mathbb{Z}^d \setminus \{0\}, \quad \hat{\pi}(k) = - \sum_{k,l} \frac{k_j k_l}{|k|^2} \widehat{u_j u_l}(k)$$

and a constant function (for $k = 0$), which is irrelevant as it does not change the value of the pressure force $-\nabla \pi$ (in other words, the mean value of the solution is preserved). Therefore, the velocity field does indeed solve the projected equation (4).

We point out that there is a slight difference between solutions defined on $\mathbb{T}^d$ and periodic flows on $\mathbb{R}^d$, as these last solutions, such as (5), may be driven by an exterior pressure. In that case, the pressure force $-\nabla \pi$ is periodic, although the pressure itself is not.

## 1.3 Outline of the paper and summary of the results

In this subsection, we give a summary of our main results, as well as a short overview of the paper. We consider three different models of incompressible fluids which present similar challenges: the Euler equations, the Navier-Stokes equations and the ideal MHD system.

<footnote n="3">³The statement of [16] is written for the Navier-Stokes system, but it can easily be applied to the Euler equations by ignoring the viscosity term.</footnote>
## Euler equations

In this paragraph, we introduce the formal ideas behind our main result, Theorem 3.6. Consider a smooth bounded solution $u$ of the Euler system which has bounded derivatives. As we have explained above, the pressure field is the solution of a Poisson equation

$$-\Delta\pi = \partial_j\partial_k(u_j u_k).$$

The problem lies in the fact that the solution of this elliptic problem is not unique, as it is given, in $S'$, up to the addition of a harmonic polynomial. Therefore, in order to recover the pressure force $-\nabla\pi$, we must add to (2) the gradient of a harmonic polynomial $Q \in \mathbb{R}[x]$, so as to obtain

$$\partial_t u + \mathbb{P}\operatorname{div}(u \otimes u) + \nabla Q = 0.$$

The presence of this polynomial is sometimes dismissed as being irrelevant, equation (2) giving the “canonical” choice of the pressure corresponding to a loosely stated far field condition. However, the question we ask, and solve, is much deeper, as we wish to know what property of the flow $u$ this choice is related to. As we will see in Theorem 3.6 below, the cancellation $\nabla Q = 0$ of this polynomial is equivalent to the condition

$$(7) \qquad \forall t, \quad u(t) - u(0) \in S'_h,$$

where $S'_h$ is the subspace of $S'$ consisting of distributions whose Fourier transform satisfies a smallness criterion at $\xi = 0$ (see Definition 2.5 below). In particular, $S'_h$ contains no polynomial function, as the Fourier transform of a polynomial is a sum of derivatives of the Dirac mass $\delta_0$.

Condition (7) is fairly natural: it means, if $u$ is e.g. smooth, that $\partial_t u \in S'_h$. Therefore, if we can prove that $\mathbb{P}\operatorname{div}(u \otimes u)$ is always an element of $S'_h$, we see that (7) will indeed be equivalent to the cancellation of the polynomial $\nabla Q = 0$.

The proof will be divided into two parts. In the first one, we work with regular in time solutions, and follow the ideas we have presented above. A key point of the argument will be to show that the operator $\mathbb{P}\operatorname{div}$ has its range in $S'_h$. In the second part of the proof, we will resort to a regularization procedure to adapt our method to functions which have very low time regularity. In fact, we will be able to deal with weak solutions in $C^0(L^\infty)$.

On another note, we point out that the initial value problem for bounded solutions has a few unusual difficulties, which are discussed in Subsection 3.1. These complications are the result of initial datum only being defined up to an additive constant in the weak formulation of the Euler system. However, these technicalities are not at the core of the proof.

As we have explained above, unlike the Euler equations, the projected problem (4) is well-posed in the space $C^0([0, T]; B_{\infty, 1}^1)$, at least for $T > 0$ small enough. As a corollary of our main result, the Euler system (1) is also well-posed in that space, provided we require the solutions to satisfy condition (7). Corollary 3.15 below contains a full well-posedness statement.

## Navier-Stokes equations

The techniques developed for the Euler equations also apply to the incompressible Navier-Stokes system, which reads

$$(8) \qquad \left\{ \begin{array}{l} \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = \Delta u \\ \operatorname{div}(u) = 0. \end{array} \right.$$

As before, we may formally apply the Leray projection operator to the momentum equation and obtain the associated problem

$$(9) \qquad \partial_t u + \mathbb{P}\operatorname{div}(u \otimes u) = \Delta u.$$
However, the use of $\mathbb{P}$ implies that a Poisson equation has to be solved. As before with the Euler equations, a harmonic polynomial $\nabla Q \in \mathbb{R}[x]$ may appear as an additional term in (9), so that problems (8) and (9) are not equivalent. In Theorem 4.1, we prove that a solution $u$ in the critical space $C^0(B_{\infty,\infty}^{-1}) \cap L^2(L^\infty)$ of the Navier-Stokes equations (8) solves (9) if and only if the condition

$$ \forall t \ge 0, \quad u(t) - u(0) \in S'_h $$

is satisfied. Although the proof of Theorem 4.1 is close to that of Theorem 3.6, the additional difficulty of working in a space of functions with even lower regularity makes the regularization argument a bit more delicate.

## Ideal MHD

Finally, the methods used in this article can also be applied to the ideal magnetohydrodynamic (MHD for short) system, which generalizes the Euler equations as

$$ (10) \qquad \begin{cases} \partial_t u + \operatorname{div}(u \otimes u - b \otimes b) + \nabla (\pi + \frac{1}{2}|b|^2) = 0 \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = 0 \\ \operatorname{div}(u) = \operatorname{div}(b) = 0. \end{cases} $$

In the MHD system (10), the new unknown $b(t,x) \in \mathbb{R}^d$ is the magnetic field generated by the electrical current in the fluid. As we further explain in Section 5 below, the theory of these equations in $L^p$-based spaces (with $p \neq 2$) requires working with a symmetrized version of (10), the so-called Elsässer system, which reads

$$ (11) \qquad \begin{cases} \partial_t \alpha + \operatorname{div}(\beta \otimes \alpha) + \nabla \pi_1 = 0 \\ \partial_t \beta + \operatorname{div}(\alpha \otimes \beta) + \nabla \pi_2 = 0 \\ \operatorname{div}(\alpha) = \operatorname{div}(\beta) = 0. \end{cases} $$

In (11) above, the Elsässer variables $(\alpha, \beta)$ are linked to the “physical” variables $(u, b)$ by the following change of variables:

$$ (12) \qquad \alpha = u + b \quad \text{and} \quad \beta = u - b, $$

and the functions $\pi_1$ and $\pi_2$ are possibly distinct scalar functions. The problem is that, if $(\alpha, \beta)$ is a solution of the Elsässer system with “pressure functions” $\pi_1$ and $\pi_2$, then the magnetic field $b = (\alpha - \beta)/2$ solves

$$ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = \frac{1}{2} \nabla (\pi_2 - \pi_1). $$

The left hand side of this equation is divergence-free, so that $Q = \pi_2 - \pi_1$ must be a harmonic polynomial. If $\nabla Q \neq 0$, then the Elsässer variables (12) do not solve the ideal MHD system (10), as we had already noted in [8] (see Section 4 for more on this topic). In Section 5, we state and prove Theorem 5.4, an equivalence criterion between these two systems: the Elsässer variables solve (10) if and only if the magnetic field satisfies

$$ \forall t, \quad b(t) - b_0 \in S'_h. $$

As with the Euler equations, Theorem 5.4 will allow us to give a full well-posedness result for bounded solutions of the ideal MHD system: this is Corollary 5.5. Also note that the arguments deployed for the Navier-Stokes equations can be applied to partially dissipative (for instance viscous but non-resistive) fluids, as explained in Remark 4.3.
## Structure of the paper

Before moving onwards, let us quickly give an outline of the paper.

We open the article by summarizing, in Section 2, a few tools we borrow from Littlewood-Paley analysis. Section 3 is the core of our work, and contains our discussion of the Euler equations. In Section 4, we turn our attention to the Navier-Stokes problem and in Section 5, we apply our methods the ideal MHD problem. The manuscript ends with an appendix containing a few harmonic analysis results, these concerning Fourier multipliers and the inclusion $S'_h \oplus \mathbb{R}[x] \subsetneq S'$.

## Acknowledgements

I am extremely grateful to my advisor, Francesco Fanelli, for his continuous support to this work, and his patience with my getting somewhat sidetracked by this problem. Many thanks also go to my co-workers for their kindness and help. This work has been partially supported by the project CRISIS (ANR-20-CE40-0020-01), operated by the French National Research Agency (ANR).

## Notations

In this paragraph, we give a short list of some of the notations we will be using.

All function spaces are defined on the whole Euclidean space unless otherwise specified. Therefore, we will omit $\mathbb{R}^d$ in the notation. For example, we note $L^\infty(\mathbb{R}^d) = L^\infty$. We note $\mathbb{R}[x]$ the space of polynomial functions on $\mathbb{R}^d$.

If $X$ is a Banach space and $T > 0$, we will often consider spaces of functions defined on $[0, T[$ and with values in $X$. For simplicity of notation, we often note $C^0([0, T[; X) = C_T^0(X)$ and likewise for other types of regularity with respect to the time variable (such as $C^1$ or $L_{\text{loc}}^2$).

We will note $C$ any generic constant, which may be different from one line to the next, but the value of which will remain irrelevant.

Unless otherwise specified, all derivatives are taken in the sense of distributions. The derivative symbols $\nabla$, $\Delta$ and $\text{div}$ refer only to the space variable $x \in \mathbb{R}^d$.

In the absence of a subscript, brackets $\langle \cdot, \cdot \rangle$ should be understood in the sense of the $D' \times D$ or the $S' \times S$ duality.

# 2 Littlewood-Paley analysis

In this short section, we recall the main ideas of Littlewood-Paley theory and the definitions of Besov spaces. We refer to the second chapter of [1] for more details.

First of all, we introduce the Littlewood-Paley decomposition based on a dyadic partition of unity with respect to the Fourier variable. We fix a smooth radial function $\chi$ supported in the ball $B(0, 2)$, equal to 1 in a neighborhood of $B(0, 1)$ and such that $r \mapsto \chi(re)$ is nonincreasing over $\mathbb{R}_+$ for all unitary vectors $e \in \mathbb{R}^d$. Set $\varphi(\xi) = \chi(\xi) - \chi(2\xi)$ and $\varphi_m(\xi) := \varphi(2^{-m}\xi)$ for all $m \ge 0$. The homogeneous dyadic blocks $(\dot{\Delta}_m)_{m \in \mathbb{Z}}$ are defined by Fourier multiplication, namely

$$ \forall m \in \mathbb{Z}, \quad \dot{\Delta}_m = \varphi(2^{-m}D), $$

whereas the non-homogeneous dyadic blocks $(\Delta_m)_{m \in \mathbb{Z}}$ are defined by

$$ \Delta_m := 0 \quad \text{if } m \le -2, \quad \Delta_{-1} := \chi(D) \quad \text{and} \quad \Delta_m := \varphi(2^{-m}D) \quad \text{if } m \ge 0. $$

The main interest of the Littlewood-Paley decomposition (see (13) below) is the way the dyadic blocks interact with derivatives, and more generally with homogeneous Fourier multipliers.
**Lemma 2.1** (Bernstein inequalities). Let $0 < r < R$. A constant $C$ exists so that, for any nonnegative integer $k$, any couple $(p, q)$ in $[1, +\infty]^2$, with $p \le q$, and any function $u \in L^p$, we have, for all $\lambda > 0$,

$$ \begin{array}{l} \text{Supp } \hat{u} \subset B(0, \lambda R) \quad \Rightarrow \quad \|\nabla^k u\|_{L^q} \le C^{k+1} \lambda^{k+d} \left(\frac{1}{p} - \frac{1}{q}\right) \|u\|_{L^p}; \\ \text{Supp } \hat{u} \subset \{\xi \in \mathbb{R}^d | r\lambda \le |\xi| \le R\lambda\} \quad \Rightarrow \quad C^{-k-1} \lambda^k \|u\|_{L^p} \le \|\nabla^k u\|_{L^p} \le C^{k+1} \lambda^k \|u\|_{L^p}. \end{array} $$

**Remark 2.2.** It appears from the proof of the first Bernstein inequality (see Lemma 2.1 in [1]) that it may be generalized to Fourier multipliers whose symbols are at least of class $C^{2d}$. For symbols that are homogeneous functions (many of which are singular at $\xi = 0$), we will use Lemma 2.3 below (see Lemma 2.2 in [1]).

**Lemma 2.3.** Let $\sigma$ be a homogeneous function of degree $N \in \mathbb{Z}$. There exists a constant depending only on $\sigma$ and on the dyadic decomposition function $\chi$ such that, for all $p \in [1, +\infty]$,

$$ \forall m \in \mathbb{Z}, \forall f \in L^p, \quad \|\dot{\Delta}_m \sigma(D)f\|_{L^p} \le C 2^{mN} \|\dot{\Delta}_m f\|_{L^p}. $$

**Remark 2.4.** The dependency of the constant $C$ on the dyadic decomposition function is implicit in [1], but it clearly appears in the proof that dealing with finite annuli is crucial to the lemma. As a matter of fact, the result, as it is in [1], does not hold in general for functions whose spectral support contains large frequencies $|\xi| \to +\infty$. We refer to Appendix A for a more precise discussion. This limitation of Lemma 2.3 makes of the Littlewood-Paley decomposition (13) a crucial part of our analysis.

Knowledge of the dyadic blocks formally allows to reconstruct any function: this is the Littlewood-Paley decomposition. Formally, we have

$$ (13) \qquad \text{Id} = \sum_{m \ge -1} \Delta_m \quad \text{and} \quad \text{Id} = \sum_{j \in \mathbb{Z}} \dot{\Delta}_j. $$

However, although the first decomposition holds in the space $S'$, the second does not so, as can be seen from the fact that it fails for polynomials (but not only! See Appendix B). This fact will be absolutely crucial in our analysis. We note $S'_h$ the space for which this second decomposition is true. More precisely, we have the following definition.

**Definition 2.5.** Let $S'_h$ be the space of those tempered distributions $u \in S'$ such that

$$ (14) \qquad \chi(\lambda D) u \underset{\lambda \to +\infty}{\longrightarrow} 0 \quad \text{in } S'. $$

Note that this space does not depend on the precise choice of the low frequency cut-off $\chi$.

**Remark 2.6.** Two definitions of the space $S'_h$ coexist, as the convergence (14) is sometimes required to be in the norm topology of $L^\infty$, as in [1]. We follow Section 1.5.1 in [9] and Definition 2.1.1 in [6]. However, although the two definitions are not equivalent, our arguments work with both of them, so that our results remain valid in all cases.

**Lemma 2.7.** The space $L^\infty \cap S'_h$ is closed in $L^\infty$ for the strong topology.

*Proof.* Let $(f_n)_{n \ge 0}$ be a converging sequence of functions in $L^\infty \cap S'_h$ whose limit is $f \in L^\infty$. We have, for all $g \in S'$,

$$ \begin{aligned} |\langle \chi(\lambda D) f, g \rangle| & \le |\langle \chi(\lambda D)(f - f_n), g \rangle| + |\langle \chi(\lambda D) f_n, g \rangle| \\ & \le \|\chi(\lambda D)(f - f_n)\|_{L^\infty} \|g\|_{L^1} + |\langle \chi(\lambda D) f_n, g \rangle|. \end{aligned} $$

The fact that the $\chi(\lambda D) f_n$ converge to 0 in $S'$ as $\lambda \to +\infty$ shows that we have,

$$ \forall n \ge 0, \quad \limsup_{\lambda \to +\infty} |\langle \chi(\lambda D) f, g \rangle| \le C \|(f - f_n)\|_{L^\infty} \|g\|_{L^1}. $$

This term has limit 0 as $n \to +\infty$ so that $f$ indeed lies in $L^\infty \cap S'$.

□
Whether a function belongs or not to $S'_h$ is entirely determined by its behavior at $|x| \to +\infty$, as all compactly supported distributions are in $S'_h$. On another note, $L^p$ functions, with $p < +\infty$, also lie in $S'_h$ (see Lemma 2.1 above). The space $S'_h$ can be seen as a set of functions whose “average value on $\mathbb{R}^d$” is zero, meaning that their Fourier transform satisfies some smallness condition at $\xi = 0$. Even though a precise notion of “average value” may not always exist, our intuition is conformed by a few examples.

**Example 2.8.** Firstly, as is pointed out in [16] (see the proof of Theorem 11.1 pp. 109-111), the Morrey-type condition

$$ (15) \qquad \sup_{x \in \mathbb{R}^d} \frac{1}{\lambda^d} \int_{|x-y| \le \lambda} |f(y)|^2 dy \underset{\lambda \to +\infty}{\longrightarrow} 0 $$

is sufficient to assert that $f \in S'_h$, even though it is not optimal. For example, trigonometric functions with no constant component, such as $\sin(x_1)$, lie in $S'_h$ without satisfying (15).

**Example 2.9.** Less trivially, the sign function $\sigma = 1_{\mathbb{R}_+} - 1_{\mathbb{R}_-}$ is$^4$ an element of $S'_h$. Let us prove this assertion. The Fourier transform of $\sigma$ is given (up to an irrelevant multiplicative constant) by the principal value distribution p.v. $(1/\xi)$. Therefore, for all $\lambda > 0$ and $\phi \in S$, we have

$$ \langle \chi(\lambda D)\sigma, \phi \rangle \approx \lim_{\epsilon \to 0^+} \int_{\epsilon}^{+\infty} \chi(\lambda\xi) (\hat{\phi}(\xi) - \hat{\phi}(-\xi)) \frac{d\xi}{\xi} := \lim_{\epsilon \to 0^+} \mathcal{I}_{\lambda, \epsilon}. $$

By changing variables in the integral and using a Taylor expansion, we obtain

$$ |\mathcal{I}_{\lambda, \epsilon}| = \left| \int_{\lambda}^{+\infty} \chi(\epsilon\xi) \left( \hat{\phi}\left(\frac{\epsilon}{\lambda}\xi\right) - \hat{\phi}\left(-\frac{\epsilon}{\lambda}\xi\right) \right) \frac{d\xi}{\xi} \right| \le 2\frac{\epsilon}{\lambda} \left\| \partial_{\xi}\hat{\phi} \right\|_{L^{\infty}} \int_{\lambda}^{+\infty} \chi(\epsilon\xi)d\xi. $$

Finally, by using the fact that $\chi$ is compactly supported, say on some bounded interval $[-K, K]$, we get

$$ \langle \chi(\lambda D)\sigma, \phi \rangle = O\left(\frac{1}{\lambda}\right) \quad \text{as } \lambda \to +\infty. $$

This provides the convergence $\chi(\lambda D)\sigma \to 0$ in the space $S'$.

Finally, we define the class of non-homogeneous and homogeneous Besov spaces.

**Definition 2.10.** Let $s \in \mathbb{R}$ and $1 \le p, r \le +\infty$. The non-homogeneous Besov space $B_{p,r}^s = B_{p,r}^s(\mathbb{R}^d)$ is defined as the subset of tempered distributions $u \in S'$ for which

$$ \|u\|_{B_{p,r}^s} := \left\| \left(2^{ms} \|\Delta_m u\|_{L^p}\right)_{m \ge -1} \right\|_{L^r} < +\infty. $$

The non-homogeneous space $B_{p,r}^s$ is Banach for all values of $(s, p, r)$.

**Definition 2.11.** Let $1 \le p, r \le +\infty$ and $s \in \mathbb{R}$ such that the following condition holds:

$$ (16) \qquad s \le \frac{d}{p} \quad \text{and} \quad r = 1 \text{ if } s = \frac{d}{p}. $$

We define the homogeneous Besov space $\dot{B}_{p,r}^s = \dot{B}_{p,r}^s(\mathbb{R}^d)$ as the set of those distributions $u \in S'_h$ such that

$$ \|u\|_{\dot{B}_{p,r}^s} = \left\| \left(2^{ms} \|\dot{\Delta}_m u\|_{L^p}\right)_{m \in \mathbb{Z}} \right\|_{l^r(\mathbb{Z})} < +\infty. $$

Condition (16) insures that the homogeneous Besov space $\dot{B}_{p,r}^s$ is Banach. The Bernstein inequalities provide the embedding $\dot{B}_{p,1}^{d/p} \hookrightarrow L^\infty$, so the exponent $d/p$ will be said to be critical. In the sequel, we will refer to (16) as a supercriticality condition.

<footnote>$^4$Here and below, we note $1_A$ the characteristic function of a subset $A \subset \mathbb{R}^d$.</footnote>
In the functional analysis literature, the homogeneous Besov spaces are defined as a subspace of the quotient $S'/\mathbb{R}[x]$. The following lemma states that, under condition (16), there is no real difference with our previous definition (see Section II in [2] for details on these issues).

**Lemma 2.12.** Consider $(s, p, r) \in \mathbb{R} \times [1, +\infty]^2$ such that condition (16) is fulfilled. Consider $u \in S'$ such that we have

$$ \left\| \left( 2^{ms} \| \dot{\Delta}_m u \|_{L^p} \right)_{m \in \mathbb{Z}} \right\|_{l^r(\mathbb{Z})} < +\infty. $$

Then the function $u$ can be recovered from the homogeneous Littlewood-Paley decomposition up to a polynomial: there exists a $Q \in \mathbb{R}[x]$ such that

$$ u = Q + \sum_{m \in \mathbb{Z}} \dot{\Delta}_m u, \quad \text{with convergence in } S'. $$

**Remark 2.13.** The supercriticality condition (16) is absolutely necessary to Lemma 2.12. In Appendix B, we exhibit a function $f \in L^\infty$ which does not lie in $S'_h \oplus \mathbb{R}[x]$. In particular, the homogeneous Besov space $\dot{B}^0_{\infty, \infty}$ cannot be realized as subspace of $S'_h$. Although this seems to be common knowledge, we have been unable to find a proof in the mathematical literature.

Finally, we end this section by stating an integrability lemma which gives sense to the notion of weak solution in incompressible fluid mechanics. This is Proposition 1.2.1 of [4] (see also Lemma 3.11 of [8] for a proof based on Fourier analysis).

**Lemma 2.14.** Let $T \in S'$ be a tempered distribution such that $\langle T, \phi \rangle = 0$ for all divergence-free $\phi \in \mathcal{D}$. Then there exists a $S \in S'$ such that $T = \nabla S$.

# 3 Euler equations

Our first concern will be the bounded solutions of the homogeneous Euler equations. These read

$$ (17) \qquad \begin{cases} \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = 0 \\ \operatorname{div}(u) = 0. \end{cases} $$

We will start by commenting a bit on the weak bounded solutions of these equations, before stating and proving the main theorem of this paper.

## 3.1 Bounded weak solutions

We start by defining the notion of bounded weak solution to the Euler problem. Although this is done in a usual way, we will see that the lack of integrability of these solutions already poses problems with regard to the understanding of initial data.

**Definition 3.1.** Let $T > 0$ and $u_0 \in L^\infty$ be a divergence-free initial datum. A function $u \in L^2_{\text{loc}}([0, T[; L^\infty)$ is said to be a bounded weak solution to the Euler problem (17) related to the initial datum $u_0$ if, for all divergence-free test function $\phi \in \mathcal{D}([0, T[\times \mathbb{R}^d; \mathbb{R}^d)$,

$$ (18) \qquad \int_0^T \int \{ \partial_t \phi \cdot u + \nabla \phi : u \otimes u \} \, dx \, dt + \int u_0 \cdot \phi(0) \, dx = 0. $$

This definition of weak solution is entirely consistent with the Euler equations (17), even though the pressure function does not appear explicitly in the integral equation (18), since Lemma 2.14 insures the existence of $\pi$ based on (18), this function being unique up to an irrelevant constant summand.
**Remark 3.2.** Before moving on, some remarks are in order concerning the initial datum problem. Because we use divergence-free test functions $\phi$ in the weak form of the momentum equation (18), we have, for any constant $C \in \mathbb{R}^d$,

$$ \int (u_0 + C) \cdot \phi(0) \, dx = \int u_0 \cdot \phi(0) \, dx. $$

This means that if $u \in C_T^1(L^\infty)$ is a weak solution to the Euler problem related to the initial datum $u_0 \in L^\infty$, according to Definition 3.1, then $u$ is also a weak solution related to the different initial datum $u_0 + C$, which also is divergence-free. In other words, regardless of regularity, the weak solution solves the weak form of the Euler equations (17) for infinitely many different bounded initial data that differ from the initial value flow $u(0)$.

The weak form (18) of the momentum equation is tested with smooth divergence-free compactly supported functions. These functions are orthogonal, in the sense of the $S' \times S$ duality, to all gradients. Therefore, any (regular) flow $u$ which solves the Euler problem is also a weak solution (according to Definition 3.1) related to the initial data $u(0) + \nabla g$ (for any $g \in S'$).

In the $L^p$ framework (with $p < +\infty$), this ambiguity totally disappears, as long as we require the initial data to be $L^p$. Indeed, any divergence-free $f \in L^p$ which is also the gradient of some tempered distribution $f = \nabla g$ must be a harmonic polynomial, since we would then have $\Delta g = 0$. Since there is no nonzero polynomial in $L^p$, this implies $f = 0$.

Our problem lies in the fact that the space $L^\infty$ intersects non-trivially with $\mathbb{R}[x]$, so that there are nonzero divergence-free functions which are also gradients: constant functions.

**Remark 3.3.** Another problem, which we pointed out in [8], is that, because we test with divergence-free functions, there is no need for a weak solution (as in Definition 3.1) to be continuous with respect to time, even in the $D'$ topology. Take for instance $u(t, x) = 1_{[1, +\infty]}(t)V$, where $V \in \mathbb{R}^d$ is a constant vector. Then $u$ is a weak solution of (17), but is not continuous with respect to time. This problem only arises in the $L^\infty$ framework: $L^p$ regularity of the solutions (with $p < +\infty$) implies continuity with respect to time in some weak topology, since, in that case, we may indeed apply the Leray projection to obtain (19) below.

As we said in the introduction, bounded solutions of the Euler equations (17) are not unique, even under $C^\infty$ smoothness assumptions.

## 3.2 The projected problem

In this section, we focus on the projected problem: formally, one may apply the Leray projection operator to kill the pressure term and obtain an equation on $u$ only, namely

$$ (19) \qquad \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u) = 0. $$

However, as pointed out in the introduction, the Leray projection $\mathbb{P}$ is not properly defined on $L^\infty$ as a Fourier multiplier. We must therefore clarify the meaning of equation (19) when the solutions are solely bounded $u \in L_T^2(L^\infty)$. We follow remarks of Pak and Park (see [20], equations (20) and (21) p. 1161) who point out that the operator $\Delta_{-1}\partial_j(-\Delta)^{-1}\partial_k\partial_l$ is well-defined and bounded on $L^\infty$. To do this, they study the convolution kernel $\Gamma$ of this operator.

**Proposition 3.4.** Consider $j, k, l \in \{1, ..., d\}$. Let $E$ be the fundamental solution of the Laplacian on $\mathbb{R}^d$ given by

$$ E(x) = \frac{C(d)}{|x|^{d-2}} \quad \text{if } d \ge 3 \quad \text{and} \quad E(x) = C(d) \log |x| \quad \text{if } d = 2, $$

where the constant $C(d) > 0$ is such that $-\Delta E = \delta_0$. Let $\psi \in S$ be such that the low frequency block $\Delta_{-1}$ is given by the convolution $\Delta_{-1}u = \psi * u$ (therefore $\hat{\psi} = \chi$). Finally, consider $\theta \in D$
a nonnegative function whose value is 1 on a neighborhood of the origin. We define a function $\Gamma \in L^1$ by setting

$$ \Gamma_{j,k,l} = \psi * \partial_j \partial_k \partial_l ((1-\theta)E) + \partial_j \partial_k \partial_l \psi * (\theta E). $$

Moreover, this definition does not depend on the value of the function $\theta$.

*Proof.* This proof that $\Gamma \in L^1$ can be found in [20] (see equations (20) and (21) p. 1161). It relies on the fact that all functions in the convolutions defining $\Gamma$ are $L^1$-integrable, since

$$ E(x) \underset{|x| \to 0^+}{=} O\left(\frac{1}{|x|^{d-2}}\right) \quad \text{and} \quad \partial_j \partial_k \partial_l E(x) \underset{|x| \to +\infty}{=} O\left(\frac{1}{|x|^{d+1}}\right). $$

The fact that $\Gamma$ does not depend on $\theta$ is fairly straightforward. Let $\theta' \in \mathcal{D}$ be another function satisfying the same requirements. We have

$$ \Gamma_{j,k,l} = \partial_j \partial_k \partial_l \psi * (\theta'E) + \partial_j \partial_k \partial_l \psi * ((\theta - \theta')E) + \psi * \partial_j \partial_k \partial_l ((1-\theta)E). $$

By integrating by parts in the second summand, we find that it is equal to

$$ \partial_j \partial_k \partial_l \psi * ((\theta - \theta')E) = \psi * \partial_j \partial_k \partial_l ((\theta - \theta')E). $$

In this last convolution, both factors $\psi$ and $\partial_j \partial_k \partial_l ((\theta - \theta')E)$ are smooth and integrable, since $\theta - \theta'$ is compactly supported away from the origin $x = 0$, and so the integration by parts is possible. Using this, we may conclude:

$$ \begin{aligned} \Gamma_{j,k,l} &= \partial_j \partial_k \partial_l \psi * (\theta'E) + \psi * \partial_j \partial_k \partial_l * ((\theta - \theta')E + (1-\theta)E) \\ &= \psi * \partial_j \partial_k \partial_l ((1-\theta')E) + \partial_j \partial_k \partial_l \psi * (\theta'E) = \Gamma'_{j,k,l}. \end{aligned} $$

□

Thanks to this, we may define the operator $\mathbb{P} \mathrm{div}$ we need. Let $f \in L^\infty(\mathbb{R}^d; \mathbb{R}^d \otimes \mathbb{R}^d)$ be a matrix valued function, $\chi \in \mathcal{D}$ be a cut-off function as in Proposition 3.4 and $\Gamma = \Gamma_{j,k,l}$ defined accordingly. We define $\mathbb{P} \mathrm{div}(f)$ by separating the low frequencies, where a convolution by $\Gamma$ is used, from the high ones, where $\mathbb{P} \mathrm{div}$ is well-defined as a Fourier multiplier:

$$ [\mathbb{P} \mathrm{div}(f)]_j := \left( \sum_k \Delta_{-1} \partial_k f_{k,j} - \sum_{k,l} \Gamma_{j,k,l} * f_{k,l} \right) + \left[ (\mathrm{Id} - \Delta_{-1}) \mathbb{P} \mathrm{div}(f) \right]_j $$

Because $\Gamma \in L^1$, we see that we define a bounded operator on non-homogenous Besov spaces: for all $r \in [1, +\infty]$ and $s \in \mathbb{R}$,

$$ (20) \qquad \mathbb{P} \mathrm{div} : B_{\infty,r}^s \to B_{\infty,r}^{s-1} $$

which allows us to, in turn, define the notion of weak solution of the projected equations (19).

**Definition 3.5.** Let $T > 0$ and $u_0 \in L^\infty$ be a divergence-free initial datum. A vector field $u \in L_{\mathrm{loc}}^2([0, T; L^\infty)$ is said to be a weak solution of the projected problem (19) related to the initial datum $u_0$ if, for all $\phi \in \mathcal{D}([0, T \times \mathbb{R}^d; \mathbb{R}^d)$, we have

$$ \int_0^T \int u \cdot \partial_t \phi \, dx \, dt + \int_0^T \langle \mathbb{P} \mathrm{div}(u \otimes u), \phi \rangle \, dt = \int u_0 \cdot \phi(0) \, dx. $$

Note that, unlike the solutions of the Euler problem, those of the projected equations, as defined above, are neccesarily continuous with respect to time in the $B_{\infty,\infty}^{-1}$ topology.
## 3.3 Main result

In this subsection, we state and comment our main result. The next two subsections will be devoted to proving it. The proof will start with the simpler case where the solutions are $C^1$ with respect to the time variable, before dealing with the full statement of Theorem 3.6.

**Theorem 3.6.** Consider $T > 0$. Let $u_0 \in L^\infty$ be a divergence-free initial datum and $u \in C^0([0, T]; L^\infty)$ be a weak solution of the Euler problem (according to Definition 3.1) for some pressure field $\pi$ and related to the initial datum $u_0$. The following conditions are equivalent:

(i) the flow $u$ solves the projected problem (19) in the sense of Definition 3.5 with the initial datum $u(0)$ that satisfies $u(0) - u_0 = Cst$;

(ii) for all $t \in [0, T[$, we have $u(t) - u(0) \in S'_h$.

**Remark 3.7.** Let us start by a remark concerning the class of solutions on which we work. Existence of $C_T^0(B_{\infty,1}^1) \cap C_T^1(B_{\infty,1}^0)$ solutions has been proved in [20], so it would seem natural to give a theorem that applies to $C_T^0(W^{1,\infty}) \cap C_T^1(L^\infty)$ solutions. However, $C^1$ regularity with respect to time stems from the fact that the authors of [20] work with $C_T^0(B_{\infty,1}^1)$ solutions of the projected problem (19), so that $\partial_t u = -P \operatorname{div}(u \otimes u)$ lies in $C_T^0(B_{\infty,1}^0)$.

If $u \in C_T^0(B_{\infty,1}^1)$ is only a solution of the Euler equations, one cannot repeat this operation because the pressure term $\nabla\pi$ may well not be regular with respect to time. Recall that the uniform flow (5) solves the Euler problem even if $f(t)$ is not $C^1$.

In that regard, working with $C_T^1(L^\infty)$ solutions would be somewhat artificial. In contrast, continuity in the time variable is a very reasonable assumption, in the light of the existence of paradoxical solutions (that dissipate kinetic energy) in the class $C^0(L^2)$, see [10]. In other words, there is no point in going below time-continuous regularity.

**Remark 3.8.** We note that condition (ii) of the theorem is in fact a Galilean condition: its validity is independent of the inertial reference frame in which the velocity is computed. This shows that the equivalence issue has something to do with a fundamental property of the solution, and is not a mere artifact of the Galilean nature of the equations.

## 3.4 Equivalence of the two formulations: smooth in time solutions

In this subsection, we prove Theorem 3.6 under a comfortable assumption that the solutions are $C^1(L^\infty)$.

**Proposition 3.9.** Consider $T > 0$. Let $u_0 \in L^\infty$ be a divergence-free initial datum and $u \in C^1([0, T]; L^\infty)$ be a weak solution of the Euler problem (according to Definition 3.1) for some pressure field $\pi$ and related to the initial datum $u_0$. Then conditions (i) and (ii) of Theorem 3.6 are equivalent.

The proof of this proposition is split in several steps. We start by reformulating the problem in a way that lets the Leray projector appear in the equations. This is the purpose of the following lemma.

**Lemma 3.10.** Let $u$ be as in Proposition 3.9. For all times $t \in ]0, T[$, there exists a polynomial $Q(t) \in \mathbb{R}[x]$ such that

$$ (21) \qquad \partial_t u + P \operatorname{div}(u \otimes u) + \nabla Q(t) = 0. $$

Moreover, $Q(t)$ must be of degree at most one: $\deg(Q(t)) \le 1$.
**Remark 3.11.** Of course, since deg($\nabla Q$) ≤ 0, then $\nabla Q = g(t)$ depends only on the time variable. Inspired by the example of the uniform flow (5), it may seem very tempting to find $f(t)$ such that $f'(t) = g(t)$ and ask if $u - f(t)$ is a solution of the projected problem. The nonlinearity of the equations implies the answer is, of course, no, as

$$ \partial_t(u - f) + \mathbb{P} \operatorname{div}((u - f) \otimes (u - f)) + \mathbb{P} \operatorname{div}(f \otimes u) = 0. $$

This last summand is zero if and only if the curl matrix of $\operatorname{div}(f \otimes u)$ is also zero, that is if $(f(t) \cdot \nabla)\omega \equiv 0$, where the vorticity matrix $\omega$ of the flow is defined by $[\omega]_{ij} = \partial_j u_i - \partial_i u_j$. This can only happen if the vorticity has no dependence on space in the direction of $f(t)$.

**Proof of Lemma 3.10.** Since $u \in C_T^1(L^\infty)$ is a solution of the Euler equations for the pressure $\pi$, we may write

$$ \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = 0 \quad \text{in } \mathcal{D}'([0, T[ \times \mathbb{R}^d). $$

By taking the divergence of this equation, we see that the pressure solves the elliptic equation

$$ -\Delta\pi = \partial_j\partial_k(u_j u_k), $$

in which there is an implicit sum on the repeated indices. We wish to invert this equation in order to recover $\pi$. In order to avoid the singularity of the inverse Laplacian operator $(-\Delta)^{-1}$ at the frequency $\xi = 0$, we do so for homogeneous dyadic blocks: for any $m \in \mathbb{Z}$, we have

$$ \begin{aligned} (22) \qquad \dot{\Delta}_m \nabla \pi &= \dot{\Delta}_m \nabla (-\Delta)^{-1} \partial_j \partial_k (u_j u_k) \\ &= -\dot{\Delta}_m (\mathrm{Id} - \mathbb{P}) \operatorname{div}(u \otimes u). \end{aligned} $$

Now, the operator appearing in righthand side of this equation is a Fourier multiplier whose symbol is a homogeneous function of degree 1, localized at the block $\dot{\Delta}_m$ and applied to the function $u \otimes u \in C_T^1(L^\infty)$. We may therefore apply Lemma 2.3 to obtain, at all times $t \in [0, T[$,

$$ (23) \qquad \| \dot{\Delta}_m (\mathrm{Id} - \mathbb{P}) \operatorname{div}(u \otimes u) \|_{L^\infty} \le C 2^m \| \dot{\Delta}_m (u \otimes u) \|_{L^\infty} \le C 2^m \|u\|_{L^\infty}^2. $$

This, along with Lemma 2.12, shows that the functions $\nabla\pi$ and $-(\mathrm{Id}-\mathbb{P})\operatorname{div}(u\otimes u)$ define elements of the homogeneous Besov space $\dot{B}_{\infty,\infty}^{-1}$ up to polynomial functions: for every time $t \in [0, T[$, we may fix $\nabla Q(t), \nabla R(t) \in \mathbb{R}[x]$ such that

$$ \begin{aligned} (24) \qquad -(\mathrm{Id} - \mathbb{P}) \operatorname{div}(u \otimes u) &= \nabla R(t) - \sum_{m \in \mathbb{Z}} \dot{\Delta}_m (\mathrm{Id} - \mathbb{P}) \operatorname{div}(u \otimes u) \\ &= \nabla \pi + \nabla Q(t). \end{aligned} $$

Both polynomials in (24) are gradients because all other terms in the equation are gradients. We have shown that (21) holds. The assertion on the degree of $Q(t)$ is obtained by applying the low frequency block $\Delta_{-1}$ to (21) and noting that

$$ -\nabla Q = \partial_t \Delta_{-1} u + \Delta_{-1} \mathbb{P} \operatorname{div}(u \otimes u) \in C_T^0(L^\infty) $$

is at all times bounded in view of (20), thus ending the proof of the lemma. $\square$

Before moving onwards, we study the Leray projection more closely. The following proposition shows that, for any $f \in L^\infty$, the function $(\mathrm{Id}-\mathbb{P})\operatorname{div}(f)$ is in $S'_h$. As a consequence, the polynomial $\nabla R(t)$ appearing in (24) must be zero.

**Proposition 3.12.** Let $\Gamma$ be as defined in Proposition 3.4. Then we have, for all $p \in [1, +\infty]$ and all $f \in L^p$, the estimate

$$ \|\chi(\lambda D)(\Gamma * f)\|_{L^p} = O\left(\frac{1}{\lambda} \log(\lambda)\right) \quad \text{as } \lambda \to +\infty. $$

The factor $\log(\lambda)$ may be dispensed with whenever $d \ge 3$.
**Remark 3.13.** The $O(\lambda^{-1})$ estimate (neglecting the logarithmic term when $d=2$) seems rather natural, considering that convolution by $\Gamma$ is formally the Fourier multiplication by a homogeneous symbol of order 1. However, the first Bernstein inequality is insufficient to obtain this result, as, though it may be generalized to Fourier multipliers, it requires their symbol to be smooth.

*Proof of Proposition 3.12.* We recall that the Fourier multiplier $\Delta_{-1} = \chi(D)$ is equal to the convolution operator $f \mapsto \psi * f$ (see Proposition 3.4). Therefore, we have, for any $\lambda > 0$,

$$ \chi(\lambda D) = \psi_{\lambda}, \quad \text{where } \psi_{\lambda}(x) = \frac{1}{\lambda^d} \psi\left(\frac{x}{\lambda}\right). $$

Taking advantage of the fact that the function $\Gamma$ does not depend on the precise choice of the cut-off $\theta$, we take $\theta_{\lambda}(x) = \chi(\lambda^{-\alpha}x)$ for some $\alpha > 0$. We therefore have, for $\lambda$ large enough that $\chi(\lambda D)\Delta_{-1} = \chi(\lambda D)$,

$$ \chi(\lambda D)\Gamma = \psi_{\lambda} * \partial_{j}\partial_{k}\partial_{l}\left((1 - \theta_{\lambda})E\right) + \partial_{j}\partial_{k}\partial_{l}\psi_{\lambda} * (\theta_{\lambda}E). $$

To prove the proposition, the Hausdorff-Young convolution inequality asserts that it is enough to show that the $L^1$ norm of both these convolution products is $O(\lambda^{-1} \log(\lambda))$. We start by studying the second one:

$$ \|\partial_j \partial_k \partial_l \psi_\lambda * (\theta_\lambda E)\|_{L^1} \le \|\partial_j \partial_k \partial_l \psi_\lambda\|_{L^1} \|\theta_\lambda E\|_{L^1}. $$

On the one hand, the three derivatives $\partial_j\partial_k\partial_l$ provide a $\lambda^{-3}$ decay, as

$$ \|\partial_j \partial_k \partial_l \psi_\lambda\|_{L^1} = \frac{1}{\lambda^3} \int \left| (\partial_j \partial_k \partial_l \psi) \left(\frac{x}{\lambda}\right) \right| \frac{dx}{\lambda^d} = O\left(\frac{1}{\lambda^3}\right). $$

On the other hand, the fundamental solution $E(x)$ is locally integrable, because it has a $O(|x|^{d-2})$ singularity at $x=0$ when $d \ge 3$, and a $O(\log|x|)$ one if $d=2$. Since the function $\theta_{\lambda}$ is supported in the ball $B(0, 2\lambda^{\alpha})$, we have another inequality: in the case where $d \ge 3$, we get

$$ \|\theta_{\lambda} E\|_{L^1} \le C \int_{|x| \le \lambda^{\alpha}} \frac{dx}{|x|^{d-2}} = O(\lambda^{2\alpha}), $$

and in the case where $d=2$, we instead have

$$ \begin{aligned} \|\theta_{\lambda} E\|_{L^1} &\le C \int_{|x| \le \lambda^{\alpha}} \log|x| \, dx \le C \int_{0}^{\lambda^{\alpha}} r \log(r) \, dr \\ &= \frac{C}{2} r^2 \left( \log(r) - \frac{1}{2} \right) \Big|_{r=0}^{\lambda^{\alpha}} = O\left(\lambda^{2\alpha} \log(\lambda)\right). \end{aligned} $$

We conclude that, in all dimensions $d \ge 2$, the convolution product $\partial_j\partial_k\partial_l\psi_\lambda * (\theta_\lambda E)$ has a $L^1$ norm that tends to zero as long as $\alpha < 3/2$, at the speed $O(\lambda^{2\alpha-3}\log(\lambda))$.

We next look at the other convolution product $\psi_\lambda * \partial_j\partial_k\partial_l((1-\theta_\lambda)E)$. Here, we will take advantage of the integrability the third derivatives of $E(x)$ possess at $|x| \to +\infty$. However, this in itself is not enough, as we aim at showing decay as $\lambda \to +\infty$. We will also have to use the fact that the support of the cutoff $1-\theta_\lambda$ shrinks as $\lambda$ becomes large. More precisely, we have the estimate, which holds in any dimension $d \ge 2$,

$$ |\partial_j \partial_k \partial_l ((1 - \theta_\lambda)E)(x)| \le C \frac{1 - \mathbb{1}_{B(0, 2\lambda^\alpha)}(x)}{|x|^{d+1}} := M_\lambda(x). $$

By using this on the convolution product, we find that

$$ \|\psi_{\lambda} * \partial_{j}\partial_{k}\partial_{l}\left((1 - \theta_{\lambda})E\right)\|_{L^{1}} \le C \|\psi_{\lambda}\|_{L^{1}} \|M_{\lambda}\|_{L^{1}} \le C \|M_{\lambda}\|_{L^{1}}. $$
Finally, we may bound this last integral by

$$ \|M_{\lambda}\|_{L^1} \le C \int_{|x| \ge \lambda^{\alpha}} \frac{dx}{|x|^{d+1}} = C \int_{\lambda^{\alpha}}^{+\infty} \frac{dr}{r^2} = O\left(\frac{1}{\lambda^{\alpha}}\right). $$

Putting both estimates together, we have a low frequency inequality for the kernel $\Gamma$. In the limit $\lambda \to +\infty$,

$$ \|\chi(\lambda D)\Gamma\|_{L^1} = O\left(\frac{1}{\lambda^{\alpha}} + \frac{1}{\lambda^{3-2\alpha}}\log(\lambda)\right), $$

and this gives convergence to 0 as long as $0 < \alpha < 3/2$. Taking $\alpha = 1$ (the optimal value) ends proving our statement. $\square$

Proposition 3.12 has the following consequence.

**Corollary 3.14.** Let $f \in B_{\infty, \infty}^0(\mathbb{R}^d; \mathbb{R}^d \otimes \mathbb{R}^d)$ be a field of matrices. Then we have

$$ (25) \qquad \mathbb{P} \operatorname{div}(f) = \sum_{m \in \mathbb{Z}} \dot{\Delta}_m \mathbb{P} \operatorname{div}(f) \quad \text{with convergence in } \mathcal{S}'. $$

We therefore define a bounded operator $\mathbb{P} \operatorname{div} : B_{\infty, \infty}^0 \to \dot{B}_{\infty, \infty}^{-1}$. The same statement holds for the operator $\mathbb{Q} = \mathrm{Id} - \mathbb{P}$.

*Proof.* From Lemma 2.12 and inequality (23), we see that the sum in the statement converges to an element of $\dot{B}_{\infty, \infty}^{-1}$. It remains to show that it is equal to the left hand side. Now, the Fourier transform of both sides of (25) are equal away from the frequency $\xi = 0$. Therefore, their difference must be spectrally supported at $\xi = 0$, and so must be a polynomial.

On the other hand, we already know from Proposition 3.12 that $(\mathrm{Id} - \mathbb{P})\operatorname{div}(f)$ defines an element of $\mathcal{S}'_h$, and the same is true for $\operatorname{div}(f)$, thanks to the first Bernstein inequality (Lemma 2.1), as

$$ \|\chi(\lambda D) \operatorname{div}(f)\|_{L^\infty} = \|\chi(\lambda D) \Delta_{-1} \operatorname{div}(f)\|_{L^\infty} = O\left(\frac{1}{\lambda}\right) \quad \text{as } \lambda \to +\infty. $$

We deduce that both sides in (25) are in $\mathcal{S}'_h$, and we know that they differ by a polynomial: they must therefore be equal. $\square$

We now have all the necessary elements to prove Proposition 3.9.

*Proof of Proposition 3.9.* With Lemma 3.10 and Corollary 3.14 at our disposal, we are ready to complete the proof of Proposition 3.9, as it is equivalent for $u$ to solve the projected problem (19) and for the polynomial $\nabla Q$ in Lemma 3.10 to be zero. Now, by Corollary 3.14, we see that $\partial_t u$ decomposes as

$$ \partial_t u = -\mathbb{P} \operatorname{div}(u \otimes u) - \nabla Q \in \mathcal{S}'_h \oplus \mathbb{R}[x], $$

and so $\nabla Q \equiv 0$ if and only if $\partial_t u(t) \in \mathcal{S}'_h$ at all times $t \in ]0, T[$.

Start by assuming that condition (ii) of the Proposition 3.9 holds, so that $u(t) - u(0) \in \mathcal{S}'_h$ for all $t \in [0, T[$. By differentiating with respect to time, we see that $\partial_t u(t)$ is also in $\mathcal{S}'_h$ for all $t \in ]0, T[$. Indeed, since $u \in C_T^1(L^\infty)$ the difference quotients $h^{-1}(u(t+h) - u(t))$ will converge to its time derivative $\partial_t u(t)$ for the norm topology of $L^\infty$, and the fact that the difference quotients are in $\mathcal{S}'_h \cap L^\infty$ insures that $\partial_t u(t)$ also is, because $\mathcal{S}'_h \cap L^\infty$ is a closed subspace of $L^\infty$ (by Lemma 2.7). This implies that $\nabla Q \equiv 0$ and that $u$ solves the projected problem (19) on $]0, T[ \times \mathbb{R}^d$.

The fact that it does so with initial datum $u(0)$, and in the sense of Definition 3.5, simply stems from the $C^1$ regularity with respect to time.

Finally, we show that $V = u_0 - u(0)$ is a constant function. Any solution of the projected problem (19) must also be a solution of the Euler problem (17) with the same initial datum. Since
u solves both problems (in the sense of Definitions 3.1 and 3.5), this implies that the weak form (18) of the momentum equation must hold for both initial data, so

$$ \int (u_0 - u(0)) \cdot \phi \, dx = 0 $$

for all divergence-free $\phi \in \mathcal{D}(\mathbb{R}^d; \mathbb{R}^d)$, and so $V \in L^\infty$ is the gradient of some tempered distribution $V = \nabla h$. But since $V$ is also divergence-free, $h$ must be a harmonic polynomial, so that $V$ is also polynomial. The fact that $V \in L^\infty$ forces $V$ to be constant.

Next, we instead assume that condition (i) holds. By integrating with respect to time, we have

$$ u(t) - u(0) = \int_{0}^{t} \partial_{t} u(s) \mathrm{d}s, $$

this integral being defined as an element of the Banach space $L^\infty$ (e.g. as a limit of Riemann sums). The fact that $S'_h \cap L^\infty$ is closed in $L^\infty$ for the strong topology implies that we must also have $u(t) - u(0) \in S'_h$. $\square$

## 3.5 Equivalence of the two formulations: low time regularity

In this paragraph, we fully prove Theorem 3.6 by working with the more general class of $C_T^0(L^\infty)$ solutions of (17). The obvious problem is that the derivative $\partial_t u$ might not exist as a function of time for these solutions. We will use a regularization procedure to circumvent this issue.

*Proof of Theorem 3.6.* Because we deal with solutions which have low time regularity, we take a mollification sequence $(K_\epsilon(t))_{\epsilon>0}$ such that $K_1$ is supported in the compact interval $[-1, 1]$ and

$$ K_{\epsilon}(t) = \frac{1}{\epsilon} K_{1} \left( \frac{t}{\epsilon} \right). $$

Next, we extend all functions $u$ and $\pi$ to functions on $t \in \mathbb{R}$ by setting them to zero on $]-\infty, 0[$. For the sake of simplicity, we continue to note $u$ and $\pi$ the extensions. This allows us to incorporate the initial data condition in the righthand side of the equations, and see that $(u, \pi)$ solves

$$ \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = \delta_0(t) \otimes u_0(x) \quad \text{in } \mathcal{D}'(-\infty, T[\times \mathbb{R}^d]), $$

where the tensor product $\delta_0(t) \otimes u_0(x)$ of these two distributions of the time and space variables is defined by

$$ \forall \phi \in \mathcal{D}(-\infty, T[\times \mathbb{R}^d; \mathbb{R}^d]), \quad \langle \delta_0(t) \otimes u_0(x), \phi(t, x) \rangle = \int u_0(x) \cdot \phi(0, x) \, dx. $$

For all $\epsilon > 0$, we note $(u_\epsilon, \pi_\epsilon) = K_\epsilon * (u, \pi)$, where the convolution is done with respect to the time variable $t \in] -\infty, T - \epsilon[$, as the kernel $K_\epsilon$ is supported in $[-\epsilon, \epsilon]$. We obtain, by convoluting with respect to time, an equality in the sense of $\mathcal{D}'(-\infty, T - \epsilon[\times \mathbb{R}^d)$:

$$ \begin{aligned} \partial_t u_\epsilon + K_\epsilon * \operatorname{div}(u \otimes u) + \nabla \pi_\epsilon &= (K_\epsilon * \delta_0)(t) \otimes u_0(x) \\ &= K_\epsilon(t)u_0(x). \end{aligned} \quad (26) $$

By taking the divergence of this equation and the $m$-th homogeneous dyadic block $\dot{\Delta}_m$, we see that

$$ \forall m \in \mathbb{Z}, \quad \dot{\Delta}_m \nabla \pi_\epsilon = \sum_{j,k} \dot{\Delta}_m \nabla (-\Delta)^{-1} \partial_j \partial_k K_\epsilon * (u_j u_k). $$

Because, for all $t \in \mathbb{R}$, the functions $K_\epsilon * (u_j u_k)(t)$ are bounded, Corollary 3.14 insures that we can sum the previous equation over $m \in \mathbb{Z}$ to get, as in (24),

$$ \nabla \pi_{\epsilon} = \nabla Q_{\epsilon}(t) - K_{\epsilon} * (\mathrm{Id} - \mathbb{P}) \operatorname{div}(u \otimes u), $$
for some polynomial $Q_{\epsilon}(t) \in \mathbb{R}[x]$. By substituting this expression in (26), we find that

$$ (27) \qquad \partial_t u_{\epsilon} + K_{\epsilon} * \mathbb{P} \operatorname{div}(u \otimes u) + \nabla Q_{\epsilon} = K_{\epsilon} u_0, \quad \text{for } t \in ]-\infty, T[. $$

By taking the limit $\epsilon \to 0^+$ in the previous equation, we see that all terms converge in each $\mathcal{D}'[] - \infty, T - \eta[\times \mathbb{R}^d)$ space, with $\eta > 0$, to some limit. Therefore, the $\nabla Q_{\epsilon}$ must also have a limit as $\epsilon \to 0^+$, which we will note $\nabla Q$, and which satisfies

$$ \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u) + \nabla Q = \delta_0(t) \otimes u_0(x) \quad \text{in } \mathcal{D}'[] - \infty, T[\times \mathbb{R}^d). $$

By taking the time convolution of this last equation by $K_{\epsilon}(t)$, we see that we in fact have $\nabla Q_{\epsilon} = K_{\epsilon} * \nabla Q$. The whole proof hinges on finding under what condition $\nabla Q = 0$ in $\mathcal{D}'[]0, T[\times \mathbb{R}^d)$.

We start by assuming that condition (i) holds, so that $\nabla Q = 0$ on $]0, T[\times \mathbb{R}^d$. This implies that $\nabla Q_{\epsilon}(t) = 0$ for $t \in [\epsilon, T - \epsilon]$, since $K_{\epsilon}$ is compactly supported in $[-\epsilon, \epsilon]$. Similarly, the term containing the initial datum $u_0$ in (26) vanishes as soon as $t \ge \epsilon$. Therefore, for $t \in [\epsilon, T - \epsilon]$,

$$ \partial_t u_{\epsilon}(t) = -K_{\epsilon} * \mathbb{P} \operatorname{div}(u \otimes u) \in S'_{h}. $$

Therefore, by integrating with respect to time, we find that

$$ u_{\epsilon}(t) - u_{\epsilon}(\epsilon) = \int_{\epsilon}^{t} \partial_{t} u_{\epsilon}(s) ds \in L^{\infty} \cap S'_{h}. $$

Since $u_{\epsilon} \in C^{\infty}(L^{\infty})$, this last integral is well-defined as an element of the Banach space $L^{\infty} \cap S'_{h}$ (see Lemma 2.7) as, e.g. a limit of Riemann sums. We wish to take the limit $\epsilon \to 0^+$ in this last equation and use the fact that $L^{\infty} \cap S'_{h}$ is closed in $L^{\infty}$.

Thanks to the convergence $u_{\epsilon} \to u$ in the space $C^0[]0, T[; L^{\infty})$, which is equipped with the topology of uniform convergence on compact sets of $]0, T[$, we deduce on the one hand that we have pointwise convergence $u_{\epsilon}(t) \to u(t)$. On the other hand, to compute the limit of $u_{\epsilon}(\epsilon)$, we write

$$ \begin{aligned} \|u_{\epsilon}(\epsilon) - u(0)\|_{L^{\infty}} &\le \|u_{\epsilon}(\epsilon) - u(\epsilon)\|_{L^{\infty}} + \|u(\epsilon) - u(0)\|_{L^{\infty}} \\ &= \|u_{\epsilon}(\epsilon) - u(\epsilon)\|_{L^{\infty}} + o(1). \end{aligned} $$

Next, by writing explicitly the convolution integrals involved in $u_{\epsilon}(\epsilon) - u(\epsilon)$ and using the fact that the kernel $K_{\epsilon}$ is supported in $[-\epsilon, \epsilon]$, we get that

$$ u_{\epsilon}(\epsilon) - u(\epsilon) = \int_{0}^{2\epsilon} K_{\epsilon}(\epsilon - s) u(s) ds - u(\epsilon) = \int_{0}^{2\epsilon} K_{\epsilon}(\epsilon - s) \{u(s) - u(\epsilon)\} ds, $$

The function $u \in C^0([0, T[; L^{\infty})$ is uniformly continuous on any compact interval $[0, \eta]$. Therefore, we may take the $L^{\infty}$ norm of the above equation to obtain

$$ \|u_{\epsilon}(\epsilon) - u(\epsilon)\|_{L^{\infty}} \le \|K_{\epsilon}\|_{L^1} \sup_{s \in [0, 2\epsilon]} \|u(s) - u(\epsilon)\|_{L^{\infty}} \to 0 \quad \text{as } \epsilon \to 0^{+}. $$

This proves that $u_{\epsilon}(t) - u_{\epsilon}(\epsilon)$ converges to $u(t) - u(0)$ as $\epsilon \to 0^+$. Because $L^{\infty} \cap S'_{h}$ is closed in $L^{\infty}$, we finally deduce that $u(t) - u(0) \in S'_{h}$ for all $0 \le t < T$.

Now, let us assume that $u(t) - u(0) \in S'_{h}$ for all $t \in [0, T[$. We take the convolution of $u(t) - u(0)$ with $K_{\epsilon}$ to find that $u_{\epsilon}(t) - u(0)$ also lies in $L^{\infty} \cap S'_{h}$, but only for $t \in ]\epsilon, T - \epsilon[$, as the condition $u(t) - u(0) \in S'_{h}$ does not hold on the extension for $t \notin [0, T[$, as $u(0)$ need not be in $S'_{h}$. Since $u_{\epsilon}(t) - u(0)$ is a $C^1$ function with respect to time, we can differentiate. The derivative $\partial_t u_{\epsilon}$ will then be found as the $L^{\infty}$ limit of the difference quotients, and will therefore also be an element of $L^{\infty} \cap S'_{h}$,

$$ \forall t \in ]\epsilon, T - \epsilon[, \quad \partial_t u_{\epsilon}(t) \in S'_{h}. $$
This implies that the polynomial $\nabla Q_{\epsilon}(t)$ from (27) must also be in $S'_{h}$ if $t \in ]\epsilon, T - \epsilon[$, and so must be zero on that time interval. Therefore, $\nabla Q \equiv 0$ in $D'(]0, T[ \times \mathbb{R}^d)$. This means that $u$ solves the projected system (19), we just have to check the assertion concerning the initial datum for the velocity.

Since the function $u$ solves the projected equation, we may use it to prove that $u$ is in fact $C_T^1(\dot{B}_{\infty, \infty}^{-1})$ regular. The $C^1$ time regularity implies that $u$ will be a weak solution of the projected equation with initial datum $u(0)$, as in Definition 3.5.

Lastly, since $u$ solves the projected problem with initial datum $u(0)$, it must solve the Euler system with the datum $u(0)$. Therefore, $u$ is a weak solution of the Euler equations, as in Definition 3.1 with both initial data $u_0$ and $u(0)$, and so $u_0 - u(0)$ must be a constant function, as shown in Subsection 3.4 above. $\square$

## 3.6 A full well-posedness result

Theorem 3.6 above provides us with a full well-posedness result for the Euler system in the space $C_T^0(B_{\infty,1}^1)$.

**Corollary 3.15.** Consider $u_0 \in B_{\infty,1}^1$ a divergence-free initial datum. There exists a time $T > 0$ such that the Euler problem (17) has a unique solution $u \in C^0([0, T[; B_{\infty,1}^1)$ (in the sense of Definition 3.1) related to the initial datum $u_0$ that satisfies

$$ (28) \qquad u(0) = u_0 \quad \text{and} \quad u(t) - u(0) \in S'_{h} \text{ for } t \in [0, T[. $$

Moreover, this solutions lies in the space $C^1([0, T[; B_{\infty,1}^0)$.

**Remark 3.16.** Note that condition (28) is a rather natural one for a well-posedness result, as $u(t) - u(0) \in S'_{h}$ it implies that the flow is, unlike the Poiseuille-type flow (5), not driven by an exterior action (recall that $S'_{h}$ is a space of functions that are “on average” zero at $|x| \to +\infty$). A flow that is left to its own devices must be deterministic.

*Proof.* Thanks to Theorem 3.6, the proof is straightforward. With the help of the results of [20], we have a $T > 0$ such that the projected problem has a unique solution $u \in C_T^0(B_{\infty,1}^1) \cap C_T^1(B_{\infty,1}^0)$ with initial datum $u_0$. This solution must also solve the original Euler problem with the same initial datum.

On the other hand, by Theorem 3.6, any solution $v \in C_T^0(B_{\infty,1}^1) \hookrightarrow C_T^0(L^\infty)$ that fulfills condition (28) must also solve the projected problem (19) with the initial datum $u_0$, and so must coincide with $u$ on $[0, T[ \times \mathbb{R}^d$. $\square$

## 4 Navier-Stokes equations

In this section, we consider the Leray projection of the Navier-Stokes equations: these read

$$ (29) \qquad \left\{ \begin{array}{l} \partial_t u + \operatorname{div}(u \otimes u) + \nabla \pi = \Delta u \\ \operatorname{div}(u) = 0. \end{array} \right. $$

As has been well-known for the better part of a century, these equations can be solved (locally in time) by studying their Leray projection

$$ (30) \qquad \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u) = \Delta u. $$

However, using the operator $\mathbb{P}$ must be done carefully: as was the case with the Euler equations, not all solutions of system (29) solve (30), the uniform flow (5) being an example of such an oddity. As we have done before for the Euler problem, we provide an equivalence result for the Navier-Stokes equations.
We will work in the critical space $u \in C_T^0(B_{\infty,\infty}^{-1}) \cap L_T^2(L^\infty)$. Let us comment a bit on this assumption. First of all, the space $B_{\infty,\infty}^{-1}$ is rather large, as it contains all translation invariant subspaces $X \subset S'$ that scale appropriately to the the Navier-Stokes system (recall that $X \hookrightarrow \dot{B}_{\infty,\infty}^{-1} \hookrightarrow B_{\infty,\infty}^{-1}$ by Proposition 5.31 in [1]). In addition, reaching as far as the regularity exponent $s = -1$ is appropriate, as, on the one hand, local well-posedness of the Navier-Stokes system is known for initial data in $BMO^{-1}$ [15], and on the other, uniqueness has also been proved for $C^0(\dot{B}_{\infty,\infty}^{-1})$ solutions under the condition that they are of finite energy [5]. Finally the existence of finite time blowup results in $\dot{B}_{\infty,\infty}^{-1}$ for some Generalized Navier-Stokes systems (see [19] for the case of cheap Navier-Stokes) show that the requirements on $u$ are not too stringent.

For the sake of conciseness, we omit the full definitions of weak solutions of (29) and (30), as they are nearly identical to Definitions 3.1 and 3.5.

**Theorem 4.1.** Let $T > 0$ and $u_0 \in B_{\infty,\infty}^{-1}$ be a divergence-free function. Let $u \in C^0([0, T[; B_{\infty,\infty}^{-1}) \cap L_{\text{loc}}^2([0, T[; L^\infty)$ be a weak solution of the Navier-Stokes problem (29) related to the initial datum $u_0$. The following conditions are equivalent:

(i) the flow $u$ solves the projected problem (30) with the initial datum $u(0)$ that satisfies $u(0) - u_0 = Cst$;

(ii) for all $t \in [0, T[$, we have $u(t) - u(0) \in S'_h$.

**Remark 4.2.** Although we have stated Theorem 4.1 for $C_T^0(B_{\infty,\infty}^{-1}) \cap L_T^2(L^\infty)$ solutions, it appears clearly from the proof that time continuity is only needed for the low frequency part of the velocity $\Delta_{-1}u \in C_T^0(L^\infty)$. On the other hand, the $L_T^2(L^\infty)$ assumption is mainly required to define the convective term $\mathbb{P}\text{div}(u \otimes u)$ and apply Proposition 3.12. Therefore, our theorem remains valid at all regularity levels $C_T^0(B_{\infty,\infty}^s) \cap L_T^2(L^\infty)$, for any $s \in \mathbb{R}$. Of course, the $L_T^2(L^\infty)$ hypothesis is redundant if $s > 0$.

*Proof.* The proof of this theorem has many similarities with the one above for the Euler system, so we will cover it a bit more quickly. Let $\pi$ be the pressure function associated to $u$. Because $u$ has low regularity with respect to the time variable, we regularize. If $K_\epsilon(t)$ is the mollifier from the proof of Theorem 3.6, we prolong $(u, \pi)$ to a function on $]-\infty, T[ \times \mathbb{R}^d$ and set $(u_\epsilon, \pi_\epsilon) = K_\epsilon*(u, \pi)$. By doing as in the proof of Proposition 3.12 above, we find that there is, for every $t \in ]-\infty, T - \epsilon[$, a polynomial $Q_\epsilon(t) \in \mathbb{R}[x]$ such that

$$ (31) \qquad \partial_t u_\epsilon + K_\epsilon * \mathbb{P} \text{div}(u \otimes u) + \nabla Q_\epsilon = \Delta u_\epsilon + K_\epsilon(t) u_0(x) \quad \text{in } \mathcal{D}'(]-\infty, T - \epsilon[ \times \mathbb{R}^d). $$

By letting $\epsilon \to 0^+$ in the above, we see that $\nabla Q_\epsilon$ converges to some $\nabla Q$ in the sense of distributions and that

$$ \partial_t u + \mathbb{P} \text{div}(u \otimes u) + \nabla Q = \Delta u + \delta_0(t) \otimes u_0(x) \quad \text{in } \mathcal{D}'(]-\infty, T - \epsilon[ \times \mathbb{R}^d), $$

so that we are concerned with the conditions under which $\nabla Q = 0$ in $]0, T[ \times \mathbb{R}^d$. As before with the Euler system, we have $\nabla Q_\epsilon = K_\epsilon * \nabla Q$.

We start by assuming that condition (i) holds, so that $\nabla Q = 0$ on $]0, T[ \times \mathbb{R}^d$. This implies that $\nabla Q_\epsilon = 0$ for all $\epsilon \le t \le T - \epsilon$, since $K_\epsilon$ is compactly supported in $[-\epsilon, \epsilon]$. Likewise, the term containing the initial datum $u_0$ vanishes as soon as $t \ge \epsilon$. This implies that, for $t \in [\epsilon, T - \epsilon]$, $\partial_t u_0(t) = -K_\epsilon * \mathbb{P} \text{div}(u \otimes u) + \Delta u_0 \in S'_h$.

Therefore, by integrating with respect to time, we find that

$$ (32) \qquad u_\epsilon(t) - u_\epsilon(\epsilon) = \int_\epsilon^t \partial_t u_\epsilon(s) ds \in L^\infty \cap S'_h. $$
We want to take the limit $\epsilon \to 0^+$ in this last line and take advantage of the closure of $L^\infty \cap S'_h$ in $L^\infty$, however $u$ only has $L_T^2(L^\infty)$ regularity, so we are not guaranteed that the lefthand side will converge in $L^\infty$ at any fixed value of $\epsilon \le t \le T - \epsilon$. Instead, by taking the non-homogeneous low frequency block $\Delta_{-1}$, we may work with $\Delta_{-1}u$, which is in $C_T^0(L^\infty)$. Applying the operator $\Delta_{-1}$ to (32), we see that we in fact have

$$ \forall \epsilon > 0, \forall t \in [\epsilon, T - \epsilon], \quad \Delta_{-1}u_{\epsilon}(t) - \Delta_{-1}u_{\epsilon}(\epsilon) \in L^{\infty} \cap S'_{h}. $$

We have convergence of the low frequency blocks $\Delta_{-1}u_{\epsilon} \to \Delta_{-1}u$ in the space $C^0([0, T[; L^\infty)$, which is endowed with the topology of uniform convergence on compacts subsets of $]0, T[$, and therefore, by arguing exactly as in the proof of Theorem 3.6 above,

$$ \begin{aligned} \|\Delta_{-1}u_{\epsilon}(\epsilon) - \Delta_{-1}u(0)\|_{L^{\infty}} &\le \|\Delta_{-1}u(\epsilon) - \Delta_{-1}u(0)\|_{L^{\infty}} + \sup_{s \in [0,2\epsilon]} \|\Delta_{-1}u(s) - \Delta_{-1}u(\epsilon)\|_{L^{\infty}} \\ &\to 0 \quad \text{as } \epsilon \to 0^+. \end{aligned} $$

Thanks to the closure of $L^\infty \cap S'_h$ in the norm topology of $L^\infty$, we finally get that $u(t) - u(0) \in S'_h$ for all $0 \le t < T$.

Secondly, assume that $u(t) - u(0) \in S'_h$ for all $t \in [0, T[$. The convolution product $u_{\epsilon}(t) - u(0)$ also lies in $L^\infty \cap S'_h$ for $t \in ]\epsilon, T - \epsilon[$, and so

$$ \forall t \in ]\epsilon, T - \epsilon[, \quad \partial_t u_{\epsilon}(t) \in S'_{h}. $$

Consequently, we obtain cancellation $\nabla Q_{\epsilon}(t) = 0$ of the polynomials in (31) for all times $\epsilon \le t \le T - \epsilon$. Therefore, $\nabla Q \equiv 0$ in $D'([0, T[\times \mathbb{R}^d)$ and $u$ is a solution of the projected system (30) on $]0, T[$. All that is left to do is to make sure the initial datum and initial velocity field differ from an additive constant.

We start by noting that, since $u$ solves the projected system, we may use it to prove that $u$ possesses regularity with respect to time: $\partial_t u \in L_T^1(\dot{B}_{\infty, \infty}^{-2})$. Next, we see that for all $\phi \in D([0, T[\times \mathbb{R}^d; \mathbb{R}^d)$ and $f \in W_T^{1,1}(\dot{B}_{\infty, \infty}^{-2}) \cap L_T^2(L^\infty)$,

$$ \int_0^T \int \partial_t \phi \cdot f \, dx \, dt + \int_0^T \langle \partial_t f, \phi \rangle \, dt = - \int f(0) \cdot \phi(0) \, dx, $$

as both members of this equation are continuous functions of $f$ in the $W_T^{1,1}(\dot{B}_{\infty, \infty}^{-2}) \cap L_T^2(L^\infty)$ topology. This implies that we may test the projected equation

$$ \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u) = \Delta u $$

with $\phi \in D([0, T[\times \mathbb{R}^d; \mathbb{R}^d)$ and integrate by parts with respect to the time variable to show that $u$ is a weak solution of the projected problem (30) for the initial datum $u(0)$. By repeating the same arguments we used for the Euler problem, we finally find that $u(0) - u_0$ must be a bounded harmonic function, and therefore is constant.

<figure><img src="image_2.png" /></figure>

**Remark 4.3.** This proof can easily be generalized to similar equations that describe incompressible fluids. For example, we could consider the non-resistive magnetohydrodynamic system (the next section contains a short discussion of incompressible magnetofluids)

$$ (33) \qquad \begin{cases} \partial_t u + \operatorname{div}(u \otimes u - b \otimes b) + \nabla \pi = \Delta u \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = 0 \\ \operatorname{div}(u) = 0. \end{cases} $$
As with the Navier-Stokes equations, we could ask which solutions of this system also solve the projected equation

$$ egin{cases} \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u - b \otimes b) = \Delta u \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = 0. \end{cases} $$

By doing so, we find that Theorem 4.1 may be generalized to weak solutions $(u, b) \in C^0(B_{\infty, \infty}^{-1}) \cap L^2(L^\infty) \times C^0(L^\infty)$ of (33), and that solving the projected system is equivalent to the condition $u(t) - u(0) \in S'_h$ holding for all times $t$.

We point out that problem (33) has proven to be very challenging, and it is still unknown whether local solutions even exist in the critical Besov spaces $(u, b) \in C^0(\dot{B}_{p,1}^{-1+d/p}) \cap L^1(\dot{B}_{p,1}^{1+d/p}) \times C^0(\dot{B}_{p,1}^{d/p})$ for $p > 2d$. In fact, the existence of global finite energy weak solutions remains to be proved. The well-posedness theory of this system was initiated in Sobolev spaces by Fefferman, McCormick, Robinson and Rodrigo [12], [13], and extended to Besov spaces in [7] and [17].

# 5 Elsässer variables in ideal magnetohydrodynamics

In this section, we study the ideal magnetohydrodynamic equations. These describe ideal conducting fluids which are subject to the magnetic field generated through their own motion. The ideal MHD equations read

$$ (34) \qquad \left\{ \begin{aligned} \partial_t u + \operatorname{div}(u \otimes u - b \otimes b) + \nabla (\pi + \frac{1}{2} |b|^2) &= 0 \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) &= 0 \\ \operatorname{div}(u) &= 0 \\ \operatorname{div}(b) &= 0. \end{aligned} \right. $$

In the above, $u$ and $b$ are the velocity and magnetic fields of the fluid, and $\pi$ is the usual hydrodynamic pressure. When there is no magnetic field, these equations reduce to the Euler system (17).

Formally, system (34) is overdetermined, as it is a system of $2d + 2$ equations whereas there are $2d + 1$ unknowns $(u, b, \pi)$. However, the divergence condition $\operatorname{div}(b) = 0$ is in fact redundant, because it automatically follows from the magnetic field equation, provided that the initial datum $\operatorname{div}(b_0) = 0$ is also divergence-free:

$$ \partial_t \operatorname{div}(b) = \partial_k \partial_j (u_k b_j - b_k u_j) = 0, $$

where, once again, there is an implicit summation on both repeated indices $j, k = 1, \dots, d$.

By using a suitable change of variables, we may see system (34) as a system of transport equations: define the Elsässer variables by

$$ (35) \qquad \alpha = u + b \quad \text{and} \quad \beta = u - b. $$

By adding and subtracting the momentum and magnetic field equations, we see that these variables solve the new problem

$$ (36) \qquad \left\{ \begin{aligned} \partial_t \alpha + \operatorname{div}(\beta \otimes \alpha) + \nabla \pi_1 &= 0 \\ \partial_t \beta + \operatorname{div}(\alpha \otimes \beta) + \nabla \pi_2 &= 0 \\ \operatorname{div}(\alpha) = \operatorname{div}(\beta) &= 0. \end{aligned} \right. $$

In the previous system, $\pi_1$ and $\pi_2$ are two (a priori distinct) scalar functions which enforce the independent divergence-free conditions $\operatorname{div}(\alpha) = \operatorname{div}(\beta) = 0$. In contrast to (34), the Elsässer system (36) has the same number $2d + 2$ of equations as of unknowns, which are $(\alpha, \beta, \pi_1, \pi_2)$.
This difference is explained by noting that the structure of system (36) implies that we must have, e.g. under mild integrability assumptions (see [8] and Theorem 5.4 below), the equality of the “pressure” functions $\pi_1 = \pi_2$. This is consistent with the fact that, whenever we sum and subtract the first two equations of (34), we obtain (36) with $\pi_1 = \pi_2 = \pi + \frac{1}{2}|b|^2$.

As a set of transport equations, the Elsässer system is much easier to deal with than the “classical” ideal MHD system, since (34) only is (up to the non-local pressure term) a quasi-linear symmetric hyperbolic system<fnref n="5" />. As a matter of fact, all well-posedness results obtained for the ideal MHD equations have used, in one way or another, this change of variables (see [23], [24], [18], [8]).

The catch is that, although systems (34) and (36) are closely related, they are not, strictly speaking, equivalent. In [8] (see Section 4), we had shown that equivalence between the two systems holds in an $L^p$ framework (with $p < +\infty$), and that it does not in the $L^\infty$ one. In fact, it is simple enough to find examples of functions $(\alpha, \beta)$ which solve system (36) but are such that the functions

$$ (37) \qquad u = \frac{1}{2}(\alpha + \beta) \quad \text{and} \quad b = \frac{1}{2}(\alpha - \beta) $$

do not solve (34). As we will see below, the same problem as in our study of the Euler equations arise: the space $L^\infty$ contains non-trivial polynomials.

## 5.1 Weak solutions

In this paragraph, we define the appropriate notions of weak solutions of systems (34) and (36). We start by the usual MHD system (34) before doing so for the Elsässer system.

**Definition 5.1.** Let $T > 0$ and $u_0, b_0 \in L^\infty$ be two divergence-free functions. A couple $(u, b) \in L_{\text{loc}}^2([0, T[; L^\infty)$ is said to be a weak solution of (34), related to the initial data $(u_0, b_0)$ if

(i) the momentum equation is satisfied in the weak sense: for all divergence-free test function $\phi \in \mathcal{D}([0, T[\times \mathbb{R}^d; \mathbb{R}^d)$, we have

$$ \int_0^T \int \{u \cdot \partial_t \phi + (u \otimes u - b \otimes b) : \nabla \phi\} dx dt + \int u_0 \cdot \phi(0) dx = 0; $$

(ii) the magnetic field equation is satisfied in the weak sense: for all $\varphi \in \mathcal{D}([0, T[\times \mathbb{R}^d; \mathbb{R}^d)$,

$$ \int_0^T \int \{b \cdot \partial_t \varphi + (u \otimes b - b \otimes u) : \nabla \varphi\} dx dt + \int b_0 \cdot \varphi(0) dx = 0; $$

(iii) the divergence-free condition $\text{div}(u) = 0$ holds in $\mathcal{D}'([0, T[\times \mathbb{R}^d)$.

**Definition 5.2.** Let $T > 0$ and $\alpha_0, \beta_0 \in L^\infty$ be two divergence-free functions. A couple $(\alpha, \beta) \in L_{\text{loc}}^2([0, T[; L^\infty)$ is said to be a weak solution of (36) related to the initial data $(\alpha_0, \beta_0)$ if

(i) the equation for $\alpha$ is satisfied in the weak sense: for all divergence-free $\phi \in \mathcal{D}([0, T[\times \mathbb{R}^d)$, we have

$$ \int_0^T \int \{\alpha \cdot \partial_t \phi + (\beta \otimes \alpha) : \nabla \phi\} dx dt + \int \alpha_0 \cdot \phi(0) dx = 0, $$

and likewise for $\beta$;

<footnote n="5">⁵In contrast with transport equations, the basic theory of such systems only exists in $L^2$ based spaces. We refer to [3] for more insight on the this: a symmetric hyperbolic system with constant coefficients is well-posed in $L^p$, $p \neq 2$, if and only if the matrix coefficients all commute.</footnote>
(ii) both divergence-free conditions $\mathrm{div}(\alpha) = \mathrm{div}(\beta) = 0$ hold in $\mathcal{D}'(]0, T[ \times \mathbb{R}^d)$.

**Remark 5.3.** As above with the Euler system, we have the same ambiguity concerning initial data. If $(u, b)$ is as in Definition 5.1, then there is no need for $u_0$ and $u(0)$ to be the same, even if the solution lies in $C_T^1(L^\infty)$. Likewise, there is no reason for $u$ to be continuous with respect to time. However, the magnetic field has a much more pleasant behavior: it automatically lies in $W_T^{1,1}(W^{-1,\infty})$, and we have $b_0 = b(0)$.

As for the Elsässer variables $(\alpha, \beta)$ in Definition 5.2, they share the same problems the Euler solutions may have: both $\alpha$ and $\beta$ may be discontinuous with respect to time.

## 5.2 Equivalence of the formulations

**Theorem 5.4.** Let $T > 0$, $\alpha_0, \beta_0 \in L^\infty$ and $(\alpha, \beta) \in C^0([0, T[; L^\infty)$. Define $(u, b)$ as in (37), and $(u_0, b_0)$ accordingly. The assertions below are true.

1. Assume that $(u, b)$ is a weak solution of (34) that is related to the initial data $(u_0, b_0)$, in the sense of Definition 5.1. Then $(\alpha, \beta)$ solves the Elsässer system (36) with initial data $(\alpha_0, \beta_0)$ and $\pi_1 = \pi_2 = \pi + \frac{1}{2}|b|^2$.

2. Assume that $(\alpha, \beta)$ is a weak solution of (36) for the initial data $(\alpha_0, \beta_0)$. Then the following statements are equivalent:

(i) the functions $(u, b)$ solve the “classical” MHD system (34) with initial data $(u_0, b(0))$, and the difference $b(0) - b_0$ is a constant function;

(ii) we have, for all times $t \in [0, T[, b(t) - b(0) \in S'_h$.

In addition, if one of these equivalent conditions is fulfilled, then the pressure functions $\pi_1$ and $\pi_2$ are equal up to a constant summand.

*Proof.* Because the proof of this theorem bears strong similarities with that of Theorem 3.6, we only give an outline of the arguments.

If $(u, b)$ is a weak solution of (34), then adding and subtracting the weak forms of the momentum and magnetic field equations makes it clear that $(\alpha, \beta)$ solves the Elsässer system with initial data $(\alpha_0, \beta_0)$ with $\pi_1 = \pi_2$.

On the other hand, assume that $(\alpha, \beta)$ is a solution of the Elsässer system (36), according to Definition 5.2, and with the initial data $(\alpha_0, \beta_0)$. As in the proof of Theorem 3.6, extend the solution $(\alpha, \beta, \pi_1, \pi_2)$ to $\mathbb{R} \times \mathbb{R}^d$ and continue to note $(\alpha, \beta, \pi_1, \pi_2)$ the extension. By subtracting both equations for $\alpha$ and $\beta$, we see that the magnetic field satisfies

$$ \partial_t b + \mathrm{div}(u \otimes b - b \otimes u) = \frac{1}{2} \nabla(\pi_2 - \pi_1) + \delta_0(t) \otimes b_0(x). $$

As with $C_T^0(L^\infty)$ solutions of the Euler system, we apply the same mollification sequence $(K_\epsilon)_{\epsilon>0}$ in order to deal with low regularity in the time variable. We have, by noting $(u_\epsilon, b_\epsilon) = K_\epsilon * (u, b)$ the convolution with respect to time,

$$ \partial_t b_\epsilon + K_\epsilon * \mathrm{div}(u \otimes b - b \otimes u) = \frac{1}{2} K_\epsilon * \nabla(\pi_2 - \pi_1) + K_\epsilon(t) b_0(x) \quad \text{in } ]-\infty, T - \epsilon[ \times \mathbb{R}^d. $$

By taking the divergence of this equation, we see that $K_\epsilon * \Delta(\pi_2 - \pi_1) = 0$, hence the functions $Q_\epsilon(t) = K_\epsilon * (\pi_2 - \pi_1)(t) \in \mathbb{R}[x]$ are harmonic polynomials. As in the proof of Theorem 3.6, we see that $\nabla Q_\epsilon = K_\epsilon * \nabla Q$ for some distribution $Q \in \mathcal{D}'(\mathbb{R} \times \mathbb{R}^d)$. We must find a condition under which we have $\nabla Q = 0$ in $]0, T[ \times \mathbb{R}^d$.
We now notice that the quadratic term in the magnetic field equation lies in $S'_h$, as shown by the first Bernstein inequality: for all times $t \in ]0, T[$,

$$ \|\chi(\lambda D) \operatorname{div}(u \otimes b - b \otimes u)\|_{L^{\infty}} = O\left(\frac{1}{\lambda}\right) \quad \text{as } \lambda \to +\infty. $$

Therefore, by arguing with $b$ exactly as with $u$ at the end of the proof of Theorem 3.6, we get the equivalence of points (i) and (ii). □

## 5.3 Further remarks on the ideal MHD system

As with the Euler equations, well-posedness of a projected version of the ideal MHD system has been studied in the critical Besov space $B_{\infty,1}^1$ in [18]. In the ideal MHD equations (34), the pressure only appears in the momentum equation. This means that, in order to apply the Leray projection we must require the velocity field $u$ to satisfy a special condition. The resulting system is

$$ (38) \qquad \begin{cases} \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u - b \otimes b) = 0 \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = 0, \end{cases} $$

whose well-posedness has been established in [18]. The following result is the MHD analog of Corollary 3.15.

**Corollary 5.5.** Let $(u_0, b_0) \in B_{\infty,1}^1$ be a set of divergence-free initial data. There exists a time $T > 0$ such that the ideal MHD system (34) has a unique solution $(u, b) \in C^0([0, T[; B_{\infty,1}^1)$, associated to some pressure function and related to the initial data $(u_0, b_0)$, such that

$$ u(0) = u_0 \quad \text{and} \quad u(t) - u(0) \in S'_h \text{ for } t \in [0, T[. $$

Moreover, this unique solution has $C^1$ regularity with respect to time $(u, b) \in C^1([0, T[; B_{\infty,1}^0)$.

*Proof.* Existence and uniqueness of $C_T^0(B_{\infty,1}^1)$ solutions has been proved in [18] for the projected MHD problem

$$ (39) \qquad \begin{cases} \partial_t u + \mathbb{P} \operatorname{div}(u \otimes u - b \otimes b) = 0 \\ \partial_t b + \operatorname{div}(u \otimes b - b \otimes u) = 0. \end{cases} $$

Solutions of this system are obviously solutions of the original MHD problem (34), so that we need only prove the uniqueness result. Let $(u, b)$ be a weak solution of (34) that fulfills the assumptions of our statement. By replicating (with almost no modification) the proof of Theorem 3.6, we see that $(u, b)$ solves the projected system (38), whose $C_T^0(B_{\infty,1}^1)$ solutions are unique, with initial data $(u_0, b_0)$. □

**Remark 5.6.** The projected system (38) is trivially equivalent to a projected system for the Elsässer variables, namely

$$ (40) \qquad \begin{cases} \partial_t \alpha + \mathbb{P} \operatorname{div}(\beta \otimes \alpha) = 0 \\ \partial_t \beta + \mathbb{P} \operatorname{div}(\alpha \otimes \beta) = 0. \end{cases} $$

However, solutions $(\alpha, \beta) \in C_T^0(L^\infty)$ of the Elsässer system, as in Definition 5.2, solve these projected equations, with appropriate initial data, if and only if

$$ (\alpha(0), \beta(0)) = (\alpha_0, \beta_0) \quad \text{and} \quad (\alpha(t) - \alpha(0), \beta(t) - \beta(0)) \in S'_h \text{ for } t \in [0, T[, $$
so that Theorem 5.4 remains a necessary step to recast the Elsässer system (34) into the classical MHD one. Relations between these different systems are summarized in the following diagram.

<figure><img src="image_3.png" /></figure>

In this diagram, Els stands for the Elsässer system (36) while MHD stands for the classical MHD system (34). The names $\mathbb{P}(\text{Els})$ and $\mathbb{P}(\text{MHD})$ refer to, respectively, the projected equations (40) and (38). The arrows are labeled with the conditions required to pass from a system to another, bare arrows need no conditions.

## A Appendix: Fourier multipliers at high frequencies

As explained in the introduction, the Leray projector is a singular integral operator, so Calderón-Zygmund theory provides boundedness $\mathbb{P}: L^p \to L^p$ for $1 < p < +\infty$. In the limit cases $p \in \{1, +\infty\}$, this is no longer true, and one has to resort to other functional spaces.

A fairly obvious reason for this breakdown is the singularity of the symbol $m(\xi)$, which is not continuous at $\xi = 0$. If $f \in L^1$, then $m(\xi)\hat{f}(\xi)$ cannot be the Fourier transform of an $L^1$ function (or, in fact, of a finite measure) since it is discontinuous at $\xi = 0$ if $\hat{f}(0) \neq 0$.

To short-circuit this problem, a straightforward idea is to cut-off low frequencies, and study the $L^1$ boundedness of the operator

$$T = (\mathrm{Id} - \Delta_{-1})m(D).$$

Unfortunately, it turns out that $T$ is not bounded on $L^1$, as we will explain in Proposition A.1. This shows that the Leray projector may also have an unpleasant behavior due to high frequencies. That boundedness issues arise because of both low and high frequencies makes the Littlewood-Paley decomposition an extremely natural tool to deal with endpoint exponents, as annuli are isolated from both $\xi = 0$ and $|\xi| \to +\infty$.

**Proposition A.1.** Let $\chi \in \mathcal{D}$ be such that $\chi(\xi) = 1$ for $|\xi|$ small enough and consider a bounded, non-constant and homogeneous function $\sigma(\xi)$ of degree zero. Then, the operator

$$T = (\mathrm{Id} - \chi(D))\sigma(D)$$

is not bounded in the $L^1 \to L^1$ topology.

*Proof.* We assume, in order to obtain a contradiction, that the operator $T: L^1 \to L^1$ is bounded. By scaling, we see that the family of operators

$$(41) \qquad T_{\lambda} = (\mathrm{Id} - \chi(\lambda D))\sigma(\lambda D) = (\mathrm{Id} - \chi(\lambda D))\sigma(D), \quad \text{where } \lambda > 0,$$

is uniformly bounded on $L^1$. Therefore, for any $f \in L^1$, the family of functions $(T_{\lambda}f)_{\lambda>0}$ is bounded in the space $\mathcal{M}$ of finite Borel measures on $\mathbb{R}^d$. Now, $\mathcal{M}$ is the dual of the space $C_0$ of continuous functions which have limit 0 at $|x| \to +\infty$ (we refer to Theorem 6.19, pp. 129–132 in [21] for a proof of this fact). Since the space $C_0$ is separable, we obtain, up to an extraction, convergence of the sequence $(T_{\lambda}f)_{\lambda>0}$ in the weak-(*) topology of $\mathcal{M}$, as $\lambda \to +\infty$. There exists a $\mu \in \mathcal{M}$ and an increasing sequence $\lambda_k \nearrow +\infty$ such that

$$T_{\lambda_k} f \stackrel{*}{\rightharpoonup} \mu \quad \text{in } \mathcal{M}.$$
Formally, we expect the Fourier transform of $\mu$ to be $\hat{\mu}(\xi) = \sigma(\xi)\hat{f}(\xi)$. This would be the contradiction we seek, as $\hat{\mu}(\xi)$ is a continuous function, while $\sigma(\xi)\hat{f}(\xi)$ cannot be so if $\hat{f}(0) \neq 0$, as the symbol $\sigma(\xi)$ is discontinuous at $\xi = 0$.

To show that we indeed have $\hat{\mu} = \sigma\hat{f}$, we take $\phi \in S$ such that $\hat{\phi}$ is compactly supported away from $\xi = 0$. Then, for $\lambda$ large enough, the support of $\chi(\lambda\xi)$ is disjoint from that of $\hat{\phi}(\xi)$. Therefore, for such $\lambda$, we have

$$ \langle \sigma(\xi) \hat{f}(\xi), (1 - \chi(\lambda\xi)) \hat{\phi}(\xi) \rangle = \langle \sigma(\xi) \hat{f}(\xi), \hat{\phi}(\xi) \rangle. $$

This shows that the difference $\hat{\mu}(\xi) - \sigma(\xi)\hat{f}(\xi)$ is supported at $\xi = 0$ and ends the proof of the proposition. □

**Remark A.2.** In the proof of the previous proposition, we have in fact proven that the operator $T$ is not bounded in the $L^1 \to \mathcal{M}$ topology.

# B Appendix: On the space $S'_h$

This paragraph revolves around the definition of homogeneous Besov spaces. As we have pointed out above, in the functional analysis literature, these are usually defined as spaces of distributions modulo polynomials

$$ (42) \qquad \dot{B}_{p,r}^s = \{ f \in S' / \mathbb{R}[x], \quad \|f\|_{\dot{B}_{p,r}^s} < +\infty \}, $$

where the $\dot{B}_{p,r}^s$ norm is as in Definition 2.11. We refer to [22] for background on these spaces. In contrast, when studying PDEs, one rather defines Besov spaces as the set of $f \in S'_h$ such that $\|f\|_{\dot{B}_{p,r}^s}$ is finite, these definitions being equivalent as long as the supercriticality condition (16) is satisfied. However, this is no longer true for subcritical exponents, the main reason for this being that the natural map $S'_h \hookrightarrow S'/\mathbb{R}[x]$ is not onto.

We point out that, although all this seems well-known, our personal research has not allowed us to find a proof of this last fact in the mathematical literature. In addition, as we have noted in Remark 2.6 above, several non-equivalent definitions of the space $S'_h$ coexist. In Proposition B.1 below, we prove that the strict inclusion $S'_h \oplus \mathbb{R}[x] \subsetneq S'$ holds, even with the weakest definition possible of $S'_h$ (that is with Definition 2.5 above).

**Proposition B.1.** We have a strict inclusion $(L^\infty \cap S'_h) \oplus \mathbb{R}1 \subsetneq L^\infty$, where we have noted $1 : x \mapsto 1$ the constant function. In particular, we also deduce that $S'_h \oplus \mathbb{R}[x] \subsetneq S'$.

**Remark B.2.** It is fairly easy to obtain a strict inclusion $(L^\infty \cap S'_h) \oplus \mathbb{R}1 \subsetneq L^\infty$ if the space $S'_h$ is rather defined by the strong convergence

$$ \chi(\lambda D) f \to 0 \quad \text{in } L^\infty, $$

as the sign function $\sigma = 1_{\mathbb{R}_+} - 1_{\mathbb{R}_-}$ provides an example of a function for which the convergence $\chi(\lambda D)\sigma \to 0$ takes place in $S'$ but not in $L^\infty$. Indeed, if $\psi_\lambda$ is the kernel of the $\chi(\lambda D)$ operator, as in Proposition 3.4, then dominated convergence yields

$$ \chi(\lambda D)\sigma(x) = \int_{-\infty}^{+\infty} \sigma(x-y)\psi_\lambda(y)dy \to \pm \int_{-\infty}^{+\infty} \psi(y)dy \quad \text{as } x \to \pm\infty. $$

In particular, we always have $\|\chi(\lambda D)\sigma\|_{L^\infty} = 1$ and the convergence $\chi(\lambda D)\sigma \to 0$ cannot hold with respect to the uniform norm.
**Remark B.3.** As a consequence of Proposition B.1, we see that the subcritical homogeneous Besov space $\dot{B}_{\infty, \infty}^0$, defined as in (42), cannot be realized as a subspace of $S'_h$. In other words, there is no linear map $j: \dot{B}_{\infty, \infty}^0 \to E$ to a subspace $E \subset S'_h$ such that the following diagram commutes:

<figure><img src="image_4.png" /></figure>

In the above, $p$ is the natural projection $E \subset S' \to S'/\mathbb{R}[x]$. It is simply a matter of noting that any function $f \in L^\infty$ such that $f \notin (L^\infty \cap S'_h) \oplus \mathbb{R}1$ defines an element of $L^\infty/\mathbb{R}1 \hookrightarrow \dot{B}_{\infty, \infty}^0$ which cannot be mapped to an element of $S'_h$ by $j$.

*Proof of Proposition B.1.* Before starting the proof, we make a few comments on the problem. Recall, from Definition 2.5, that the space $S'_h$ is the set of (tempered) distributions $f$ such that we have the weak convergence

$$ \chi(\lambda D)f \to 0 \quad \text{in } S'. $$

Now, for any $f \in L^\infty$, the sequence of functions $(\chi(\lambda D)f)_{\lambda>0}$ is bounded in $L^\infty$. Since $L^1$, the predual space of $L^\infty$, is separable, we thus have the convergence of an extraction in the weak-(*) topology: there is a sequence $\lambda_k \nearrow +\infty$ and an accumulation point $Q \in L^\infty$ such that

$$ \chi(\lambda_k D)f \stackrel{*}{\rightharpoonup} Q \quad \text{in } L^\infty. $$

Since the support of the Fourier transform $\chi(\lambda\xi)\hat{f}(\xi)$ decreases to zero, the accumulation point $Q$ must be spectrally supported at $\xi = 0$, which implies it is a constant function $Q \in \mathbb{R}1$. Therefore, we see that $f \in (L^\infty \cap S'_h) \oplus \mathbb{R}1$ if and only if $Q$ is the only accumulation point of the $\chi(\lambda D)f$ in the weak-(*) topology of $L^\infty$.

Taking these arguments into consideration, we will construct a function $f$ such that the $\chi(\lambda D)f$ have at least two different accumulation points. Understanding the nature of the operator $\chi(\lambda D)$ will be of great help: it is a convolution operator whose kernel is $\psi_\lambda(x) = \lambda^{-d}\psi(x/\lambda)$, where $\psi$ is as in Proposition 3.4. Since most of the mass of $\psi_\lambda$ is concentrated in a ball of radius $O(\lambda)$, we need to find a function $f$ whose average value over balls of radius $O(\lambda)$ will oscillate between two values as $\lambda \to +\infty$.

This being said, we define a function $f$ that is suited to our purposes: let, for $m \ge 0$, $C_m$ be the annulus given by

$$ C_m = \{x \in \mathbb{R}^d, \quad 2^{m^2} \le |x| \le 2^{(m+1)^2}\} $$

and set

$$ f = \sum_{m \ge 0} (-1)^m \mathbb{1}_{C_m}. $$

We will prove that both $-1$ and $1$ are accumulation points of the $\chi(\lambda D)f$ as $\lambda \to +\infty$. In order to simplify matters, we take advantage of the fact that the weak-(*) topology of the unit ball $B_1(L^\infty)$ is metrizable: let $(g_n)_{n \ge 1}$ be a sequence of $L^1$ functions that form a dense set in the unit ball $B_1(L^1)$, and define

$$ (43) \qquad \forall f_1, f_2 \in B_1(L^\infty), \quad d(f_1, f_2) = \sum_{n=1}^{\infty} \frac{1}{2^n} \left| \langle f_2 - f_1, g_n \rangle_{L^\infty \times L^1} \right|. $$

Then the metric topology of $(B_1(L^\infty), d)$ is the same as the weak-(*) one, and all the $\chi(\lambda D)f$ are (possibly up to an irrelevant multiplicative constant) elements of $B_1(L^\infty)$. We must therefore prove that, for any $\epsilon > 0$ and $\Lambda > 0$, there are values of $\lambda \ge \Lambda$ such that $d(\chi(\lambda D)f, \pm 1) \le \epsilon$.
We first restrict our attention to a finite number of the $g_n$ and a bounded part of $\mathbb{R}^d$. Let $N \ge 1$ such that $\sum_{n>N} 2^{-n} \le \epsilon$ and fix a radius $R > 0$ such that

$$ (44) \qquad \max_{1 \le n \le N} \| \mathbf{1}_{|x| \ge R} g_n(x) \|_{L^1} \le \epsilon. $$

We will have reached our purpose if we show that, by taking $\lambda$ large enough, the function $\chi(\lambda D)f$ can be made $\epsilon$-close to $\sigma = \pm 1$ on the ball $B_R$ and for the uniform norm, as, in that case,

$$ (45) \qquad \begin{aligned} d(\chi(\lambda D)f, \sigma) &\le \epsilon + \sum_{n=1}^{N} \frac{1}{2^n} \left( \|\chi(\lambda D)f - \sigma\|_{L^\infty(B_R)} \|g_n\|_{L^1} + 2\|g_n\|_{L^1(B_R)} \right) \\ &\le 2\epsilon + \|\chi(\lambda D)f - \sigma\|_{L^\infty(B_R)} \end{aligned} $$

We pick a sign $\sigma = \pm 1$ and we fix a $M \ge 0$ such that $R \le M^2$ and $\sigma = (-1)^M$. Split $f$ into three parts: a first one on $|x| \le 2^{M^2}$, a second on $|x| > 2^{(M+1)^2}$, both of which will have a negligible contribution to $\chi(\lambda D)f$ for $\lambda$ of appropriate size, and a main term. We obtain

$$ \begin{aligned} f &= \sum_{m<M} (-1)^m \mathbf{1}_{C_m} + (-1)^M \mathbf{1}_{C_M} + \sum_{m>M} (-1)^m \mathbf{1}_{C_m} \\ &= ((-1)^{M+1} \mathbf{1}_{B_M} + \sum_{m<M} (-1)^m \mathbf{1}_{C_m}) + (-1)^M \mathbf{1}_{B_{M+1}} + \sum_{m>M} (-1)^m \mathbf{1}_{C_m} \\ &:= G_M + \sigma \mathbf{1}_{B_{M+1}} + H_M, \end{aligned} $$

where, for the sake of simplicity, we have noted $B_M = B_{2M^2}$ the ball $|x| \le 2^{M^2}$ and likewise for $B_{M+1}$. The first summand $G_M$ yields simple bounds. By the Hausdorff-Young convolution inequality, we have

$$ (46) \qquad \|\chi(\lambda D)G_M\|_{L^\infty} \le \|G_M\|_{L^1} \|\psi_\lambda\|_{L^\infty} \le 2 \left( \frac{2^{M^2}}{\lambda} \right)^d. $$

This upper bound will be smaller than $2\epsilon$ provided that $\lambda$ satisfies $\lambda \ge \epsilon^{-1/d} 2^{M^2}$. On the other hand, to bound the last summand $H_M$, we need to know how convolution by $\psi_\lambda$ and $H_M$ will affect $\chi(\lambda D)f(x)$ in the ball $|x| \le R$. Fix $A > 0$ such that

$$ (47) \qquad \| \mathbf{1}_{|y| \ge A} \psi(y) \|_{L^1} = \| \mathbf{1}_{|y| \ge \lambda A} \psi_\lambda(y) \|_{L^1} \le \epsilon. $$

We are interested in estimating the convolution product

$$ \psi_\lambda * H_M(x) = \int H_M(y) \psi_\lambda(x - y) dy $$

for $|x| \le R$. Since the function $H_M(y)$ is supported at $|y| \ge 2^{(M+1)^2}$, the condition $2^{(M+1)^2} - \lambda A \ge R$ implies that $|x - y| \ge \lambda A$, and so

$$ (48) \quad \forall |x| \le R, \quad 2^{(M+1)^2} - \lambda A \ge R \quad \Rightarrow \quad |\psi_\lambda * H_M(x)| \le \|H_M\|_{L^\infty} \|\mathbf{1}_{|y| \ge \lambda A} \psi_\lambda(y)\|_{L^1} \le \epsilon. $$

We may now focus on the main term, namely $\sigma\chi(\lambda D)\mathbf{1}_{B_{M+1}}$. We split the convolution product according to whether $\psi_\lambda$ is large or not. For any $|x| \le R$, we have

$$ (49) \quad \psi_\lambda * \mathbf{1}_{B_{M+1}}(x) = \int_{|x-y| > \lambda A} \mathbf{1}_{|y| \le 2^{(M+1)^2}} \psi_\lambda(x-y) dy + \int_{|x-y| \le \lambda A} \mathbf{1}_{|y| \le 2^{(M+1)^2}} \psi_\lambda(x-y) dy. $$
The first integral is bounded by $\epsilon$, thanks to (47). As for the second one, the condition $2^{(M+1)^2} - \lambda A \ge R$, which we have already used in (48), implies that we always have $|y| \le 2^{(M+1)^2}$ in the second integral. Therefore

$$ (50) \qquad \int_{|x-y| \le \lambda A} \mathbb{1}_{|y| \le 2^{(M+1)^2}} \psi_{\lambda}(x-y) dy = \int_{|x-y| \le \lambda A} \psi_{\lambda}(x-y) dy \in [1-\epsilon, 1+\epsilon]. $$

Putting all of (46), (48), (49) and (50) together, we see that we have

$$ (51) \qquad \forall |x| \le R, \qquad |\chi(\lambda D)f(x) - \sigma| \le C\epsilon $$

under the condition that $\lambda$ satisfies

$$ (52) \qquad \frac{2^{M^2}}{\epsilon^{1/d}} \le \lambda \le \frac{1}{A} (R + 2^{(M+1)^2}). $$

The respective growth of $2^{M^2}$ and $2^{(M+1)^2}$ insures that this condition is not void for large $M$: for any $\epsilon > 0$ and any sign $\sigma = \pm 1$, we may indeed find arbitrarily large $\lambda$ such that (52) holds. By using this in inequality (45), we finally get

$$ d(f, \sigma) \le C\epsilon, $$

which in turn proves that $\sigma$ is an accumulation point of the sequence in the weak-(*) topology. $\square$

**Remark B.4.** The function $f \notin (L^\infty \cap S'_h) \oplus \mathbb{R}1$ we have constructed in the proof above is not continuous. However, this is by no means related to the nature of the our argument: the function $\chi(D)f$ has exactly the same properties while being analytic and bounded.

## References

[1] H. Bahouri, J.-Y. Chemin and R. Danchin: “Fourier analysis and nonlinear partial differential equations”. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Scinences), Springer, Heidelberg, 2011.

[2] G. Bourdeau: *Réalisation des espaces de Besov homogènes*. Arkiv för Mathematik, **26**, 1988, pp. 41–54.

[3] P. Brenner: *The Cauchy problem for symmetric hyperbolic systems in $L_p$. Math. Scand.* **19** (1966), pp. 27–37.

[4] J.-Y. Chemin: “Perfect incompressible fluids”. Oxford Lecture Series in Mathematics and its Applications, **15**, The Clarendon Press, Oxford University Press, New York, 1998.

[5] J.-Y. Chemin: *Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel*. J. Anal. Math. 77 (1999), pp. 27–50.

[6] J.-Y. Chemin: *Localization in Fourier space and Navier-Stokes system*. (Lecture notes, Di Giorgi center), Phase space analysis of partial differential equations. Vol. I, 53–135, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004.

[7] J.-Y. Chemin, D. McCormick, J. Robinson and J. Rodrigo: *Local existence for the non-resistive MHD equations in Besov spaces*. Adv. Math. 286 (2016), pp. 1–31.

[8] D. Cobb and F. Fanelli: *Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions*. arXiv:2009.11230 (2020). Submitted.

[9] R. Danchin: *Fourier Analysis Methods for PDEs*. (Lecture notes, Wuhan, Beijing), 2005. Available at https://perso.math.u-pem.fr/danchin.raphael/cours/courschine.pdf

[10] C. De Lellis and L. Székelyhidi Jr. *Dissipative continuous Euler flows*. Invent. Math. 193 (2013), n. 2, pp. 377–407.
[11] T. M. Elgindi and N. Masmoudi: *$L^\infty$ ill-posedness for a class of equations arising in hydrodynamics*. Arch. Ration. Mech. Anal. **235** (2020), no. 3, 1979–2025.

[12] C. Fefferman, D.S. McCormick, J.C. Robinson and J.L. Rodrigo: *Higher order commutator estimates and local existence for the non-resistive MHD equations and related models*. J. Funct. Anal. **267** (2014), n. 4, pp. 1035–1056.

[13] C. Fefferman, D.S. McCormick, J.C. Robinson and J.L. Rodrigo: *Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces*. Arch. Ration. Mech. Anal. **223** (2017), n. 2, pp. 677–691.

[14] T. Gallay: *Infinite energy solutions of the two-dimensional Navier-Stokes equations*. (Lecture notes), Annales de la Faculté des Sciences de Toulouse **26** (2017), 979–1027.

[15] H. Koch and D. Tataru: *Well-posedness for the Navier-Stokes equations*. Adv. Math. **157** (2001), n. 1, pp. 22–35.

[16] P. G. Lemarié-Rieusset: *"Recent developments in the Navier-Stokes problem"*. Chapman & Hall/CRC Research Notes in Mathematics, n. 431. Chapman & Hall/CRC, Boca Raton, FL, 2002.

[17] J. Li, W. Tan and Z. Yin: *Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces*. Adv. Math. **317** (2017), pp. 786–798.

[18] C. Miao and B. Yuan: *Well-posedness of the ideal MHD system in critical Besov spaces*. Methods Appl. Anal., **13** (2006), n. 1, pp. 89–106.

[19] S. Montgomery-Smith: *Finite time blow up for a Navier-Stokes like equation*. Proc. Amer. Math. Soc. **129** (2001), n. 10, pp. 3025–3029.

[20] H. C. Pak and Y. J. Park: *Existence of solutions for the Euler equations in a critical Besov space $B_{\infty,1}^{1}(\mathbb{R}^n)$*. Comm. Partial Differential Equations **29** (2004), n. 7-8, pp. 1149–1166.

[21] W. Rudin: *Real and complex analysis*. Third edition. McGraw-Hill Book Co., New York, 1987.

[22] Y. Sawano: *Homogeneous Besov spaces*. Kyoto J. Math. **60** (2020), n. 1, pp. 1–43.

[23] P. G. Schmidt: *On a magnetohydrodynamic problem of Euler type*. J. Differential Equations, **74** (1988), n. 2, pp. 318–335.

[24] P. Secchi: *On the equations of ideal incompressible magnetohydrodynamics*. Rend. Sem. Mat. Univ. Padova, **90** (1993), pp. 103–119.

[25] E. M. Stein: "Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals". With the assistance of Timothy S. Murphy. Princeton Mathematical Series, **43**. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.