# Special macroscopic modes and hypocoercivity

Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau, Stéphane Mischler, Clément Mouhot, Christian Schmeiser

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Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau, Stéphane Mischler, Clément Mouhot, et al. Special macroscopic modes and hypocoercivity. Journal of the European Mathematical Society, In press. hal-03222748v4

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Special macroscopic modes and hypocoercivity

Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau, Stéphane Mischler,
Clément Mouhot, Christian Schmeiser

► To cite this version:

Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau, Stéphane Mischler, Clément Mouhot, et al. Special macroscopic modes and hypocoercivity. Journal of the European Mathematical Society, In press, 10.4171/JEMS/1502. hal-03222748v5

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https://hal.science/hal-03222748v5

Submitted on 10 Jul 2024

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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
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Distributed under a Creative Commons Attribution 4.0 International License
Kleber Carrapatoso · Jean Dolbeault · Frédéric Hérau · Stéphane

Mischler · Clément Mouhot · Christian Schmeiser

Special macroscopic modes and hypocoercivity

**Abstract.** We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

**Keywords.** Hypocoercivity; linear kinetic equations; collision operator; transport operator; collision invariant; local conservation laws; micro/macro decomposition; commutators; hypoellipticity; confinement potential; rotations; rotational invariance; symmetries; global conservation laws; spectral gap; Poincaré-Korn inequality; Witten-Laplace operator; partially harmonic potential; time-periodic solutions; classification of steady states; special macroscopic modes

1. Introduction

Since the publication of Boltzmann's paper [5] in 1876, the existence of time-periodic steady states of the inhomogeneous Boltzmann equation in the whole Euclidean space

Kleber Carrapatoso: Centre de mathématiques Laurent Schwartz, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France; kleber.carrapatoso@polytechnique.edu

Jean Dolbeault: Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR n° 7534), Universités PSL & Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France; dolbeaul@ceremade.dauphine.fr

Frédéric Hérau: Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, 2, rue de la Houssinière BP 92208 F-44322, France; frederic.herau@univ-nantes.fr

Stéphane Mischler: Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR n° 7534), Universités PSL & Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France; mischler@ceremade.dauphine.fr

Clément Mouhot: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK; C.Mouhot@dpmms.cam.ac.uk

Christian Schmeiser: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria; Christian.Schmeiser@univie.ac.at

Mathematics Subject Classification (2020): Primary 82C40; Secondary 76P05; 35Q83; 82C70
Kleber Carrapatoso · Jean Dolbeault · Frédéric Hérau · Stéphane

Mischler · Clément Mouhot · Christian Schmeiser

# Special macroscopic modes and hypocoercivity

**Abstract.** We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

**Keywords.** Hypocoercivity; linear kinetic equations; collision operator; transport operator; collision invariant; local conservation laws; micro/macro decomposition; commutators; hypoellipticity; confinement potential; rotations; rotational invariance; symmetries; global conservation laws; spectral gap; Poincaré-Korn inequality; Witten-Laplace operator; partially harmonic potential; time-periodic solutions; classification of steady states; special macroscopic modes

## 1. Introduction

Since the publication of Boltzmann's paper [5] in 1876, the existence of time-periodic steady states of the inhomogeneous Boltzmann equation in the whole Euclidean space

Kleber Carrapatoso: Centre de mathématiques Laurent Schwartz, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France; kleber.carrapatoso@polytechnique.edu

Jean Dolbeault: Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR n° 7534), Universités PSL & Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France; dolbeaul@ceremade.dauphine.fr

Frédéric Hérau: Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, UMR 6629, 2, rue de la Houssinière BP 92208 F-44322, France; frederic.herau@univ-nantes.fr

Stéphane Mischler: Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR n° 7534), Universités PSL & Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France; mischler@ceremade.dauphine.fr

Clément Mouhot: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK; C.Mouhot@dpmms.cam.ac.uk

Christian Schmeiser: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria; Christian.Schmeiser@univie.ac.at

*Mathematics Subject Classification (2020): Primary 82C40; Secondary 76P05; 35Q83; 82C70*
is known, in presence of an external harmonic potential. As explained in [10, p. 147], “equilibrium is not necessarily achieved in an harmonic field. [...] [D]ensity, velocity and temperature oscillate with the natural frequency of the field or with twice such a frequency.” Beyond such remarks, the classification of the steady states according to the symmetries of the domain or the symmetries of the external potential remained untouched for more than a century, although some special solutions were known [10,44]. When symmetry partially or completely breaks, this turns out to be a difficult issue. With symmetry, special modes have to be taken into account in some configurations and local collision laws of the collision operator add significant difficulties to the understanding of the convergence in asymptotic regimes in all cases, even if there is no particular symmetry.

Without external potential and for a bounded domain, the problem has been studied in [12]. In presence of a given external potential, the question was so far open and our first result is to classify all steady solutions for linear kinetic equations with collision operators satisfying the local conservation laws of physics. Even more difficult is the problem of the stability of the (possibly time-periodic) steady states and the proof of the convergence to such states, with an exponential rate, for inhomogeneous kinetic equations. The question goes back to the celebrated H-theorem of Boltzmann, but became quantitative only recently with the theory of hypocoercivity. All results involving an external potential deal with collision operators admitting only one collision invariant, up to a few attempts like [15,16] which discard special modes, with non-constructive methods. Our second result gives the very first answer to the question of the convergence rate in the whole space for an external potential without any a priori symmetry, using an entirely new scheme made of a cascade of several hypocoercive estimates. Alternatively we also propose a commutator method in the spirit of [27,46].

Even when the potential has no specific symmetry, which forbids the existence of any special mode other than the standard stationary solution, the fact that the collision operator admits several collision invariants is a source of difficulties: when the potential is almost symmetric, convergence rates get deteriorated and the geometric properties of the potential have therefore to be taken into account. The notion of steady states, defined as the set of attractors in large time asymptotics, is widely used in physics, and corresponds in our case to minimizers of the mathematical entropy (that is, the physical entropy, up to the sign). In this paper we shall speak of special macroscopic modes in relation with special symmetries of the potential.

## 1.1. Equation and assumptions

Consider the kinetic equation

$$ \partial_t f = \mathcal{L} f := \mathcal{T} f + \mathcal{C} f, \quad f|_{t=0} = f_0, \qquad (1.1) $$

for the unknown distribution function $f = f(t, x, v)$ depending on the time variable $t \ge 0$, the position variable $x \in \mathbb{R}^d$, and the velocity variable $v \in \mathbb{R}^d$, where $d \ge 1$ is an arbitrary
is known, in presence of an external harmonic potential. As explained in [10, p. 147], “equilibrium is not necessarily achieved in an harmonic field. [...] [D]ensity, velocity and temperature oscillate with the natural frequency of the field or with twice such a frequency.” Beyond such remarks, the classification of the steady states according to the symmetries of the domain or the symmetries of the external potential remained untouched for more than a century, although some special solutions were known [10,44]. When symmetry partially or completely breaks, this turns out to be a difficult issue. With symmetry, special modes have to be taken into account in some configurations and local collision laws of the collision operator add significant difficulties to the understanding of the convergence in asymptotic regimes in all cases, even if there is no particular symmetry.

Without external potential and for a bounded domain, the problem has been studied in [12]. In presence of a given external potential, the question was so far open and our first result is to classify all steady solutions for linear kinetic equations with collision operators satisfying the local conservation laws of physics. Even more difficult is the problem of the stability of the (possibly time-periodic) steady states and the proof of the convergence to such states, with an exponential rate, for inhomogeneous kinetic equations. The question goes back to the celebrated H-theorem of Boltzmann, but became quantitative only recently with the theory of hypocoercivity. All results involving an external potential deal with collision operators admitting only one collision invariant, up to a few attempts like [15,16] which discard special modes, with non-constructive methods. Our second result gives the very first answer to the question of the convergence rate in the whole space for an external potential without any a priori symmetry, using an entirely new scheme made of a cascade of several hypocoercive estimates. Alternatively we also propose a commutator method in the spirit of [27,46].

Even when the potential has no specific symmetry, which forbids the existence of any special mode other than the standard stationary solution, the fact that the collision operator admits several collision invariants is a source of difficulties: when the potential is almost symmetric, convergence rates get deteriorated and the geometric properties of the potential have therefore to be taken into account. The notion of steady states, defined as the set of attractors in large time asymptotics, is widely used in physics, and corresponds in our case to minimizers of the mathematical entropy (that is, the physical entropy, up to the sign). In this paper we shall speak of special macroscopic modes in relation with special symmetries of the potential.

## 1.1. Equation and assumptions

Consider the kinetic equation

$$ \partial_t f = \mathcal{L} f := \mathcal{T} f + \mathcal{C} f, \quad f|_{t=0} = f_0, \qquad (1.1) $$

for the unknown distribution function $f = f(t, x, v)$ depending on the time variable $t \ge 0$, the position variable $x \in \mathbb{R}^d$, and the velocity variable $v \in \mathbb{R}^d$, where $d \ge 1$ is an arbitrary
dimension. The transport operator $\mathcal{T}$ is given by

$$ \mathcal{T} f := -v \cdot \nabla_x f + \nabla_x \phi \cdot \nabla_v f $$

with a stationary, position dependent potential $\phi: \mathbb{R}^d \to \mathbb{R}$. We assume that the linear collision operator $\mathcal{C}$ is acting only along the velocity variable $v \in \mathbb{R}^d$, is self-adjoint in $L^2(\mu^{-1})$, with weight given by the local Maxwellian function

$$ \forall v \in \mathbb{R}^d, \quad \mu(v) := \frac{e^{-|v|^2/2}}{(2\pi)^{d/2}}, \qquad (1.2) $$

and has the $(d+2)$-dimensional kernel of collision invariants given by

$$ \mathrm{Ker} \mathcal{C} = \mathrm{Span} \{\mu, v_1 \mu, \dots, v_d \mu, |v|^2 \mu\}, \qquad (H0) $$

corresponding to the local conservation of mass, momentum and energy. Here $L^2(\mu^{-1})$ is the subspace of $L_{\mathrm{loc}}^2(\mathbb{R}^d, dv)$ of the functions $f$ such that

$$ \|f\|_{L^2(\mu^{-1})} := \left( \int_{\mathbb{R}^d} \frac{|f|^2}{\mu(v)} dv \right)^{1/2} $$

is finite.

We assume that $\mathcal{C}$ satisfies the following spectral gap property (which is a quantitative version of the spatially homogeneous linearized H-theorem)

$$ - \int_{\mathbb{R}^d} (\mathcal{C} f(v)) f(v) \mu(v)^{-1} dv \geq \mathcal{C} \|f - \Pi f\|_{L^2(\mu^{-1})}^2 \qquad (H1) $$

for some constant $c_{\mathcal{C}} > 0$ and all $f$ in the domain of $\mathcal{C}$, where $\Pi$ denotes the $L^2(\mu^{-1})$-orthogonal projection onto $\mathrm{Ker} \mathcal{C}$. Moreover, we suppose that for any polynomial function $p(v) : \mathbb{R}^d \to \mathbb{R}$ of degree at most 4, the function $p \mu$ is in the domain of $\mathcal{C}$ and

$$ C(p) := \|\mathcal{C}(p \mu)\|_{L^2(\mu^{-1})} < \infty. \qquad (H2) $$

We provide examples of collision operators satisfying these conditions in Appendix C.1, including the linearized Boltzmann and Landau operators.

Throughout the paper, we assume that the potential $\phi: \mathbb{R}^d \to \mathbb{R}$ is such that $\rho(x) := e^{-\phi(x)}$ is a centred probability density, i.e.,

$$ \int_{\mathbb{R}^d} \rho(x) \, dx = 1 \quad \text{and} \quad \int_{\mathbb{R}^d} x \rho(x) \, dx = 0. \qquad (H3) $$

We also assume that $\phi$ is of class $C^2(\mathbb{R}^d; \mathbb{R})$, and for all $\varepsilon > 0$, there exists a constant $C_{\varepsilon}$ such that

$$ \forall x \in \mathbb{R}^d, \quad |\nabla_x^2 \phi(x)| \leq \varepsilon |\nabla_x \phi(x)|^2 + C_{\varepsilon}, \qquad (H4) $$

where $\nabla_x^2 \phi$ denotes the Hessian matrix of $\phi$. We further assume that the measure $\rho(x)$ dx satisfies the Poincaré inequality with a constant $c_P > 0$,

$$ c_P \int_{\mathbb{R}^d} |\varphi - \langle \varphi \rangle|^2 \rho \, dx \leq \int_{\mathbb{R}^d} |\nabla_x \varphi|^2 \rho \, dx, \qquad (H5) $$
dimension. The transport operator $\mathcal{T}$ is given by

$$ \mathcal{T} f := -v \cdot \nabla_x f + \nabla_x \phi \cdot \nabla_v f $$

with a stationary, position dependent potential $\phi: \mathbb{R}^d \to \mathbb{R}$. We assume that the linear collision operator $\mathcal{C}$ is acting only along the velocity variable $v \in \mathbb{R}^d$, is self-adjoint in $L^2(\mu^{-1})$, with weight given by the local Maxwellian function

$$ \forall v \in \mathbb{R}^d, \quad \mu(v) := \frac{e^{-|v|^2/2}}{(2\pi)^{d/2}}, \qquad (1.2) $$

and has the $(d+2)$-dimensional kernel of collision invariants given by

$$ \mathrm{Ker} \mathcal{C} = \mathrm{Span} \{\mu, v_1 \mu, \dots, v_d \mu, |v|^2 \mu\}, \qquad (H0) $$

corresponding to the local conservation of mass, momentum and energy. Here $L^2(\mu^{-1})$ is the subspace of $L_{\mathrm{loc}}^2(\mathbb{R}^d, dv)$ of the functions $f$ such that

$$ \|f\|_{L^2(\mu^{-1})} := \left( \int_{\mathbb{R}^d} \frac{|f|^2}{\mu(v)} dv \right)^{1/2} $$

is finite.

We assume that $\mathcal{C}$ satisfies the following spectral gap property (which is a quantitative version of the spatially homogeneous linearized H-theorem)

$$ -\int_{\mathbb{R}^d} (\mathcal{C} f(v)) f(v) \mu(v)^{-1} dv \geq \mathcal{C} \|f - \Pi f\|_{L^2(\mu^{-1})}^2 \qquad (H1) $$

for some constant $c_{\mathcal{C}} > 0$ and all $f$ in the domain of $\mathcal{C}$, where $\Pi$ denotes the $L^2(\mu^{-1})$-orthogonal projection onto $\mathrm{Ker} \mathcal{C}$. Moreover, we suppose that for any polynomial function $p(v) : \mathbb{R}^d \to \mathbb{R}$ of degree at most 4, the function $p \mu$ is in the domain of $\mathcal{C}$ and

$$ C(p) := \|\mathcal{C}(p \mu)\|_{L^2(\mu^{-1})} < \infty. \qquad (H2) $$

We provide examples of collision operators satisfying these conditions in Appendix C.1, including the linearized Boltzmann and Landau operators.

Throughout the paper, we assume that the potential $\phi: \mathbb{R}^d \to \mathbb{R}$ is such that $\rho(x) := e^{-\phi(x)}$ is a centred probability density, i.e.,

$$ \int_{\mathbb{R}^d} \rho(x) \, dx = 1 \quad \text{and} \quad \int_{\mathbb{R}^d} x \rho(x) \, dx = 0. \qquad (H3) $$

We also assume that $\phi$ is of class $C^2(\mathbb{R}^d; \mathbb{R})$, and for all $\varepsilon > 0$, there exists a constant $C_{\varepsilon}$ such that

$$ \forall x \in \mathbb{R}^d, \quad |\nabla_x^2 \phi(x)| \leq \varepsilon |\nabla_x \phi(x)|^2 + C_{\varepsilon}, \qquad (H4) $$

where $\nabla_x^2 \phi$ denotes the Hessian matrix of $\phi$. We further assume that the measure $\rho(x)$ dx satisfies the Poincaré inequality with a constant $c_P > 0$,

$$ c_P \int_{\mathbb{R}^d} |\varphi - \langle \varphi \rangle|^2 \rho \, dx \leq \int_{\mathbb{R}^d} |\nabla_x \varphi|^2 \rho \, dx, \qquad (H5) $$
for all $\varphi \in L^2(\rho)$, where

$$ \langle \varphi \rangle := \int_{\mathbb{R}^d} \varphi \rho \, dx $$

is the average of $\varphi$. Here $L^2(\rho)$ is the subspace of $L^2_{\text{loc}}(\mathbb{R}^d, \text{d}x)$ of the functions $\varphi$ such that $\|\varphi\|_{L^2(\rho)}^2 = \int_{\mathbb{R}^d} |\varphi|^2 \rho \, \text{d}x$ is finite.

We assume moment bounds on $\rho$, namely

$$ \int_{\mathbb{R}^d} (|x|^4 + |\phi|^2 + |\nabla_x \phi|^4) \rho \, dx \le C_\phi \quad (H6) $$

for some constant $C_\phi > 0$. We also introduce the normalization

$$ \langle \nabla_x^2 \phi \rangle = \int_{\mathbb{R}^d} \nabla_x^2 \phi \rho \, dx = \text{Id}_{d \times d}, \quad (H7) $$

where $\text{Id}_{d \times d}$ the identity matrix of size $d$. The assumption that $\langle \nabla_x^2 \phi \rangle$ is diagonal is not a restriction since it can be obtained through a rotation in position space. Note that the same rotation in velocity space leaves the kinetic equation invariant and all assumptions made so far remain valid. The stronger assumption (H7) is made for notational simplicity, and a discussion of the general case is given in Appendix C.3.

The potential

$$ \phi(x) := (1 + |a_\gamma x|^2)^{\gamma/2} - Z_\gamma, $$

with $\gamma > 1$ and real normalization constants $a_\gamma$, $Z_\gamma$, satisfies (H3)–(H4)–(H5)–(H6)–(H7). See Appendix C.2 for other examples. No sign is assumed on $f$: one should think of $f$ as a real valued fluctuation around the equilibrium in the nonlinear Boltzmann or Landau equation (see Appendix C.1). Throughout this article we shall refer to (H1) and (H5) as spectral gap properties, and to (H2) and (H6) as bounded moment properties. These are the structural assumptions on $\mathcal{C}$ and $\phi$ for our theory.

Finally, since we are concerned with large time asymptotic behaviour, we require that the evolution equation (1.1) is well-posed, a condition which is satisfied by our standard examples of application, and assume that

$$ t \mapsto e^{t\mathcal{L}} \text{ is a strongly continuous semi-group on the space } L^2(\mathcal{M}^{-1}), \quad (H8) $$

where $\mathcal{M}$ is the global Maxwellian equilibrium function given by

$$ \forall (x, v) \in \mathbb{R}^d \times \mathbb{R}^d, \quad \mathcal{M}(x, v) := \rho(x) \mu(v) = \frac{e^{-\frac{1}{2}|v|^2 - \phi(x)}}{(2\pi)^{d/2}}, \quad (1.3) $$

and the space $L^2(\mathcal{M}^{-1})$ is the subspace of $L^2_{\text{loc}}(\mathbb{R}^d \times \mathbb{R}^d, \text{d}x \, \text{d}v)$ of the functions $f$ such that

$$ \|f\|_{L^2(\mathcal{M}^{-1})} := \left( \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|f|^2}{\mathcal{M}} \, \text{d}x \, \text{d}v \right)^{1/2} $$

is finite.
for all $\varphi \in L^2(\rho)$, where

$$ \langle \varphi \rangle := \int_{\mathbb{R}^d} \varphi \rho \, dx $$

is the average of $\varphi$. Here $L^2(\rho)$ is the subspace of $L^2_{\text{loc}}(\mathbb{R}^d, \text{d}x)$ of the functions $\varphi$ such that $\|\varphi\|_{L^2(\rho)}^2 = \int_{\mathbb{R}^d} |\varphi|^2 \rho \, \text{d}x$ is finite.

We assume moment bounds on $\rho$, namely

$$ \int_{\mathbb{R}^d} (|x|^4 + |\phi|^2 + |\nabla_x \phi|^4) \rho \, \text{d}x \le C_{\phi} \quad (H6) $$

for some constant $C_{\phi} > 0$. We also introduce the normalization

$$ \langle \nabla_x^2 \phi \rangle = \int_{\mathbb{R}^d} \nabla_x^2 \phi \rho \, \text{d}x = \text{Id}_{d \times d}, \quad (H7) $$

where $\text{Id}_{d \times d}$ the identity matrix of size $d$. The assumption that $\langle \nabla_x^2 \phi \rangle$ is diagonal is not a restriction since it can be obtained through a rotation in position space. Note that the same rotation in velocity space leaves the kinetic equation invariant and all assumptions made so far remain valid. The stronger assumption (H7) is made for notational simplicity, and a discussion of the general case is given in Appendix C.3.

The potential

$$ \phi(x) := (1 + |a_{\gamma} x|^2)^{\gamma/2} - Z_{\gamma}, $$

with $\gamma > 1$ and real normalization constants $a_{\gamma}, Z_{\gamma}$, satisfies (H3)–(H4)–(H5)–(H6)–(H7). See Appendix C.2 for other examples. No sign is assumed on $f$: one should think of $f$ as a real valued fluctuation around the equilibrium in the nonlinear Boltzmann or Landau equation (see Appendix C.1). Throughout this article we shall refer to (H1) and (H5) as spectral gap properties, and to (H2) and (H6) as bounded moment properties. These are the structural assumptions on $\mathcal{C}$ and $\phi$ for our theory.

Finally, since we are concerned with large time asymptotic behaviour, we require that the evolution equation (1.1) is well-posed, a condition which is satisfied by our standard examples of application, and assume that

$$ t \mapsto e^{t\mathcal{L}} \text{ is a strongly continuous semi-group on the space } L^2(\mathcal{M}^{-1}), \quad (H8) $$

where $\mathcal{M}$ is the global Maxwellian equilibrium function given by

$$ \forall (x, v) \in \mathbb{R}^d \times \mathbb{R}^d, \quad \mathcal{M}(x, v) := \rho(x) \mu(v) = \frac{e^{-\frac{1}{2}|v|^2 - \phi(x)}}{(2\pi)^{d/2}}, \quad (1.3) $$

and the space $L^2(\mathcal{M}^{-1})$ is the subspace of $L^2_{\text{loc}}(\mathbb{R}^d \times \mathbb{R}^d, \text{d}x \, \text{d}v)$ of the functions $f$ such that

$$ \|f\|_{L^2(\mathcal{M}^{-1})} := \left( \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|f|^2}{\mathcal{M}} \, \text{d}x \, \text{d}v \right)^{1/2} $$

is finite.
1.2. The main result

From here on, we assume the normalization conditions (H3)–(H7). The function $\mathcal{M}$ defined by (1.3) is a stationary solution of (1.1) but it is not the unique attractor of the time- dependent solutions of (1.1), even up to a mass normalization. Let us introduce a larger class of steady states. *Special macroscopic modes* of (1.1) are the solutions $F = F(t, x, v)$ to the system

$$ \mathcal{C}F = 0, \quad \partial_t F = \mathcal{T}F. \qquad (1.4) $$

Of course we read from (H0) that $F = \alpha \mathcal{M}$, $\alpha \in \mathbb{R}$, is a special macroscopic mode but we also look for solutions to (1.4) that can be written as

$$ F = (r(t, x) + m(t, x) \cdot v + e(t, x) \mathcal{E}(v)) \mathcal{M}, \qquad (1.5) $$

for some functions $r, m$ and $e$ with values respectively in $\mathbb{R}$, $\mathbb{R}^d$ and $\mathbb{R}$, with

$$ \mathcal{E}(v) := \frac{|v|^2 - d}{\sqrt{2d}}. \qquad (1.6) $$

The energy mode $F = \beta \mathcal{H} \mathcal{M}$, $\beta \in \mathbb{R}$, is another stationary solution to (1.4) where $\mathcal{H}$ defined by

$$ \mathcal{H}(x, v) := \frac{1}{2} (|v|^2 - d) + \phi(x) - \langle \phi \rangle \qquad (1.7) $$

is the Hamiltonian energy associated with the characteristics of the transport equation $\partial_t f = \mathcal{T}f$. As we shall see in Section 2.1, it turns out that the linear combination of global Maxwellian equilibrium functions and energy modes are the only special macroscopic modes for “generic potentials”. Other special macroscopic modes are available under additional symmetry properties of $\phi$ as observed by L. Boltzmann in [5]. These modes deserve some explanations.

The set of infinitesimal rotations compatible with $\phi$ defined as

$$ \mathcal{R}_{\phi} := \{x \mapsto Ax : A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R}) \text{ s.t. } \forall x \in \mathbb{R}^d, \nabla_x \phi(x) \cdot Ax = 0\} \qquad (1.8) $$

is identified with a subset of the space of skew-symmetric matrices

$$ \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R}) := \{A \in \mathfrak{M}_{d \times d}(\mathbb{R}) : {}^T A = -A\}. $$

In other words, $A \in \mathcal{R}_{\phi}$ if and only if $\phi$ is invariant by the rotation group $\theta \mapsto e^{\theta A}$, i.e.,

$$ \forall (\theta, x) \in \mathbb{R} \times \mathbb{R}^d, \quad \phi(e^{\theta A}x) = \phi(x). $$

The set $\mathcal{R}_{\phi}$ gives rise to the set of rotation modes compatible with $\phi$ defined by

$$ \mathfrak{R}_{\phi} := \{(x, v) \mapsto (Ax \cdot v) \mathcal{M}(x, v) : A \in \mathcal{R}_{\phi}\}. $$

Functions in $\mathfrak{R}_{\phi}$ are stationary solutions of (1.4) associated to all invariances of $\phi$ under rotation.
1.2. The main result

From here on, we assume the normalization conditions (H3)–(H7). The function $\mathcal{M}$ defined by (1.3) is a stationary solution of (1.1) but it is not the unique attractor of the time- dependent solutions of (1.1), even up to a mass normalization. Let us introduce a larger class of steady states. *Special macroscopic modes* of (1.1) are the solutions $F = F(t, x, v)$ to the system

$$ \mathcal{C}F = 0, \quad \partial_t F = \mathcal{T}F. \qquad (1.4) $$

Of course we read from (H0) that $F = \alpha \mathcal{M}$, $\alpha \in \mathbb{R}$, is a special macroscopic mode but we also look for solutions to (1.4) that can be written as

$$ F = (r(t,x) + m(t,x) \cdot v + e(t,x) \mathcal{E}(v)) \mathcal{M}, \qquad (1.5) $$

for some functions $r, m$ and $e$ with values respectively in $\mathbb{R}, \mathbb{R}^d$ and $\mathbb{R}$, with

$$ \mathcal{E}(v) := \frac{|v|^2 - d}{\sqrt{2d}}. \qquad (1.6) $$

The energy mode $F = \beta \mathcal{H} \mathcal{M}$, $\beta \in \mathbb{R}$, is another stationary solution to (1.4) where $\mathcal{H}$ defined by

$$ \mathcal{H}(x, v) := \frac{1}{2} (|v|^2 - d) + \phi(x) - \langle \phi \rangle \qquad (1.7) $$

is the Hamiltonian energy associated with the characteristics of the transport equation $\partial_t f = \mathcal{T}f$. As we shall see in Section 2.1, it turns out that the linear combination of global Maxwellian equilibrium functions and energy modes are the only special macroscopic modes for “generic potentials”. Other special macroscopic modes are available under additional symmetry properties of $\phi$ as observed by L. Boltzmann in [5]. These modes deserve some explanations.

The set of infinitesimal rotations compatible with $\phi$ defined as

$$ \mathcal{R}_{\phi} := \{x \mapsto Ax : A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R}) \text{ s.t. } \forall x \in \mathbb{R}^d, \nabla_x \phi(x) \cdot Ax = 0\} \qquad (1.8) $$

is identified with a subset of the space of skew-symmetric matrices

$$ \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R}) := \{A \in \mathfrak{M}_{d \times d}(\mathbb{R}) : {}^T A = -A\}. $$

In other words, $A \in \mathcal{R}_{\phi}$ if and only if $\phi$ is invariant by the rotation group $\theta \mapsto e^{\theta A}$, i.e.,

$$ \forall (\theta, x) \in \mathbb{R} \times \mathbb{R}^d, \quad \phi(e^{\theta A}x) = \phi(x). $$

The set $\mathcal{R}_{\phi}$ gives rise to the set of rotation modes compatible with $\phi$ defined by

$$ \mathfrak{R}_{\phi} := \{(x, v) \mapsto (Ax \cdot v) \mathcal{M}(x, v) : A \in \mathcal{R}_{\phi}\}. $$

Functions in $\mathfrak{R}_{\phi}$ are stationary solutions of (1.4) associated to all invariances of $\phi$ under rotation.
There are also some time-periodic special macroscopic modes when $\phi$ has harmonic directions. Let us define

$$E_{\phi} := \operatorname{Span}_{\mathbb{R}^d} \{ \nabla_x \phi(x) - x \}_{x \in \mathbb{R}^d \}, \quad d_{\phi} := \dim E_{\phi}. \qquad (1.9)$$

Notice that $E_{\phi}$ is a subspace of $\mathbb{R}^d$ and $d_{\phi} \le d$. Alternatively, we can characterize $d_{\phi}$ by

$$d_{\phi} = \dim \operatorname{Span}_{\mathbb{L}^2(\rho)} \left( \left\{ \partial_{x_i} \phi \right\}_{i=1,\dots,d} \cup \left\{ x_i \right\}_{i=1,\dots,d} \right) - d$$

and choose cartesian coordinates $(x_1, x_2, \dots, x_d)$ such that $\partial_{x_i} \phi = x_i$ if and only if $i \in I_{\phi} := \{d_{\phi} + 1, \dots, d\}$. We face three different cases.

▷ The case $d_{\phi} = d$ is called fully non-harmonic: $E_{\phi} = \mathbb{R}^d$ and $\phi$ has no harmonic direction and it that case, as we shall see below, there are no time-periodic solutions.

▷ In the case $1 \le d_{\phi} \le d - 1$, the potential is called partially harmonic. In the harmonic coordinates $x_{d_{\phi}+1}, \dots, x_d$, we have $\partial_{x_i} \phi = x_i$ and define harmonic directional modes by

$$\mathcal{D}_{\phi} := \operatorname{Span} \left\{ (x_i \cos t - v_i \sin t) \mathcal{M}, (x_i \sin t + v_i \cos t) \mathcal{M} : i \in I_{\phi} \right\}. \qquad (1.10)$$

Harmonic directional modes are also defined if $d_{\phi} = 0$. By convention, we set $\mathcal{D}_{\phi} := \{0\}$ if $d_{\phi} = d$. All functions in $\mathcal{D}_{\phi}$ are solutions to (1.4) which correspond to an inertia-driven oscillation of period 1 of particles in a potential well along a direction in $E_{\phi}^{\perp}$. These modes are independent of each other.

▷ In the case $d_{\phi} = 0$, the potential is called fully harmonic and $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$. In addition to the harmonic directional modes, the set of harmonic pulsating modes

$$\mathfrak{P}_{\phi} := \operatorname{Span} \left\{ \left( \frac{1}{2} (|x|^2 - |v|^2) \cos(2t) - x \cdot v \sin(2t) \right) \mathcal{M}, \left( \frac{1}{2} (|x|^2 - |v|^2) \sin(2t) + x \cdot v \cos(2t) \right) \mathcal{M} \right\} \quad (1.11)$$

is also made of solutions to (1.4). By convention, we set $\mathfrak{P}_{\phi} = \{0\}$ if $d_{\phi} \ge 1$. These macroscopic modes correspond to a radially symmetric pulsation of period $1/2$ of particles in the potential well.

Summing up the above observations, we have obtained special macroscopic modes of the form

$$F = \alpha \mathcal{M} + \beta \mathcal{H} \mathcal{M} + A x \cdot v \mathcal{M} + F_{\text{dir}} + F_{\text{pul}} \qquad (1.12)$$

where $\alpha, \beta \in \mathbb{R}$, $(x \mapsto Ax) \in \mathcal{R}_{\phi}$, $F_{\text{dir}} \in \mathcal{D}_{\phi}$ and $F_{\text{pul}} \in \mathfrak{P}_{\phi}$. With these definitions at hand, we can now state the main result of this paper.

**Theorem 1.1** (Special macroscopic modes and hypocoercivity). Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy the assumptions (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8). Then

(1) All special macroscopic modes of (1.4) are given by (1.12), i.e., are linear combinations of the Maxwellian, the energy mode, rotation modes compatible with $\phi$, and harmonic directional or pulsating modes if allowed by $\phi$.
There are also some time-periodic special macroscopic modes when $\phi$ has harmonic directions. Let us define

$$E_{\phi} := \operatorname{Span}_{\mathbb{R}^d} \{ \nabla_x \phi(x) - x \}_{x \in \mathbb{R}^d}, \quad d_{\phi} := \dim E_{\phi}. \qquad (1.9)$$

Notice that $E_{\phi}$ is a subspace of $\mathbb{R}^d$ and $d_{\phi} \le d$. Alternatively, we can characterize $d_{\phi}$ by

$$d_{\phi} = \dim \operatorname{Span}_{\mathbb{L}^2(\rho)} \left( \left\{ \partial_{x_i} \phi \right\}_{i=1,\dots,d} \cup \left\{ x_i \right\}_{i=1,\dots,d} \right) - d$$

and choose cartesian coordinates $(x_1, x_2, \dots, x_d)$ such that $\partial_{x_i} \phi = x_i$ if and only if $i \in I_{\phi} := \{d_{\phi} + 1, \dots, d\}$. We face three different cases.

▷ The case $d_{\phi} = d$ is called fully non-harmonic: $E_{\phi} = \mathbb{R}^d$ and $\phi$ has no harmonic direction and it that case, as we shall see below, there are no time-periodic solutions.

▷ In the case $1 \le d_{\phi} \le d - 1$, the potential is called partially harmonic. In the harmonic coordinates $x_{d_{\phi}+1}, \dots, x_d$, we have $\partial_{x_i} \phi = x_i$ and define harmonic directional modes by

$$\mathcal{D}_{\phi} := \operatorname{Span} \left\{ (x_i \cos t - v_i \sin t) \mathcal{M}, (x_i \sin t + v_i \cos t) \mathcal{M} : i \in I_{\phi} \right\}. \qquad (1.10)$$

Harmonic directional modes are also defined if $d_{\phi} = 0$. By convention, we set $\mathcal{D}_{\phi} := \{0\}$ if $d_{\phi} = d$. All functions in $\mathcal{D}_{\phi}$ are solutions to (1.4) which correspond to an inertia-driven oscillation of period 1 of particles in a potential well along a direction in $E_{\phi}^{\perp}$. These modes are independent of each other.

▷ In the case $d_{\phi} = 0$, the potential is called fully harmonic and $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$. In addition to the harmonic directional modes, the set of harmonic pulsating modes

$$\mathfrak{P}_{\phi} := \operatorname{Span} \left\{ \left( \frac{1}{2} (|x|^2 - |v|^2) \cos(2t) - x \cdot v \sin(2t) \right) \mathcal{M}, \left( \frac{1}{2} (|x|^2 - |v|^2) \sin(2t) + x \cdot v \cos(2t) \right) \mathcal{M} \right\} \quad (1.11)$$

is also made of solutions to (1.4). By convention, we set $\mathfrak{P}_{\phi} = \{0\}$ if $d_{\phi} \ge 1$. These macroscopic modes correspond to a radially symmetric pulsation of period $1/2$ of particles in the potential well.

Summing up the above observations, we have obtained special macroscopic modes of the form

$$F = \alpha \mathcal{M} + \beta \mathcal{H} \mathcal{M} + A x \cdot v \mathcal{M} + F_{\text{dir}} + F_{\text{pul}} \qquad (1.12)$$

where $\alpha, \beta \in \mathbb{R}$, $(x \mapsto Ax) \in \mathcal{R}_{\phi}$, $F_{\text{dir}} \in \mathcal{D}_{\phi}$ and $F_{\text{pul}} \in \mathfrak{P}_{\phi}$. With these definitions at hand, we can now state the main result of this paper.

**Theorem 1.1** (Special macroscopic modes and hypocoercivity). Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy the assumptions (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8). Then

(1) All special macroscopic modes of (1.4) are given by (1.12), i.e., are linear combinations of the Maxwellian, the energy mode, rotation modes compatible with $\phi$, and harmonic directional or pulsating modes if allowed by $\phi$.
(2) There are explicit constants $C > 0$ and $\kappa > 0$ such that, for any $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ solving (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$, there exists a unique special macroscopic mode $F$ such that

$$ \forall t \ge 0, \quad \|f(t) - F(t)\|_{L^2(\mathcal{M}^{-1})} \le C e^{-\kappa t} \|f_0 - F(0)\|_{L^2(\mathcal{M}^{-1})}. $$

The constants in the decay estimate being explicit means that the proof is constructive and provides a finite algorithm for computing $C$ and $\kappa$.

In the following, the norm and scalar product without subscript, $\| \cdot \|$ and $\langle \cdot, \cdot \rangle$, refer to the space $L^2(\mathcal{M})$, so that

$$ \|h\|^2 = \iint_{\mathbb{R}^d \times \mathbb{R}^d} |h|^2 \mathcal{M} \, dx \, dv = \langle h, h \rangle. \qquad (1.13) $$

With $h := f/\mathcal{M}$, $h_0 := f_0/\mathcal{M}$ and $h^\| := F/\mathcal{M}$, Part (2) of Theorem 1.1 amounts to

$$ \forall t \ge 0, \quad \|h(t) - h^\|(t) \le C e^{-\kappa t} \|h_0 - h^\|(0)\|. $$

When considering functions of $x$ only, the $L^2(\rho)$-norm coincides with the $L^2(\mathcal{M})$-norm. We recall that $\langle \cdot, \cdot \rangle$ stands for the average with respect to $\rho \, dx$.

In Theorem 1.1, the constants $C$ and $\kappa$ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to $\phi$ such as the rigidity constant (to be defined later) which admits a quantitative estimate. Moreover, the special macroscopic mode $F$ can be explicitly computed in terms of the initial data $f_0$ (see Section 2.2).

## 1.3. Framework, comments and methods

During the last two decades, new hypocoercive methods were developed for the study of spatially inhomogeneous kinetic equations. Many linear or nonlinear models were tackled, including Fokker-Planck, Boltzmann and Landau equations in various geometries, ranging from bounded domains to the whole Euclidean space, with or without confining potentials. The central issue is the trend to equilibrium for these equations, in the spirit of the celebrated H-Theorem by Boltzmann on the decay of the entropy, but with constructive estimates which measure the rate of convergence towards asymptotic regimes described by steady states. The set of steady states is not fully characterized by the entropy dissipation, but also depends on the transport operator and the geometric setting governed either by boundary conditions or by properties of the potential. The goal of this paper is to make the notion of steady states explicit by classifying all special macroscopic modes, and to derive quantitative estimates on the rate of convergence, with explicit constants.

Let us give a brief account of the literature. In a series of papers [20–22] on Landau, Boltzmann and Vlasov-Boltzmann equations in a periodic box, Y. Guo used micro-macro methods inspired from Grad’s 13 moments method introduced in [17]. The approach of [12] relies on the derivation of a suitable set of ordinary differential inequalities. It provides an algebraic rate of convergence to equilibrium under strong smoothness
(2) There are explicit constants $C > 0$ and $\kappa > 0$ such that, for any $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ solving (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$, there exists a unique special macroscopic mode $F$ such that

$$ \forall t \ge 0, \quad \|f(t) - F(t)\|_{L^2(\mathcal{M}^{-1})} \le C e^{-\kappa t} \|f_0 - F(0)\|_{L^2(\mathcal{M}^{-1})}. $$

The constants in the decay estimate being explicit means that the proof is constructive and provides a finite algorithm for computing $C$ and $\kappa$.

In the following, the norm and scalar product without subscript, $\| \cdot \|$ and $\langle \cdot, \cdot \rangle$, refer to the space $L^2(\mathcal{M})$, so that

$$ \|h\|^2 = \iint_{\mathbb{R}^d \times \mathbb{R}^d} |h|^2 \mathcal{M} \, dx \, dv = \langle h, h \rangle. \quad (1.13) $$

With $h := f/\mathcal{M}$, $h_0 := f_0/\mathcal{M}$ and $h^\| := F/\mathcal{M}$, Part (2) of Theorem 1.1 amounts to

$$ \forall t \ge 0, \quad \|h(t) - h^\|(t) \le C e^{-\kappa t} \|h_0 - h^\|(0) \|. $$

When considering functions of $x$ only, the $L^2(\rho)$-norm coincides with the $L^2(\mathcal{M})$-norm. We recall that $\langle \cdot, \cdot \rangle$ stands for the average with respect to $\rho \, dx$.

In Theorem 1.1, the constants $C$ and $\kappa$ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to $\phi$ such as the rigidity constant (to be defined later) which admits a quantitative estimate. Moreover, the special macroscopic mode $F$ can be explicitly computed in terms of the initial data $f_0$ (see Section 2.2).

### 1.3. Framework, comments and methods

During the last two decades, new hypocoercive methods were developed for the study of spatially inhomogeneous kinetic equations. Many linear or nonlinear models were tackled, including Fokker-Planck, Boltzmann and Landau equations in various geometries, ranging from bounded domains to the whole Euclidean space, with or without confining potentials. The central issue is the trend to equilibrium for these equations, in the spirit of the celebrated H-Theorem by Boltzmann on the decay of the entropy, but with constructive estimates which measure the rate of convergence towards asymptotic regimes described by steady states. The set of steady states is not fully characterized by the entropy dissipation, but also depends on the transport operator and the geometric setting governed either by boundary conditions or by properties of the potential. The goal of this paper is to make the notion of steady states explicit by classifying all special macroscopic modes, and to derive quantitative estimates on the rate of convergence, with explicit constants.

Let us give a brief account of the literature. In a series of papers [20–22] on Landau, Boltzmann and Vlasov-Boltzmann equations in a periodic box, Y. Guo used micro-macro methods inspired from Grad’s 13 moments method introduced in [17]. The approach of [12] relies on the derivation of a suitable set of ordinary differential inequalities. It provides an algebraic rate of convergence to equilibrium under strong smoothness
assumptions on the solution. The study of linear inhomogeneous kinetic equations with
single conservation laws, such as the linear Boltzmann or Fokker-Planck equations, and
nonlinear equations in a nonlinear but perturbative regime, took advantage of various
ideas of the theory of hypoellipticity, for instance of [29], and gave rise to robust Hil-
bertian hypocoercive methods. T. Gallay coined the word *hypocoercivity*, by analogy
with hypoellipticity, when coercivity is degenerate in the ambient space but recovered
using commutators, in the context of convergence to steady states. Hypocoercivity is
well adapted to kinetic equations with general collision operators. We refer to the mem-
oir [46] by C. Villani for an overview of the initial developments of this theory and
to [13, 14, 26, 27, 36] for various other contributions in exponentially weighted spaces.
The theory of *enlargement of spaces* of [18] allows to extend convergence rates to larger,
and physically more relevant, polynomially weighted spaces.

Usually, explicit and constructive estimates cannot be obtained via compactness argu-
ments. Such estimates are essential for applications in physics (typical time-scale for
relaxation) but also for a wide array of mathematical questions: range of validity of per-
turbation methods applied to nonlinear kinetic equations, conditions of convergence in
the study of diffusive or macroscopic limits, control of the limiting processes leading to
hydrodynamical equations when the Knudsen number tends to zero, control of the range of
parameters, time and length scales in the corresponding asymptotic regimes, etc. Among
a huge literature, we can refer for instance to [3, 45] and to [7, 23, 28] in polynomially
weighted spaces.

In this article, we focus on an important and old problem. We study kinetic equa-
tions involving an external confining potential as well as several local conservation laws
in the collision process. The linear problem was solved for a fully harmonic potential
in [15] and under full asymmetry assumptions on the potential in [16], both with non-
constructive arguments and for well-prepared initial data so that, in particular, there are no
special macroscopic modes beyond the Maxwellian stationary solution. Such an assump-
tion destroys the rich structure of special macroscopic modes and bypasses the non-trivial
consequences of the geometric properties of the potential on convergence rates. Our con-
tribution is precisely the study of these consequences, which requires new methods, by
classifying all *special macroscopic modes* and proving hypocoercivity results with con-
structive convergence rates in a natural Hilbertian structure. As in [15,16], we restrict our
analysis to the linear framework and, for simplicity, to exponentially weighted spaces, but
cover rather general confining potentials and discuss the consequences of their geometric
properties in terms of symmetry, partial symmetry or lack of symmetry under rotations.
On the one hand, the extension of our results to polynomially weighted spaces in the
spirit of [18] is probably doable. On the other hand, nonlinear stability for Boltzmann and
Landau equations with confining potentials, close to special macroscopic modes, presents
additional difficulties.

The special macroscopic modes other than the global Maxwellian stationary solutions
and the energy modes are consequences of the symmetries of the potential. Some of these
modes are known in the literature, although no systematic study seems to have been done.
From the point of view of mechanics, any function $F(x, v) = G(\mathcal{H}(x, v), A x \cdot v)$ is a sta-
assumptions on the solution. The study of linear inhomogeneous kinetic equations with
single conservation laws, such as the linear Boltzmann or Fokker-Planck equations, and
nonlinear equations in a nonlinear but perturbative regime, took advantage of various
ideas of the theory of hypoellipticity, for instance of [29], and gave rise to robust Hil-
bertian hypocoercive methods. T. Gallay coined the word *hypocoercivity*, by analogy
with hypoellipticity, when coercivity is degenerate in the ambient space but recovered
using commutators, in the context of convergence to steady states. Hypocoercivity is
well adapted to kinetic equations with general collision operators. We refer to the mem-
oir [46] by C. Villani for an overview of the initial developments of this theory and
to [13, 14, 26, 27, 36] for various other contributions in exponentially weighted spaces.
The theory of *enlargement of spaces* of [18] allows to extend convergence rates to larger,
and physically more relevant, polynomially weighted spaces.

Usually, explicit and constructive estimates cannot be obtained via compactness argu-
ments. Such estimates are essential for applications in physics (typical time-scale for
relaxation) but also for a wide array of mathematical questions: range of validity of per-
turbation methods applied to nonlinear kinetic equations, conditions of convergence in
the study of diffusive or macroscopic limits, control of the limiting processes leading to
hydrodynamical equations when the Knudsen number tends to zero, control of the range of
parameters, time and length scales in the corresponding asymptotic regimes, etc. Among
a huge literature, we can refer for instance to [3, 45] and to [7, 23, 28] in polynomially
weighted spaces.

In this article, we focus on an important and old problem. We study kinetic equa-
tions involving an external confining potential as well as several local conservation laws
in the collision process. The linear problem was solved for a fully harmonic potential
in [15] and under full asymmetry assumptions on the potential in [16], both with non-
constructive arguments and for well-prepared initial data so that, in particular, there are no
special macroscopic modes beyond the Maxwellian stationary solution. Such an assump-
tion destroys the rich structure of special macroscopic modes and bypasses the non-trivial
consequences of the geometric properties of the potential on convergence rates. Our con-
tribution is precisely the study of these consequences, which requires new methods, by
classifying all *special macroscopic modes* and proving hypocoercivity results with con-
structive convergence rates in a natural Hilbertian structure. As in [15,16], we restrict our
analysis to the linear framework and, for simplicity, to exponentially weighted spaces, but
cover rather general confining potentials and discuss the consequences of their geometric
properties in terms of symmetry, partial symmetry or lack of symmetry under rotations.
On the one hand, the extension of our results to polynomially weighted spaces in the
spirit of [18] is probably doable. On the other hand, nonlinear stability for Boltzmann and
Landau equations with confining potentials, close to special macroscopic modes, presents
additional difficulties.

The special macroscopic modes other than the global Maxwellian stationary solutions
and the energy modes are consequences of the symmetries of the potential. Some of these
modes are known in the literature, although no systematic study seems to have been done.
From the point of view of mechanics, any function $F(x, v) = G(H(x, v), Ax \cdot v)$ is a sta-
tionary solution of the transport equation where $Ax \cdot v$ is known as the *angular momentum* generated by $A \in \mathfrak{M}_{d \times d}(\mathbb{R})$. The property that there is no other stationary solution, under appropriate conditions, is known in astrophysics as *Jeans' theorem* (usually considered with a potential induced by a mean field coupling) and has to do with Noether's theorem: see [4] and references therein. Of course the only profile $G$ compatible with (H0) is $G(h, a) = p(h, a) e^{-h}$ where $p$ is a polynomial of order at most two. Proving that any stationary solution of (1.1) has to solve (1.4) is also known as a *factorization result*.

The existence of time-periodic steady states for the fully harmonic potential was shown by L. Boltzmann in [5] and is mentioned in some references: see for instance [6, 10, 19, 44]. In [19], time-periodic modes are called *breathing modes*. The consideration of partially harmonic potentials and their corresponding harmonic directional modes seems to be new. The fact that special macroscopic modes also exist for the nonlinear Boltzmann equation is discussed in Appendix C.5.

Now let us review some of the tools which are used in our paper. To estimate the convergence rate, a major difficulty is to quantify “how far” the potential $\phi$ is from having certain partial symmetries. Inspired by [11, 12], we use some Korn inequalities for bounded domains which go back to [32, 33] and adapt them to the Euclidean space, in presence of a confining potential: see [9]. A typical quantity involved in our approach is the *rigidity constant*

$$c_K := \min \left\{ \int_{\mathbb{R}^d} |\nabla \phi(x) \cdot A x|^2 \rho(x) \, dx : A \in \mathcal{R}_\phi^\perp \text{ such that } \int_{\mathbb{R}^d} |Ax|^2 \rho(x) \, dx = 1 \right\} > 0, \quad (1.14)$$

where $\mathcal{R}_\phi^\perp$ is the orthogonal complement in $L^2(\rho)$ of the set $\mathcal{R}_\phi$ defined in (1.8). The time-periodic special macroscopic modes in $\mathcal{D}_\phi$ and $\mathfrak{P}_\phi$, defined in (1.10) and (1.11) respectively, are related to the (partial) harmonicity of $\phi$ and another difficulty is to quantify “how far” the potential $\phi$ is from being (partially) harmonic. As for Korn type inequalities, the analysis relies on the finite dimension of the space $E_\phi$ defined in (1.9).

The spectral gap assumptions (H1) in $v$ and (H5) in $x$ reflect the corresponding confining properties respectively in velocity and space. The Poincaré inequality introduced in (H5) is linked with the natural Hodge Laplacian associated to the geometry, sometimes called the *Witten Laplacian*. Denote by $\nabla_x^*$ the adjoint of $\nabla_x$ in $L^2(\rho)$ acting on vector fields $\varphi: \mathbb{R}^d \to \mathbb{R}^d$ according to $\nabla_x^* \cdot \varphi = (-\nabla_x + \nabla_x \phi) \cdot \varphi$. The Witten-Laplace operator $\nabla_x^* \cdot \nabla_x$ is self-adjoint in $L^2(\rho)$, with kernel spanned by constant functions and its first non-zero eigenvalue determines the optimal Poincaré constant $c_P$. The operator

$$\Omega := \nabla_x^* \cdot \nabla_x + 1 = - \Delta_x + \nabla_x \phi \cdot \nabla_x + 1 \quad (1.15)$$

is used in the 0th-order Poincaré inequality

$$c_{P,1} \|\varphi - \langle \varphi \rangle\|^2 \le \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\|^2$$
tionary solution of the transport equation where $Ax \cdot v$ is known as the *angular momentum* generated by $A \in \mathfrak{M}_{d \times d}(\mathbb{R})$. The property that there is no other stationary solution, under appropriate conditions, is known in astrophysics as *Jeans' theorem* (usually considered with a potential induced by a mean field coupling) and has to do with Noether's theorem: see [4] and references therein. Of course the only profile $G$ compatible with (H0) is $G(h, a) = p(h, a) e^{-h}$ where $p$ is a polynomial of order at most two. Proving that any stationary solution of (1.1) has to solve (1.4) is also known as a *factorization result*.

The existence of time-periodic steady states for the fully harmonic potential was shown by L. Boltzmann in [5] and is mentioned in some references: see for instance [6, 10, 19, 44]. In [19], time-periodic modes are called *breathing modes*. The consideration of partially harmonic potentials and their corresponding harmonic directional modes seems to be new. The fact that special macroscopic modes also exist for the nonlinear Boltzmann equation is discussed in Appendix C.5.

Now let us review some of the tools which are used in our paper. To estimate the convergence rate, a major difficulty is to quantify “how far” the potential $\phi$ is from having certain partial symmetries. Inspired by [11, 12], we use some Korn inequalities for bounded domains which go back to [32, 33] and adapt them to the Euclidean space, in presence of a confining potential: see [9]. A typical quantity involved in our approach is the *rigidity constant*

$$c_K := \min \left\{ \int_{\mathbb{R}^d} |\nabla \phi(x) \cdot A x|^2 \rho(x) \, dx : A \in \mathcal{R}_\phi^\perp \text{ such that } \int_{\mathbb{R}^d} |Ax|^2 \rho(x) \, dx = 1 \right\} > 0, \quad (1.14)$$

where $\mathcal{R}_\phi^\perp$ is the orthogonal complement in $L^2(\rho)$ of the set $\mathcal{R}_\phi$ defined in (1.8). The time-periodic special macroscopic modes in $\mathcal{D}_\phi$ and $\mathfrak{P}_\phi$, defined in (1.10) and (1.11) respectively, are related to the (partial) harmonicity of $\phi$ and another difficulty is to quantify “how far” the potential $\phi$ is from being (partially) harmonic. As for Korn type inequalities, the analysis relies on the finite dimension of the space $E_\phi$ defined in (1.9).

The spectral gap assumptions (H1) in $v$ and (H5) in $x$ reflect the corresponding confining properties respectively in velocity and space. The Poincaré inequality introduced in (H5) is linked with the natural Hodge Laplacian associated to the geometry, sometimes called the *Witten Laplacian*. Denote by $\nabla_x^*$ the adjoint of $\nabla_x$ in $L^2(\rho)$ acting on vector fields $\varphi: \mathbb{R}^d \to \mathbb{R}^d$ according to $\nabla_x^* \cdot \varphi = (-\nabla_x + \nabla_x \phi) \cdot \varphi$. The Witten-Laplace operator $\nabla_x^* \cdot \nabla_x$ is self-adjoint in $L^2(\rho)$, with kernel spanned by constant functions and its first non-zero eigenvalue determines the optimal Poincaré constant $c_P$. The operator

$$\Omega := \nabla_x^* \cdot \nabla_x + 1 = - \Delta_x + \nabla_x \phi \cdot \nabla_x + 1 \quad (1.15)$$

is used in the *0th-order Poincaré inequality*

$$c_{P,1} \|\varphi - \langle \varphi \rangle\|^2 \le \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\|^2$$
which holds for some constant $c_{p,1} > 0$ under assumptions (H5) and (H4), in the spirit of Poincaré-Lions inequalities studied in [9].

We provide two proofs of Theorem 1.1. The first proof follows a micro-macro decomposition as in [20–22] and [16]. Due to the lack of a priori symmetry assumptions and the delicate interaction of local conservation laws corresponding to the collision invariants with the potential, the complexity is significantly increased. There are also deep similarities with the analysis of hyperbolic equations with damping studied in [24, 39, 40] after the seminal paper [30] by S. Kawashima and Y. Shizuta. The second proof is given under slightly more restrictive hypotheses, namely that the collision operator $\mathcal{C}$ is bounded and $\phi$ has bounded derivatives of order two and more. The method is based on commutator estimates as in [27, 36, 46] in the spirit of the hypoellipticity theory of [29]. In practice, an elegant triple cascade of commutators based on the equality $[\nabla_v, v \cdot \nabla_x] = \nabla_x$ is needed to control all macroscopic quantities.

The plan of the article is the following. In Section 2, we review all possible conservation laws and their relations with the special macroscopic modes. Then we present the so-called macroscopic equations associated to the evolution equation (1.1) and perform a change of unknown in order to work in a simplified Hilbertian framework. In Section 3, we classify all steady states of (1.1) and prove that they correspond to the special macroscopic modes. At this stage, we already use entropy-dissipation arguments in order to prove that factorization occurs and reduce the problem to (1.4). In Section 4, we prove the remaining part of Theorem 1.1, that is, the hypocoercivity result, using the micro-macro method. In Section 5 we expose the second proof based on the commutator’s method. A number of technical results are collected in two appendices. Appendix A collects some computations and intermediate lemmata needed in the proofs. For completeness, an extension to weakly coercive collision operators is given in Appendix B. Appendix C is devoted to examples and remarks, for instance on the normalization, including a spectral interpretation of Theorem 1.1, the extension of our special macroscopic modes to the fully nonlinear Boltzmann equation, and various examples of collision operators and potentials.

## 2. Conservation laws and macroscopic equations

In this section we characterize the special macroscopic modes, as defined by (1.4), for generic potentials. We also identify the global conservation laws and the macroscopic equations associated to (1.1). From here on, we assume that (H6) holds. This assumption is needed to justify the computations, which are given below only at formal level, for sake of simplicity.

### 2.1. The equations for the special macroscopic modes

We recall that by (1.5), any special macroscopic mode $F$ can be written as

$$F = h^{\|} \mathcal{M}, \quad h^{\|} = r + m \cdot v + e \mathcal{E}(v).$$
which holds for some constant $c_{p,1} > 0$ under assumptions (H5) and (H4), in the spirit of Poincaré-Lions inequalities studied in [9].

We provide two proofs of Theorem 1.1. The first proof follows a micro-macro decomposition as in [20–22] and [16]. Due to the lack of a priori symmetry assumptions and the delicate interaction of local conservation laws corresponding to the collision invariants with the potential, the complexity is significantly increased. There are also deep similarities with the analysis of hyperbolic equations with damping studied in [24, 39, 40] after the seminal paper [30] by S. Kawashima and Y. Shizuta. The second proof is given under slightly more restrictive hypotheses, namely that the collision operator $\mathcal{C}$ is bounded and $\phi$ has bounded derivatives of order two and more. The method is based on commutator estimates as in [27, 36, 46] in the spirit of the hypoellipticity theory of [29]. In practice, an elegant triple cascade of commutators based on the equality $[\nabla_v, v \cdot \nabla_x] = \nabla_x$ is needed to control all macroscopic quantities.

The plan of the article is the following. In Section 2, we review all possible conservation laws and their relations with the special macroscopic modes. Then we present the so-called macroscopic equations associated to the evolution equation (1.1) and perform a change of unknown in order to work in a simplified Hilbertian framework. In Section 3, we classify all steady states of (1.1) and prove that they correspond to the special macroscopic modes. At this stage, we already use entropy-dissipation arguments in order to prove that factorization occurs and reduce the problem to (1.4). In Section 4, we prove the remaining part of Theorem 1.1, that is, the hypocoercivity result, using the micro-macro method. In Section 5 we expose the second proof based on the commutator’s method. A number of technical results are collected in two appendices. Appendix A collects some computations and intermediate lemmata needed in the proofs. For completeness, an extension to weakly coercive collision operators is given in Appendix B. Appendix C is devoted to examples and remarks, for instance on the normalization, including a spectral interpretation of Theorem 1.1, the extension of our special macroscopic modes to the fully nonlinear Boltzmann equation, and various examples of collision operators and potentials.

## 2. Conservation laws and macroscopic equations

In this section we characterize the special macroscopic modes, as defined by (1.4), for generic potentials. We also identify the global conservation laws and the macroscopic equations associated to (1.1). From here on, we assume that (H6) holds. This assumption is needed to justify the computations, which are given below only at formal level, for sake of simplicity.

### 2.1. The equations for the special macroscopic modes

We recall that by (1.5), any special macroscopic mode $F$ can be written as

$$F = h^{\|} \mathcal{M}, \quad h^{\|} = r + m \cdot v + e \mathcal{E}(v).$$
By (1.4), we know that $\partial_t F = \mathcal{T}F$. By integrating in $v$ the evolution equation against 1, $v$, $v \otimes v$ and $v \otimes \mathcal{E}$, we obtain macroscopic equations on the macroscopic quantities $r = r(t, x)$, $m = m(t, x)$ and $e = e(t, x)$:

$$ \partial_t r = \nabla_x^* m, \qquad (2.1a) $$

$$ \partial_t m = -\nabla_x r + \sqrt{\frac{2}{d}} e \nabla_x \phi \qquad (2.1b) $$

$$ \partial_t e = -\sqrt{\frac{2}{d}} \nabla_x m, \qquad (2.1c) $$

$$ \frac{1}{\sqrt{2}d} (\partial_t e) \mathrm{Id}_{d \times d} = -\nabla_x^{\mathrm{sym}} m, \qquad (2.1d) $$

$$ 0 = \nabla_x e, \qquad (2.1e) $$

where the symmetric gradient is defined by

$$ \forall i, j = 1, \dots, d, \quad (\nabla_x^{\mathrm{sym}} m)_{ij} := \frac{1}{2} (\partial_j m_i + \partial_i m_j). \qquad (2.2) $$

From (2.1e), we deduce that $e$ does not depend on the space variable, and therefore

$$ e = \langle e \rangle := c \qquad (2.3) $$

is a function $t \mapsto c(t)$ depending on $t$ only. We recall that the average is defined as $\langle \varphi \rangle := \int_{\mathbb{R}^d} \varphi \rho \, dx$. Then we read from (2.1d) that

$$ \frac{c'}{\sqrt{2}d} \mathrm{Id}_{d \times d} = -\nabla_x^{\mathrm{sym}} m. \qquad (2.4) $$

By the Schwarz Lemma applied to $m = (m_k)_{k=1}^d$, we have

$$ (\nabla_x^2 m_k)_{i,j} = \partial_{x_i x_j}^2 m_k = \partial_{x_i} (\nabla^{\mathrm{sym}} m)_{j,k} + \partial_{x_j} (\nabla^{\mathrm{sym}} m)_{i,k} - \partial_{x_k} (\nabla^{\mathrm{sym}} m)_{i,j} \qquad (2.5) $$

for any $i, j, k = 1, \dots, d$. By differentiating (2.4) with respect to $x_i, x_j$ and $x_k$, we get that $\nabla_x^2 m = 0$, so that in particular the skew-symmetric gradient, defined by

$$ \forall i, j = 1, \dots, d, \quad (\nabla_x^{\mathrm{skew}} m)_{ij} := \frac{1}{2} (\partial_j m_i - \partial_i m_j), \qquad (2.6) $$

is constant in the $x$-variable and equal to its average. Together with (2.4), using $\nabla_x m = \nabla_x^{\mathrm{skew}} m + \nabla_x^{\mathrm{sym}} m$, we deduce that

$$ m(t, x) = \langle \nabla m \rangle x + \langle m \rangle = A(t) x + b(t) - \frac{1}{\sqrt{2}d} c'(t) x, \qquad (2.7) $$

with $A(t) := \langle \nabla_x^{\mathrm{skew}} m \rangle$ and $b(t) := \langle m \rangle$. Taking (2.3) and (2.7) into account in (2.1b) implies

$$ \nabla_x r = - \partial_t m + \sqrt{\frac{2}{d}} e \nabla_x \phi = -A'x - b' + \frac{1}{\sqrt{2}d} c''x + \sqrt{\frac{2}{d}} c \nabla \phi. $$

Taking the skew-symmetric gradient of this equation gives $0 = -A'$. Hence $A$ is a skew-symmetric matrix which does not depend on $t$ and

$$ m(t, x) = A x + b(t) - \frac{1}{\sqrt{2}d} c'(t) x. \qquad (2.8) $$
By (1.4), we know that $\partial_t F = \mathcal{T}F$. By integrating in $v$ the evolution equation against 1, $v$, $v \otimes v$ and $v \otimes \mathcal{E}$, we obtain macroscopic equations on the macroscopic quantities $r = r(t, x)$, $m = m(t, x)$ and $e = e(t, x)$:

$$ \partial_t r = \nabla_x^* m, \qquad (2.1a) $$

$$ \partial_t m = -\nabla_x r + \sqrt{\frac{2}{d}} e \nabla_x \phi \qquad (2.1b) $$

$$ \partial_t e = -\sqrt{\frac{2}{d}} \nabla_x m, \qquad (2.1c) $$

$$ \frac{1}{\sqrt{2d}} (\partial_t e) \mathrm{Id}_{d \times d} = -\nabla_x^{\mathrm{sym}} m, \qquad (2.1d) $$

$$ 0 = \nabla_x e, \qquad (2.1e) $$

where the symmetric gradient is defined by

$$ \forall i, j = 1, \dots, d, \quad (\nabla_x^{\mathrm{sym}} m)_{ij} := \frac{1}{2} (\partial_j m_i + \partial_i m_j). \qquad (2.2) $$

From (2.1e), we deduce that $e$ does not depend on the space variable, and therefore

$$ e = \langle e \rangle := c \qquad (2.3) $$

is a function $t \mapsto c(t)$ depending on $t$ only. We recall that the average is defined as $\langle \varphi \rangle := \int_{\mathbb{R}^d} \varphi \rho \, dx$. Then we read from (2.1d) that

$$ \frac{c'}{\sqrt{2d}} \mathrm{Id}_{d \times d} = -\nabla_x^{\mathrm{sym}} m. \qquad (2.4) $$

By the Schwarz Lemma applied to $m = (m_k)_{k=1}^d$, we have

$$ (\nabla_x^2 m_k)_{i,j} = \partial_{x_i x_j}^2 m_k = \partial_{x_i} (\nabla^{\mathrm{sym}} m)_{j,k} + \partial_{x_j} (\nabla^{\mathrm{sym}} m)_{i,k} - \partial_{x_k} (\nabla^{\mathrm{sym}} m)_{i,j} \qquad (2.5) $$

for any $i, j, k = 1, \dots, d$. By differentiating (2.4) with respect to $x_i, x_j$ and $x_k$, we get that $\nabla_x^2 m = 0$, so that in particular the skew-symmetric gradient, defined by

$$ \forall i, j = 1, \dots, d, \quad (\nabla_x^{\mathrm{skew}} m)_{ij} := \frac{1}{2} (\partial_j m_i - \partial_i m_j), \qquad (2.6) $$

is constant in the $x$-variable and equal to its average. Together with (2.4), using $\nabla_x m = \nabla_x^{\mathrm{skew}} m + \nabla_x^{\mathrm{sym}} m$, we deduce that

$$ m(t, x) = \langle \nabla m \rangle x + \langle m \rangle = A(t) x + b(t) - \frac{1}{\sqrt{2d}} c'(t) x, \qquad (2.7) $$

with $A(t) := \langle \nabla_x^{\mathrm{skew}} m \rangle$ and $b(t) := \langle m \rangle$. Taking (2.3) and (2.7) into account in (2.1b) implies

$$ \nabla_x r = -\partial_t m + \sqrt{\frac{2}{d}} e \nabla_x \phi = -A'x - b' + \frac{1}{\sqrt{2d}} c''x + \sqrt{\frac{2}{d}} c \nabla \phi. $$

Taking the skew-symmetric gradient of this equation gives $0 = -A'$. Hence $A$ is a skew-symmetric matrix which does not depend on $t$ and

$$ m(t, x) = A x + b(t) - \frac{1}{\sqrt{2d}} c'(t) x. \qquad (2.8) $$
Taking (2.3) and (2.8) into account, we can then take the primitive in space of (2.1b) and we immediately deduce that the macroscopic density satisfies

$$r(t,x) = r_0 - b'(t) \cdot x + \frac{1}{2\sqrt{2d}} c''(t) \xi_2(x) + \sqrt{\frac{2}{d}} c(t) \xi_{\phi}(x) \quad (2.9)$$

where $r_0$ is an integration constant,

$$\xi_2(x) := |x|^2 - \langle |x|^2 \rangle \quad (2.10)$$

and

$$\xi_{\phi}(x) := \phi - \langle \phi \rangle . \quad (2.11)$$

An integration against $\rho$ shows that $r_0 = \langle r(t, \cdot) \rangle$ is in fact independent of $t$. Inserting the expressions of $r$ and $m$ given by (2.8) and (2.9) into (2.1a) yields a differential equation satisfied by $A$, $b(t)$ and $c(t)$:

**Proposition 2.1.** Assume that $r$, $m$ and $e$ solve (2.1). With the above notations, $A(t) := \langle \nabla_x^{\text{skew}} m \rangle$, $b(t) := \langle m \rangle$ and $c(t) := \langle e \rangle$ solve

$$\frac{2 \xi_{\phi}(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2(x)}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - b'' \cdot x - \nabla_x \phi \cdot Ax = 0. \quad (2.12)$$

Equation (2.12) suggests, on the one hand, that (partial) harmonicity of the potential $\phi$ allows for non-trivial choices of $b$ and $c$, as we shall indeed see later. On the other hand, for a generic potential $\phi$ in the sense that the functions $1$, $x$, $\phi$, $\nabla_x \phi \cdot x$, $\nabla \phi$, $|x|^2$ and $\nabla_x \phi \cdot Ax$ (if $A \neq 0$) are linearly independent, equation (2.12) implies that

$$c' = 0, \quad b = 0 \quad \text{and} \quad A = 0,$$

so that $r = r_0 + c \sqrt{\frac{2}{d}} \xi_{\phi}$, $m = 0$ and $e = c$, for two constants $r_0$ and $c \in \mathbb{R}$. In other words, we have

$$h^{\|} = r_0 + \sqrt{\frac{2}{d}} c \mathcal{H} - \sqrt{\frac{2}{d}} c (\langle \phi \rangle - \frac{d}{2})$$

if $\phi$ is not a (partially) harmonic potential: any special macroscopic mode is then a linear combination of a Maxwellian function and an energy mode.

## 2.2. Global conservation laws

Consider a solution $f \in C(\mathbb{R}^+; L^2(M^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(M^{-1})$. Associated with the symmetries of the equation, there are local conservations which, after integration on the phase space $\mathbb{R}^d \times \mathbb{R}^d$, give rise to global conservation laws. These laws allow us to identify the special macroscopic modes compatible with $\phi$ which, as we shall see later, attract the solutions to the Cauchy problem.

The conservation of mass writes

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(t, x, v) \, dx \, dv = 0.$$
Taking (2.3) and (2.8) into account, we can then take the primitive in space of (2.1b) and we immediately deduce that the macroscopic density satisfies

$$r(t,x) = r_0 - b'(t) \cdot x + \frac{1}{2\sqrt{2d}} c''(t) \xi_2(x) + \sqrt{\frac{2}{d}} c(t) \xi_{\phi}(x) \quad (2.9)$$

where $r_0$ is an integration constant,

$$\xi_2(x) := |x|^2 - \langle |x|^2 \rangle \quad (2.10)$$

and

$$\xi_{\phi}(x) := \phi - \langle \phi \rangle . \quad (2.11)$$

An integration against $\rho$ shows that $r_0 = \langle r(t, \cdot) \rangle$ is in fact independent of $t$. Inserting the expressions of $r$ and $m$ given by (2.8) and (2.9) into (2.1a) yields a differential equation satisfied by $A$, $b(t)$ and $c(t)$:

**Proposition 2.1.** Assume that $r$, $m$ and $e$ solve (2.1). With the above notations, $A(t) := \langle \nabla_x^{\text{skew}} m \rangle$, $b(t) := \langle m \rangle$ and $c(t) := \langle e \rangle$ solve

$$\frac{2 \xi_{\phi}(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2(x)}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - b'' \cdot x - \nabla_x \phi \cdot Ax = 0. \quad (2.12)$$

Equation (2.12) suggests, on the one hand, that (partial) harmonicity of the potential $\phi$ allows for non-trivial choices of $b$ and $c$, as we shall indeed see later. On the other hand, for a generic potential $\phi$ in the sense that the functions $1$, $x$, $\phi$, $\nabla_x \phi \cdot x$, $\nabla \phi$, $|x|^2$ and $\nabla_x \phi \cdot A x$ (if $A \neq 0$) are linearly independent, equation (2.12) implies that

$$c' = 0, \quad b = 0 \quad \text{and} \quad A = 0,$$

so that $r = r_0 + c \sqrt{\frac{2}{d}} \xi_{\phi}$, $m = 0$ and $e = c$, for two constants $r_0$ and $c \in \mathbb{R}$. In other words, we have

$$h^{\|} = r_0 + \sqrt{\frac{2}{d}} c \mathcal{H} - \sqrt{\frac{2}{d}} c (\langle \phi \rangle - \frac{d}{2})$$

if $\phi$ is not a (partially) harmonic potential: any special macroscopic mode is then a linear combination of a Maxwellian function and an energy mode.

## 2.2. Global conservation laws

Consider a solution $f \in C(\mathbb{R}^+; L^2(M^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(M^{-1})$. Associated with the symmetries of the equation, there are local conservations which, after integration on the phase space $\mathbb{R}^d \times \mathbb{R}^d$, give rise to global conservation laws. These laws allow us to identify the special macroscopic modes compatible with $\phi$ which, as we shall see later, attract the solutions to the Cauchy problem.

The conservation of mass writes

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(t, x, v) \, dx \, dv = 0.$$
Hence $\alpha \mathcal{M}$ is a solution to (1.4) with same mass

$$ \alpha := \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_0 \, dx \, dv \qquad (2.13) $$

as $f$. With $\mathcal{H}$ defined by (1.7), the conservation of energy amounts to

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}(x, v) f(t, x, v) \, dx \, dv = 0. $$

The distribution function $\beta \mathcal{H} \mathcal{M}$, with

$$ \beta := \frac{\iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H} f_0 \, dx \, dv}{\iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}^2 \mathcal{M} \, dx \, dv}, \qquad (2.14) $$

is a solution to (1.4) with same energy as the conserved energy of $f$. With $f_1 := \mathcal{M}$ and $f_{\mathcal{H}} := \mathcal{H} \mathcal{M} / \|\mathcal{H} \mathcal{M}\|_{L^2(\mathcal{M}^{-1})}$, we have that

$$ \alpha \mathcal{M} = \langle f_0, f_1 \rangle_{L^2(\mathcal{M}^{-1})} f_1 \quad \text{and} \quad \beta \mathcal{H} \mathcal{M} = \langle f_0, f_{\mathcal{H}} \rangle_{L^2(\mathcal{M}^{-1})} f_{\mathcal{H}}. $$

Moreover, the global conservations of mass and energy write

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} (f(t, x, v) - \alpha \mathcal{M}(x, v)) \, dx \, dv = 0, $$

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}(x, v) (f(t, x, v) - \beta \mathcal{H}(x, v) \mathcal{M}(x, v)) \, dx \, dv = 0. $$

The transport operator can be written as $\mathcal{T} f = X \cdot \nabla_{x,v} f = \nabla_{x,v} \cdot (X f)$, where $X = (-v, \nabla_x \phi) = (-\nabla_v \mathcal{H}, \nabla_x \mathcal{H})$ is a divergence free vector field, in the sense that

$$ \nabla_{x,v} \cdot X = 0. \qquad (2.15) $$

As a consequence, the volume conservation in the phase space under the action of the flow induces the local mass conservation and the (2.15) symmetry gives rise to the global mass conservation. Another symmetry is associated with the fact that $\mathcal{H}$ is conserved along the characteristics of Newton's equations $\dot{x} = v$ and $\dot{v} = -\nabla_x \phi$. This is reflected by the Poisson brackets: a stationary solution $F$ satisfies

$$ 0 = \{F, \mathcal{H}\} := \nabla_v \mathcal{H} \cdot \nabla_x F - \nabla_x \mathcal{H} \cdot \nabla_v F, \qquad (2.16) $$

but by replacing $F$ by $\mathcal{H} F$, it is also clear that

$$ \{\mathcal{H} F, \mathcal{H}\} = \mathcal{H}\{F, \mathcal{H}\} + \{\mathcal{H}, \mathcal{H}\}\mathcal{H} = 0. $$

The underlying reason is that the transport dynamics involving a time-independent potential is invariant under a translation in time, which gives rise to the global conservation of energy. These considerations can be generalized. To any continuous group of transformations which leaves $\mathcal{T}$ invariant, we can associate an infinitesimal transformation $\mathcal{G}(x, v)$
Hence $\alpha \mathcal{M}$ is a solution to (1.4) with same mass

$$ \alpha := \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_0 \, dx \, dv \qquad (2.13) $$

as $f$. With $\mathcal{H}$ defined by (1.7), the conservation of energy amounts to

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}(x, v) f(t, x, v) \, dx \, dv = 0. $$

The distribution function $\beta \mathcal{H} \mathcal{M}$, with

$$ \beta := \frac{\iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H} f_0 \, dx \, dv}{\iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}^2 \mathcal{M} \, dx \, dv}, \qquad (2.14) $$

is a solution to (1.4) with same energy as the conserved energy of $f$. With $f_1 := \mathcal{M}$ and $f_{\mathcal{H}} := \mathcal{H} \mathcal{M} / \|\mathcal{H} \mathcal{M}\|_{L^2(\mathcal{M}^{-1})}$, we have that

$$ \alpha \mathcal{M} = \langle f_0, f_1 \rangle_{L^2(\mathcal{M}^{-1})} f_1 \quad \text{and} \quad \beta \mathcal{H} \mathcal{M} = \langle f_0, f_{\mathcal{H}} \rangle_{L^2(\mathcal{M}^{-1})} f_{\mathcal{H}}. $$

Moreover, the global conservations of mass and energy write

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} (f(t, x, v) - \alpha \mathcal{M}(x, v)) \, dx \, dv = 0, $$

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{H}(x, v) (f(t, x, v) - \beta \mathcal{H}(x, v) \mathcal{M}(x, v)) \, dx \, dv = 0. $$

The transport operator can be written as $\mathcal{T} f = X \cdot \nabla_{x,v} f = \nabla_{x,v} \cdot (X f)$, where $X = (-v, \nabla_x \phi) = (-\nabla_v \mathcal{H}, \nabla_x \mathcal{H})$ is a divergence free vector field, in the sense that

$$ \nabla_{x,v} \cdot X = 0. \qquad (2.15) $$

As a consequence, the volume conservation in the phase space under the action of the flow induces the local mass conservation and the (2.15) symmetry gives rise to the global mass conservation. Another symmetry is associated with the fact that $\mathcal{H}$ is conserved along the characteristics of Newton's equations $\dot{x} = v$ and $\dot{v} = -\nabla_x \phi$. This is reflected by the Poisson brackets: a stationary solution $F$ satisfies

$$ 0 = \{F, \mathcal{H}\} := \nabla_v \mathcal{H} \cdot \nabla_x F - \nabla_x \mathcal{H} \cdot \nabla_v F, \qquad (2.16) $$

but by replacing $F$ by $\mathcal{H} F$, it is also clear that

$$ \{\mathcal{H} F, \mathcal{H}\} = \mathcal{H}\{F, \mathcal{H}\} + \{\mathcal{H}, \mathcal{H}\}\mathcal{H} = 0. $$

The underlying reason is that the transport dynamics involving a time-independent potential is invariant under a translation in time, which gives rise to the global conservation of energy. These considerations can be generalized. To any continuous group of transformations which leaves $\mathcal{T}$ invariant, we can associate an infinitesimal transformation $\mathcal{G}(x, v)$
such that {$\mathcal{G}, \mathcal{H}$} = 0 and as a consequence, if $f$ solves the transport equation $\partial_t f = \mathcal{T} f$, then

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{G}(x, v) f(t, x, v) \, dx \, dv = 0. \qquad (2.17) $$

Additionally, if $v \mapsto \mathcal{G}(x, v) \mu(v)$ is in the kernel of the collision operator $\mathcal{C}$ for any $x \in \mathbb{R}^d$, then $\gamma \mathcal{G} \mathcal{M}$ is a solution of (1.4), i.e., a special macroscopic mode, for any $\gamma \in \mathbb{R}$. Then (2.17) holds for any solution $f$ of (1.1) with initial datum $f_0$ and there is a unique $\gamma \in \mathbb{R}$ such that

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{G}(x, v) (f_0(x, v) - \gamma \mathcal{G}(x, v) \mathcal{M}(x, v)) \, dx \, dv = 0. $$

More considerations on symmetries, local and global conservation laws, and Noether's theorem can be found in textbooks on classical mechanics like, for instance, [31, 42]. The case of rotational symmetries enters this framework:

When $\phi$ is invariant under a rotation, stationary rotation modes appear. Let $A \in \mathcal{R}_\phi$ as defined in (1.8) and consider the rotation group $(R_\theta)_{\theta \in \mathbb{R}}$ defined by $R_\theta := e^{\theta A}$ and a point $x_0 \in \mathbb{R}^d$ so that $\phi(R_\theta(x - x_0) + x_0) = \phi(x)$ for any $\theta \in \mathbb{R}$. By differentiation with respect to $\theta$, we get

$$ \forall x \in \mathbb{R}^d, \quad (Ax + u) \cdot \nabla_x \phi(x) = 0 $$

with $u = -Ax_0$. Integrating the above identity against $(u \cdot x) \rho$ yields $u = 0$ after an integration by parts because $\rho(x) \, dx$ is centred according to Assumption (H3). Rotation modes compatible with $\phi$ are therefore restricted to $x_0 = 0$: see [9] for similar computations. As a consequence, if we compute the Poisson bracket as defined in (2.16), we find that

$$ \{\mathcal{G}, \mathcal{H}\} = 0 \quad \text{if} \quad \mathcal{G}(x, v) = (Ax \cdot v) $$

and the conservation of the total angular momentum associated with this rotation writes

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (Ax \cdot v) f(t, x, v) \, dx \, dv = 0. $$

Given $f_0$, let us identify the corresponding special macroscopic mode.

Associated with $f_0$, we introduce the initial momentum

$$ m_0(x) := \left( \int_{\mathbb{R}^d} v f_0(x, v) \, dv \right) e^{\phi(x)} $$

and the infinitesimal rotation $x \mapsto Ax := P_\phi m_0(x)$, where $P_\phi$ is the orthogonal projection onto the vector space $\mathcal{R}_\phi$ in $L^2(\rho)$. We can then check that $x \mapsto Ax$ belongs to $\mathcal{R}_\phi$ and thus the function (rotational mode compatible with $\phi$)

$$ F_{\text{rot}}(x, v) := (Ax \cdot v) \mathcal{M} \qquad (2.18) $$

belongs to $\mathcal{R}_\phi$, so that $F_{\text{rot}}$ is a solution to (1.4) with same conserved total angular momentum as $f$. Denoting

$$ m_f(t, x) := \left( \int_{\mathbb{R}^d} v f(t, x, v) \, dv \right) e^{\phi(x)} $$
such that {$\mathcal{G}, \mathcal{H}$} = 0 and as a consequence, if $f$ solves the transport equation $\partial_t f = \mathcal{T} f$, then

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{G}(x, v) f(t, x, v) \, dx \, dv = 0. \qquad (2.17) $$

Additionally, if $v \mapsto \mathcal{G}(x, v) \mu(v)$ is in the kernel of the collision operator $\mathcal{C}$ for any $x \in \mathbb{R}^d$, then $\gamma \mathcal{G} \mathcal{M}$ is a solution of (1.4), i.e., a special macroscopic mode, for any $\gamma \in \mathbb{R}$. Then (2.17) holds for any solution $f$ of (1.1) with initial datum $f_0$ and there is a unique $\gamma \in \mathbb{R}$ such that

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathcal{G}(x, v) (f_0(x, v) - \gamma \mathcal{G}(x, v) \mathcal{M}(x, v)) \, dx \, dv = 0. $$

More considerations on symmetries, local and global conservation laws, and Noether's theorem can be found in textbooks on classical mechanics like, for instance, [31, 42]. The case of rotational symmetries enters this framework:

When $\phi$ is invariant under a rotation, stationary rotation modes appear. Let $A \in \mathcal{R}_\phi$ as defined in (1.8) and consider the rotation group $(R_\theta)_{\theta \in \mathbb{R}}$ defined by $R_\theta := e^{\theta A}$ and a point $x_0 \in \mathbb{R}^d$ so that $\phi(R_\theta(x - x_0) + x_0) = \phi(x)$ for any $\theta \in \mathbb{R}$. By differentiation with respect to $\theta$, we get

$$ \forall x \in \mathbb{R}^d, \quad (Ax + u) \cdot \nabla_x \phi(x) = 0 $$

with $u = -Ax_0$. Integrating the above identity against $(u \cdot x) \rho$ yields $u = 0$ after an integration by parts because $\rho(x) \, dx$ is centred according to Assumption (H3). Rotation modes compatible with $\phi$ are therefore restricted to $x_0 = 0$: see [9] for similar computations. As a consequence, if we compute the Poisson bracket as defined in (2.16), we find that

$$ \{\mathcal{G}, \mathcal{H}\} = 0 \quad \text{if} \quad \mathcal{G}(x, v) = (Ax \cdot v) $$

and the conservation of the total angular momentum associated with this rotation writes

$$ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (Ax \cdot v) f(t, x, v) \, dx \, dv = 0. $$

Given $f_0$, let us identify the corresponding special macroscopic mode.

Associated with $f_0$, we introduce the initial momentum

$$ m_0(x) := \left( \int_{\mathbb{R}^d} v f_0(x, v) \, dv \right) e^{\phi(x)} $$

and the infinitesimal rotation $x \mapsto Ax := P_\phi m_0(x)$, where $P_\phi$ is the orthogonal projection onto the vector space $\mathcal{R}_\phi$ in $L^2(\rho)$. We can then check that $x \mapsto Ax$ belongs to $\mathcal{R}_\phi$ and thus the function (rotational mode compatible with $\phi$)

$$ F_{\text{rot}}(x, v) := (Ax \cdot v) \mathcal{M} \qquad (2.18) $$

belongs to $\mathcal{R}_\phi$, so that $F_{\text{rot}}$ is a solution to (1.4) with same conserved total angular momentum as $f$. Denoting

$$ m_f(t, x) := \left( \int_{\mathbb{R}^d} v f(t, x, v) \, dv \right) e^{\phi(x)} $$
the momentum of $f$, the associated conservation law then reads

$$ \mathbb{P}_{\phi} m_f(t) = \mathbb{P}_{\phi} m_0 \quad \text{or equivalently} \quad \mathbb{P}(m_f - m_0) \in \mathcal{R}_{\phi}^{\perp}, \qquad (2.19) $$

where $\mathbb{P}$ is the orthogonal projection onto the vector space of all infinitesimal rotations $\mathcal{R} := \{x \mapsto Ax : A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})\}$, identified with $\mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$, in $L^2(\rho)$, and $\mathcal{R}_{\phi}^{\perp}$ is the orthogonal of $\mathcal{R}_{\phi}$, seen as a subspace of $\mathcal{R}$, for the scalar product induced by $L^2(\rho)$. We refer to Lemma A.1 in Appendix A.1 for a precise statement and a short proof.

If we denote by $(A_j)_{j \in J_{\phi}} = (x \mapsto A_j x)_{j \in J_{\phi}}$ a basis of $\mathcal{R}_{\phi}$ normalized by the condition $\|f_{\text{rot},j}\|_{L^2(\mathcal{M}^{-1})} = 1$ for any $j \in J_{\phi}$, where $f_{\text{rot},j}(x, v) := (A_j x \cdot v) \mathcal{M}(x, v)$, then the conservation of the total angular momentum implies for all $j \in J_{\phi}$:

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} [f(t, x, v) - \langle f_0, f_{\text{rot},j} \rangle_{L^2(\mathcal{M}^{-1})} f_{\text{rot},j}(x, v)] \, dx \, dv = 0. $$

Now let us turn our attention to the time-periodic special macroscopic modes and start with the harmonic directional modes, which appear when $d_{\phi} \le d - 1$. We choose a coordinate system such that $\partial_{x_i} \phi = x_i$ for $i \in I_{\phi} = \{d_{\phi} + 1, \dots, d\}$. In that case, the potential $\phi$ is such that $x \mapsto \phi(x) - \frac{1}{2} \sum_{i \in I_{\phi}} x_i^2$ depends only on $(x_1, \dots, x_{d_{\phi}})$ and for any $i \in I_{\phi}$ the harmonic directional modes, defined for all $(t, x, v) \in \mathbb{R}^{+} \times \mathbb{R}^d \times \mathbb{R}^d$ by

$$ \begin{aligned} f_{\text{dir},i}^{+}(t, x, v) &:= (x_i \cos t + v_i \sin t) \mathcal{M}(x, v), \\ f_{\text{dir},i}^{-}(t, x, v) &:= (v_i \cos t - x_i \sin t) \mathcal{M}(x, v), \end{aligned} $$

solve (1.4). A direct computation of the solution of (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$ shows that

$$ \begin{aligned} \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f \, dx \, dv &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f \, dx \, dv, \\ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f \, dx \, dv &= - \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f \, dx \, dv, \end{aligned} $$

which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1. For any $i \in I_{\phi}$, let us define

$$ \gamma_i := \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f_0 \, dx \, dv, \quad \bar{\gamma}_i := \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f_0 \, dx \, dv. $$

The function

$$ F_{\text{dir}} := \sum_{i=d_{\phi}+1}^{d} \left( \gamma_i f_{\text{dir},i}^{+} + \bar{\gamma}_i f_{\text{dir},i}^{-} \right) \qquad (2.20) $$

solves (1.4) and belongs to $\mathcal{D}_{\phi}$ as defined by (1.10). Moreover $(f - F_{\text{dir}})$ satisfies the following two global conservation laws: for any $t \ge 0$,

$$ \begin{aligned} \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i (f(t, x, v) - F_{\text{dir}}(t, x, v)) \, dx \, dv &= 0, \\ \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i (f(t, x, v) - F_{\text{dir}}(t, x, v)) \, dx \, dv &= 0. \end{aligned} $$
the momentum of $f$, the associated conservation law then reads

$$ \mathbb{P}_{\phi} m_f(t) = \mathbb{P}_{\phi} m_0 \quad \text{or equivalently} \quad \mathbb{P}(m_f - m_0) \in \mathcal{R}_{\phi}^{\perp}, \qquad (2.19) $$

where $\mathbb{P}$ is the orthogonal projection onto the vector space of all infinitesimal rotations $\mathcal{R} := \{x \mapsto Ax : A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})\}$, identified with $\mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$, in $L^2(\rho)$, and $\mathcal{R}_{\phi}^{\perp}$ is the orthogonal of $\mathcal{R}_{\phi}$, seen as a subspace of $\mathcal{R}$, for the scalar product induced by $L^2(\rho)$. We refer to Lemma A.1 in Appendix A.1 for a precise statement and a short proof.

If we denote by $(A_j)_{j \in J_{\phi}} = (x \mapsto A_j x)_{j \in J_{\phi}}$ a basis of $\mathcal{R}_{\phi}$ normalized by the condition $\|f_{\text{rot},j}\|_{L^2(\mathcal{M}^{-1})} = 1$ for any $j \in J_{\phi}$, where $f_{\text{rot},j}(x, v) := (A_j x \cdot v) \mathcal{M}(x, v)$, then the conservation of the total angular momentum implies for all $j \in J_{\phi}$:

$$ \forall t \ge 0, \quad \iint_{\mathbb{R}^d \times \mathbb{R}^d} [f(t, x, v) - \langle f_0, f_{\text{rot},j} \rangle_{L^2(\mathcal{M}^{-1})} f_{\text{rot},j}(x, v)] \, dx \, dv = 0. $$

Now let us turn our attention to the time-periodic special macroscopic modes and start with the harmonic directional modes, which appear when $d_{\phi} \le d - 1$. We choose a coordinate system such that $\partial_{x_i} \phi = x_i$ for $i \in I_{\phi} = \{d_{\phi} + 1, \dots, d\}$. In that case, the potential $\phi$ is such that $x \mapsto \phi(x) - \frac{1}{2} \sum_{i \in I_{\phi}} x_i^2$ depends only on $(x_1, \dots, x_{d_{\phi}})$ and for any $i \in I_{\phi}$ the harmonic directional modes, defined for all $(t, x, v) \in \mathbb{R}^{+} \times \mathbb{R}^d \times \mathbb{R}^d$ by

$$ \begin{aligned} f_{\text{dir},i}^{+}(t, x, v) &:= (x_i \cos t + v_i \sin t) \mathcal{M}(x, v), \\ f_{\text{dir},i}^{-}(t, x, v) &:= (v_i \cos t - x_i \sin t) \mathcal{M}(x, v), \end{aligned} $$

solve (1.4). A direct computation of the solution of (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$ shows that

$$ \begin{aligned} \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f \, dx \, dv &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f \, dx \, dv, \\ \frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f \, dx \, dv &= - \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f \, dx \, dv, \end{aligned} $$

which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1. For any $i \in I_{\phi}$, let us define

$$ \gamma_i := \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i f_0 \, dx \, dv, \quad \bar{\gamma}_i := \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i f_0 \, dx \, dv. $$

The function

$$ F_{\text{dir}} := \sum_{i=d_{\phi}+1}^{d} \left( \gamma_i f_{\text{dir},i}^{+} + \bar{\gamma}_i f_{\text{dir},i}^{-} \right) \qquad (2.20) $$

solves (1.4) and belongs to $\mathcal{D}_{\phi}$ as defined by (1.10). Moreover $(f - F_{\text{dir}})$ satisfies the following two global conservation laws: for any $t \ge 0$,

$$ \begin{aligned} \iint_{\mathbb{R}^d \times \mathbb{R}^d} x_i (f(t, x, v) - F_{\text{dir}}(t, x, v)) \, dx \, dv &= 0, \\ \iint_{\mathbb{R}^d \times \mathbb{R}^d} v_i (f(t, x, v) - F_{\text{dir}}(t, x, v)) \, dx \, dv &= 0. \end{aligned} $$
When all coordinates are harmonic ($d_\phi = 0$), then $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ due to the normalization (H7) and the harmonic pulsating modes, defined for all $(t, x, v) \in \mathbb{R}^+ \times \mathbb{R}^d \times \mathbb{R}^d$ by

$$f_{\text{pul}}^{+}(t, x, v) := \frac{1}{\sqrt{d}} (x \cdot v \cos(2t) + \frac{1}{2} (|x|^2 - |v|^2) \sin(2t)) \mathcal{M}(x, v),$$

$$f_{\text{pul}}^{-}(t, x, v) := \frac{1}{\sqrt{d}} \left( \frac{1}{2} (|x|^2 - |v|^2) \cos(2t) - x \cdot v \sin(2t) \right) \mathcal{M}(x, v),$$

solve (1.4). A direct computation of the solution of (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$ shows that

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f \, dx \, dv = -2 \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f \, dx \, dv,$$

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f \, dx \, dv = 2 \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f \, dx \, dv,$$

which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1/2. With

$$\delta := \frac{1}{\sqrt{d}} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f_0 \, dx \, dv, \quad \bar{\delta} := \frac{1}{\sqrt{d}} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f_0 \, dx \, dv,$$

the function

$$F_{\text{pul}} := \delta f_{\text{pul}}^{+} + \bar{\delta} f_{\text{pul}}^{-} \qquad (2.21)$$

solves (1.4) and belongs to $\mathfrak{P}_\phi$ as defined by (1.11). Moreover $(f - F_{\text{pul}})$ satisfies the following two global conservation laws: for any $t \ge 0$,

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) (f(t, x, v) - F_{\text{pul}}(t, x, v)) \, dx \, dv = 0,$$

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) (f(t, x, v) - F_{\text{pul}}(t, x, v)) \, dx \, dv = 0.$$

Let us consider the set of the generators of all above special macroscopic modes

$$\hat{S} := \{f_1, f_H\} \cup \{f_{\text{rot},j}\}_{j \in J_\phi} \cup \{f_{\text{dir},i}^{\pm}\}_{i \in I_\phi, \pm} \cup \{f_{\text{pul}}^{\pm}\}_{\pm}.$$

We have the following orthogonality property.

**Lemma 2.2.** The functions of $\hat{S}$ are orthonormal in $L^2(\mathcal{M})$.

*Proof of Lemma 2.2.* This follows from direct computation using standard properties of Hermite functions. ■

As a straightforward consequence of Lemma 2.2, we obtain

**Corollary 2.3.** Assume that $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ is a solution to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With the above notations, for any $f \in \hat{S}$ and any $t \ge 0$, we have

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} (f(t) - \alpha \mathcal{M} - \beta H \mathcal{M} - F_{\text{rot}} - F_{\text{dir}}(t) - F_{\text{pul}}(t)) \frac{f}{\mathcal{M}} \, dx \, dv = 0.$$
When all coordinates are harmonic ($d_\phi = 0$), then $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ due to the normalization (H7) and the harmonic pulsating modes, defined for all $(t, x, v) \in \mathbb{R}^+ \times \mathbb{R}^d \times \mathbb{R}^d$ by

$$f_{\text{pul}}^{+}(t, x, v) := \frac{1}{\sqrt{d}} (x \cdot v \cos(2t) + \frac{1}{2} (|x|^2 - |v|^2) \sin(2t)) \mathcal{M}(x, v),$$

$$f_{\text{pul}}^{-}(t, x, v) := \frac{1}{\sqrt{d}} \left( \frac{1}{2} (|x|^2 - |v|^2) \cos(2t) - x \cdot v \sin(2t) \right) \mathcal{M}(x, v),$$

solve (1.4). A direct computation of the solution of (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$ shows that

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f \, dx \, dv = -2 \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f \, dx \, dv,$$

$$\frac{d}{dt} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f \, dx \, dv = 2 \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f \, dx \, dv,$$

which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1/2. With

$$\delta := \frac{1}{\sqrt{d}} \iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) f_0 \, dx \, dv, \quad \bar{\delta} := \frac{1}{\sqrt{d}} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) f_0 \, dx \, dv,$$

the function

$$F_{\text{pul}} := \delta f_{\text{pul}}^{+} + \bar{\delta} f_{\text{pul}}^{-} \qquad (2.21)$$

solves (1.4) and belongs to $\mathfrak{P}_\phi$ as defined by (1.11). Moreover $(f - F_{\text{pul}})$ satisfies the following two global conservation laws: for any $t \ge 0$,

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} (x \cdot v) (f(t, x, v) - F_{\text{pul}}(t, x, v)) \, dx \, dv = 0,$$

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{1}{2} (|x|^2 - |v|^2) (f(t, x, v) - F_{\text{pul}}(t, x, v)) \, dx \, dv = 0.$$

Let us consider the set of the generators of all above special macroscopic modes

$$\hat{S} := \{f_1, f_{\mathcal{H}}\} \cup \{f_{\text{rot},j}\}_{j \in J_\phi} \cup \{f_{\text{dir},i}^{\pm}\}_{i \in I_\phi, \pm} \cup \{f_{\text{pul}}^{\pm}\}_{\pm}.$$

We have the following orthogonality property.

**Lemma 2.2.** The functions of $\hat{S}$ are orthonormal in $L^2(\mathcal{M})$.

*Proof of Lemma 2.2.* This follows from direct computation using standard properties of Hermite functions. ■

As a straightforward consequence of Lemma 2.2, we obtain

**Corollary 2.3.** Assume that $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ is a solution to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With the above notations, for any $f \in \hat{S}$ and any $t \ge 0$, we have

$$\iint_{\mathbb{R}^d \times \mathbb{R}^d} (f(t) - \alpha \mathcal{M} - \beta \mathcal{H} \mathcal{M} - F_{\text{rot}} - F_{\text{dir}}(t) - F_{\text{pul}}(t)) \frac{f}{\mathcal{M}} \, dx \, dv = 0.$$
## 2.3. A micro-macro decomposition

Let us consider a solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1). We get rid of the special macroscopic modes built in (2.13)–(2.14)–(2.18)–(2.20)–(2.21) and rewrite the evolution problem in $L^2(\mathcal{M})$ in terms of

$$h := \frac{f - \alpha \mathcal{M} - \beta \mathcal{H} \mathcal{M} - F_{\text{rot}} - F_{\text{dir}} - F_{\text{pul}}}{\mathcal{M}} \qquad (2.22)$$

for all $(t, x, v)$, where only $h$, $f$, $F_{\text{dir}}$ and $F_{\text{pul}}$ depend on $t$. Then $h$ satisfies

$$\partial_t h = \mathcal{L} h := \mathcal{T} h + Ch, \quad h|_{t=0} = h_0 \qquad (2.23)$$

with $\mathcal{T}$ defined as before by $\mathcal{T} h = \nabla_x \phi \cdot \nabla_v h - v \cdot \nabla_x h$ and the new collision operator

$$Ch := \mu^{-1} \mathcal{C} (\mu h) \qquad (2.24)$$

where $\mu$ is defined in (1.2).

The operator $C$ acts only on the velocity variable, is self-adjoint in $L^2(\mathcal{M})$ (when integrating in $x$, $v$) and $L^2(\mu)$ (when integrating in $v$) and

$$\text{Ker } C = \text{Span} \{1, v_1, \dots, v_d, |v|^2\}.$$

Let us consider the micro-macro decomposition

$$h = h^{\parallel} + h^{\perp}, \quad h^{\parallel} := r + m \cdot v + e \mathcal{E}(v),$$

where $h^{\parallel}$ is the $L^2(\mu)$-orthogonal projection of $h$ on $\text{Ker } C$ and $\mathcal{E}$ defined by (1.6) is a normalized Hermite polynomial of degree 2. In other words,

$$\begin{align}
r(t, x) &:= \int_{\mathbb{R}^d} h(t, x, v) \mu(v) \, dv \notag \\
m(t, x) &:= \int_{\mathbb{R}^d} v h(t, x, v) \mu(v) \, dv \tag{2.25} \\
e(t, x) &:= \int_{\mathbb{R}^d} \mathcal{E}(v) h(t, x, v) \mu(v) \, dv \notag
\end{align}$$

are the macroscopic quantities corresponding to the spatial density, the local flux and thermal energy, while $h^{\perp}$ is the microscopic part. The definition (2.25) coincides with the definition (1.5) used in Section 2.1 to define the special macroscopic modes.

With these notations (H1) reads

$$- \langle Ch, h \rangle \geq c_{\mathcal{C}} \|h^{\perp}\|^2.$$

According to Corollary 2.3, $h$ has multiple global conservation laws.
## 2.3. A micro-macro decomposition

Let us consider a solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1). We get rid of the special macroscopic modes built in (2.13)–(2.14)–(2.18)–(2.20)–(2.21) and rewrite the evolution problem in $L^2(\mathcal{M})$ in terms of

$$h := \frac{f - \alpha \mathcal{M} - \beta \mathcal{H} \mathcal{M} - F_{\text{rot}} - F_{\text{dir}} - F_{\text{pul}}}{\mathcal{M}} \quad (2.22)$$

for all $(t, x, v)$, where only $h$, $f$, $F_{\text{dir}}$ and $F_{\text{pul}}$ depend on $t$. Then $h$ satisfies

$$\partial_t h = \mathcal{L} h := \mathcal{T} h + Ch, \quad h|_{t=0} = h_0 \quad (2.23)$$

with $\mathcal{T}$ defined as before by $\mathcal{T} h = \nabla_x \phi \cdot \nabla_v h - v \cdot \nabla_x h$ and the new collision operator

$$Ch := \mu^{-1} \mathcal{C}(\mu h) \quad (2.24)$$

where $\mu$ is defined in (1.2).

The operator $C$ acts only on the velocity variable, is self-adjoint in $L^2(\mathcal{M})$ (when integrating in $x$, $v$) and $L^2(\mu)$ (when integrating in $v$) and

$$\text{Ker } C = \text{Span} \{1, v_1, \dots, v_d, |v|^2\}.$$

Let us consider the micro-macro decomposition

$$h = h^{\|} + h^{\perp}, \quad h^{\|} := r + m \cdot v + e \mathcal{E}(v),$$

where $h^{\|}$ is the $L^2(\mu)$-orthogonal projection of $h$ on $\text{Ker } C$ and $\mathcal{E}$ defined by (1.6) is a normalized Hermite polynomial of degree 2. In other words,

$$\begin{align}
r(t, x) &:= \int_{\mathbb{R}^d} h(t, x, v) \mu(v) \, dv \nonumber \\
m(t, x) &:= \int_{\mathbb{R}^d} v h(t, x, v) \mu(v) \, dv \tag{2.25} \\
e(t, x) &:= \int_{\mathbb{R}^d} \mathcal{E}(v) h(t, x, v) \mu(v) \, dv \nonumber
\end{align}$$

are the macroscopic quantities corresponding to the spatial density, the local flux and thermal energy, while $h^{\perp}$ is the microscopic part. The definition (2.25) coincides with the definition (1.5) used in Section 2.1 to define the special macroscopic modes.

With these notations (H1) reads

$$- \langle Ch, h \rangle \geq c_{\mathcal{C}} \|h^{\perp}\|^2.$$

According to Corollary 2.3, $h$ has multiple global conservation laws.
**Corollary 2.4.** Assume that $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ is a solution to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With $h$ and $(r, m, e)$ respectively defined by (2.22)–(2.23) and (2.25), we have the following properties.

▷ Conservation of total mass and total energy

$$ \langle r \rangle = 0 \quad \text{and} \quad \sqrt{\frac{d}{2}} \langle e \rangle + \langle \phi r \rangle = 0. \qquad (2.26) $$

▷ Global conservation laws associated to rotational symmetries of $\phi$

$$ \mathbb{P}_{\phi} m = 0. \qquad (2.27) $$

This also means $\mathbb{P}(m) \in \mathcal{R}_{\phi}^{\perp}$ as in (2.19).

▷ Global conservation laws corresponding to the harmonic directional modes

$$ \forall i \in I_{\phi}, \quad \langle r x_i \rangle = 0 \quad \text{and} \quad \langle m_i \rangle = 0. \qquad (2.28) $$

▷ In the fully harmonic case $d_{\phi} = 0$, global conservation laws corresponding to the harmonic pulsating modes

$$ \langle m \cdot x \rangle = 0 \quad \text{and} \quad \sqrt{\frac{d}{2}} \langle e \rangle - \langle \phi r \rangle = 0. \qquad (2.29) $$

## 2.4. The equations for the macroscopic modes

We write the evolution equations for $r$, $m$ and $e$ defined in (2.25). In the mathematical literature, such equations are sometimes called *local conservation laws* but as this might introduce confusions with the *local conservation law of the collision operator*, which we call here *collision invariants*, and the *global conservation laws* studied in Section 2.2, we shall simply refer to these equations as the *equations for the macroscopic modes* or simply the *macroscopic equations*.

Assume that $h$ solves (2.23). For any Hermite polynomial $p: \mathbb{R}^d \to \mathbb{R}$ considered as a function of the velocity variable, we compute $J_p[h] = \int_{\mathbb{R}^d} p h \mu \, dv$ using standard properties of Hermite functions:

$$ J_p[h] = r \int_{\mathbb{R}^d} p \mu \, dv + m \cdot \int_{\mathbb{R}^d} v p \mu \, dv + e \int_{\mathbb{R}^d} \mathfrak{E} p \mu \, dv + J_p[h^{\perp}] $$

and, using (2.23), we also get

$$ \begin{aligned} \partial_t J_p[h] = & - \nabla_x r \cdot \int_{\mathbb{R}^d} v p \mu \, dv - \nabla_x m : \int_{\mathbb{R}^d} v \otimes v p \mu \, dv + m \cdot \nabla_x \phi \int_{\mathbb{R}^d} p \mu \, dv \\ & - \nabla_x e \cdot \int_{\mathbb{R}^d} v \mathfrak{E} p \mu \, dv + \sqrt{\frac{2}{d}} e \nabla \phi \cdot \int_{\mathbb{R}^d} v p \mu \, dv + \int_{\mathbb{R}^d} (\mathcal{L} h^{\perp}) p \mu \, dv. \end{aligned} $$

Plugging successively $p = 1$, $v$, $\mathfrak{E}$, $v \otimes v - \text{Id}_{d \times d}$ and $v(\mathfrak{E} - \sqrt{\frac{2}{d}})$ we get

$$ \partial_t r = \nabla_x^* m, \qquad (2.30a) $$
**Corollary 2.4.** Assume that $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ is a solution to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With $h$ and $(r, m, e)$ respectively defined by (2.22)–(2.23) and (2.25), we have the following properties.

▷ Conservation of total mass and total energy

$$ \langle r \rangle = 0 \quad \text{and} \quad \sqrt{\frac{d}{2}} \langle e \rangle + \langle \phi r \rangle = 0. \qquad (2.26) $$

▷ Global conservation laws associated to rotational symmetries of $\phi$

$$ \mathbb{P}_{\phi} m = 0. \qquad (2.27) $$

This also means $\mathbb{P}(m) \in \mathcal{R}_{\phi}^{\perp}$ as in (2.19).

▷ Global conservation laws corresponding to the harmonic directional modes

$$ \forall i \in I_{\phi}, \quad \langle r x_i \rangle = 0 \quad \text{and} \quad \langle m_i \rangle = 0. \qquad (2.28) $$

▷ In the fully harmonic case $d_{\phi} = 0$, global conservation laws corresponding to the harmonic pulsating modes

$$ \langle m \cdot x \rangle = 0 \quad \text{and} \quad \sqrt{\frac{d}{2}} \langle e \rangle - \langle \phi r \rangle = 0. \qquad (2.29) $$

## 2.4. The equations for the macroscopic modes

We write the evolution equations for $r$, $m$ and $e$ defined in (2.25). In the mathematical literature, such equations are sometimes called local conservation laws but as this might introduce confusions with the local conservation law of the collision operator, which we call here collision invariants, and the global conservation laws studied in Section 2.2, we shall simply refer to these equations as the equations for the macroscopic modes or simply the macroscopic equations.

Assume that $h$ solves (2.23). For any Hermite polynomial $p: \mathbb{R}^d \to \mathbb{R}$ considered as a function of the velocity variable, we compute $J_p[h] = \int_{\mathbb{R}^d} p h \mu \, dv$ using standard properties of Hermite functions:

$$ J_p[h] = r \int_{\mathbb{R}^d} p \mu \, dv + m \cdot \int_{\mathbb{R}^d} v p \mu \, dv + e \int_{\mathbb{R}^d} \mathfrak{E} p \mu \, dv + J_p[h^{\perp}] $$

and, using (2.23), we also get

$$ \begin{aligned} \partial_t J_p[h] = & - \nabla_x r \cdot \int_{\mathbb{R}^d} v p \mu \, dv - \nabla_x m : \int_{\mathbb{R}^d} v \otimes v p \mu \, dv + m \cdot \nabla_x \phi \int_{\mathbb{R}^d} p \mu \, dv \\ & - \nabla_x e \cdot \int_{\mathbb{R}^d} v \mathfrak{E} p \mu \, dv + \sqrt{\frac{2}{d}} e \nabla \phi \cdot \int_{\mathbb{R}^d} v p \mu \, dv + \int_{\mathbb{R}^d} (\mathcal{L} h^{\perp}) p \mu \, dv. \end{aligned} $$

Plugging successively $p = 1$, $v$, $\mathfrak{E}$, $v \otimes v - \text{Id}_{d \times d}$ and $v(\mathfrak{E} - \sqrt{\frac{2}{d}})$ we get

$$ \partial_t r = \nabla_x^* m, \qquad (2.30a) $$
$$ \partial_t m = - \nabla_x r + \sqrt{\frac{2}{d}} \nabla_x^* e + \nabla_x^* \cdot E[h^\perp], \qquad (2.30b) $$

$$ \partial_t e = - \sqrt{\frac{2}{d}} \nabla_x \cdot m + \nabla_x^* \cdot \Theta[h^\perp], \qquad (2.30c) $$

$$ \partial_t E[h] = -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^\perp], \qquad (2.30d) $$

$$ \partial_t \Theta[h] = - \left(1 + \frac{2}{d}\right) \nabla_x e + \Theta[\mathcal{L} h^\perp], \qquad (2.30e) $$

where $\nabla_x^{\text{sym}} m$ is defined by (2.2) and the matrix valued function $E[h]$ and the vector valued function $\Theta[h]$ are higher-order moments of $h$ defined by

$$ E[h] := \int_{\mathbb{R}^d} (v \otimes v - \text{Id}_{d \times d}) h \, \mu \, dv = \sqrt{\frac{2}{d}} e \text{Id}_{d \times d} + E[h^\perp], \qquad (2.31a) $$

$$ \Theta[h] := \int_{\mathbb{R}^d} v (\mathcal{E}(v) - \sqrt{\frac{2}{d}}) h \, \mu \, dv = \Theta[h^\perp]. \qquad (2.31b) $$

If $f$ is a special macroscopic mode, then (2.30) is reduced to (2.1) because, in that case, $h^\perp = 0$ and $\Theta[h^\perp] = 0$.

### 3. Classification of the special macroscopic modes

In this section, we prove Part (1) of Theorem 1.1. We write $a \le b$ if there is some positive constant $c$ such that $a \le b c$ and $a \approx b$ if and only if $a \le b \le a$. Throughout this section, we assume that (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8) hold, without further notice.

#### 3.1. Statement and preliminary results

Theorem 1.1–(1) writes:

**Proposition 3.1** (Special macroscopic modes). If $f$ is a solution of (1.4), then $h$ given by (2.22) is such that $h = 0$.

We recall that $f$ is a special macroscopic mode if and only if, by definition (1.4), $\mathcal{C}f = 0$ and $\partial_t f = \mathcal{T}f$. With the definitions of (2.13)–(2.14)–(2.18)–(2.20)–(2.21), the function

$$ F := f - \alpha \mathcal{M} - \beta \mathcal{H} \mathcal{M} - F_{\text{rot}} - F_{\text{dir}} - F_{\text{pul}} $$

is also a special macroscopic mode and (1.5) implies that $h$ defined in (2.22) satisfies $h = h^\parallel = r + m \cdot v + e \mathcal{E}(v)$. According to (2.23), $h$ solves the transport equation $\partial_t h = \mathcal{T}h$ because $Ch = 0$ (with $C$ defined by (2.24)) and Section 2.1 proved that $r, m$ and $e$ solve (2.1). Proposition 3.1 means that $r = 0, m = 0$, and $e = 0$. We split the proof into several steps. To start with, since $1, v$ and $\mathcal{E}(v)$ are orthonormal Hermite polynomials, we have

$$ \|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2. \qquad (3.1) $$
$$ \partial_t m = - \nabla_x r + \sqrt{\frac{2}{d}} \nabla_x^* e + \nabla_x^* \cdot E[h^\perp], \quad (2.30b) $$

$$ \partial_t e = - \sqrt{\frac{2}{d}} \nabla_x \cdot m + \nabla_x^* \cdot \Theta[h^\perp], \quad (2.30c) $$

$$ \partial_t E[h] = -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^\perp], \quad (2.30d) $$

$$ \partial_t \Theta[h] = - \left(1 + \frac{2}{d}\right) \nabla_x e + \Theta[\mathcal{L} h^\perp], \quad (2.30e) $$

where $\nabla_x^{\text{sym}} m$ is defined by (2.2) and the matrix valued function $E[h]$ and the vector valued function $\Theta[h]$ are higher-order moments of $h$ defined by

$$ E[h] := \int_{\mathbb{R}^d} (v \otimes v - \text{Id}_{d \times d}) h \, \mu \, dv = \sqrt{\frac{2}{d}} e \text{Id}_{d \times d} + E[h^\perp], \quad (2.31a) $$

$$ \Theta[h] := \int_{\mathbb{R}^d} v (\mathcal{E}(v) - \sqrt{\frac{2}{d}}) h \, \mu \, dv = \Theta[h^\perp]. \quad (2.31b) $$

If $f$ is a special macroscopic mode, then (2.30) is reduced to (2.1) because, in that case, $h^\perp = 0$ and $\Theta[h^\perp] = 0$.

### 3. Classification of the special macroscopic modes

In this section, we prove Part (1) of Theorem 1.1. We write $a \le b$ if there is some positive constant $c$ such that $a \le b c$ and $a \approx b$ if and only if $a \le b \le a$. Throughout this section, we assume that (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8) hold, without further notice.

#### 3.1. Statement and preliminary results

Theorem 1.1–(1) writes:

**Proposition 3.1** (Special macroscopic modes). If $f$ is a solution of (1.4), then $h$ given by (2.22) is such that $h = 0$.

We recall that $f$ is a special macroscopic mode if and only if, by definition (1.4), $\mathcal{C}f = 0$ and $\partial_t f = \mathcal{T}f$. With the definitions of (2.13)–(2.14)–(2.18)–(2.20)–(2.21), the function

$$ F := f - \alpha \mathcal{M} - \beta \mathcal{H} \mathcal{M} - F_{\text{rot}} - F_{\text{dir}} - F_{\text{pul}} $$

is also a special macroscopic mode and (1.5) implies that $h$ defined in (2.22) satisfies $h = h^\parallel = r + m \cdot v + e \mathcal{E}(v)$. According to (2.23), $h$ solves the transport equation $\partial_t h = \mathcal{T}h$ because $Ch = 0$ (with $C$ defined by (2.24)) and Section 2.1 proved that $r, m$ and $e$ solve (2.1). Proposition 3.1 means that $r = 0, m = 0$, and $e = 0$. We split the proof into several steps. To start with, since $1, v$ and $\mathcal{E}(v)$ are orthonormal Hermite polynomials, we have

$$ \|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2. \quad (3.1) $$
**Lemma 3.2.** With the above notations, the function $h$ as in Proposition 3.1 satisfies

$$ \frac{d}{dt} \|h\|^2 = 0. $$

*Proof of Lemma 3.2.* It follows from $Ch = 0$, $\mathcal{T}^* = -\mathcal{T}$ and $\frac{d}{dt} \|h\|^2 = 2(h, \mathcal{T}h) = 0$. ■

Collecting the results of Section 2.1 and using $r_0 = \langle r \rangle = 0$, we get:

**Lemma 3.3.** Consider the function $h$ as in Proposition 3.1. With the above notations, let $A := \langle \nabla_X^{\text{skew}} m \rangle$, $b(t) := \langle m \rangle$ and $c(t) := \langle e \rangle$. Then we have

$$ r(t,x) = -x \cdot b'(t) + \frac{\xi_2(x)}{2\sqrt{2d}} c''(t) + \sqrt{\frac{2}{d}} \xi_\phi c(t), \quad (3.2a) $$

$$ m(t,x) = Ax + b(t) - \frac{x}{\sqrt{2d}} c'(t), \quad (3.2b) $$

$$ e(t,x) = c(t), \quad (3.2c) $$

where $A$ is a constant skew-symmetric matrix, while $b$ and $c$ are respectively vector valued and scalar functions of $t$ related by (2.12).

We recall that the functions $\xi_2$ and $\xi_\phi$ are defined respectively by (2.10) and (2.11). Equation (2.12) in Proposition (2.1) provides us with various estimates on $A$, $b$ and $c$ which are collected in Sections 3.2, 3.3 and 3.4 in order to prove Proposition 3.1 in Section 3.5.

## 3.2. Control of A

For any $A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$, let us define

$$ |A|^2 := \int_{\mathbb{R}^d} |Ax|^2 \rho(x) \, dx. $$

The vector space $\mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$ is of finite dimension: all norms are equivalent to $| \cdot |$. This is also why the rigidity constant $c_K$ given by (1.14) is positive, which implies the Korn-type inequality

$$ \forall A \in \mathcal{R}_\phi^\perp, \quad \|\nabla_x \phi \cdot A x\|^2 \ge c_K |A|^2. \quad (3.3) $$

By multiplying (2.12) by $x_k$ for $k = 1, \dots, d$, then integrating against $\rho(x)$ and performing some integrations by part, using that $\rho(x) \, dx$ is centred and that the terms involving $\nabla \phi$ vanish, we obtain

$$ \frac{1}{\sqrt{2d}} \langle 2\phi x \rangle c' + \frac{1}{2\sqrt{2d}} \langle |x|^2 x \rangle c''' - \langle x \otimes x \rangle b'' = b. \quad (3.4) $$

With the notation of (3.2), let us define $X$ and $Y$ by

$$ X := \frac{1}{\sqrt{2d}} (2\xi_{\phi} + \nabla_x \phi \cdot x - d) c + \frac{\xi_2}{2\sqrt{2d}} c'' - x \cdot b' \quad (3.5) $$

and

$$ Y := \sqrt{\frac{2}{d}} \langle \phi x \rangle c + \frac{1}{2\sqrt{2d}} \langle |x|^2 x \rangle c'' - \langle x \otimes x \rangle b'. \quad (3.6) $$
**Lemma 3.2.** With the above notations, the function $h$ as in Proposition 3.1 satisfies

$$ \frac{d}{dt} \|h\|^2 = 0. $$

*Proof of Lemma 3.2.* It follows from $Ch = 0$, $\mathcal{T}^* = -\mathcal{T}$ and $\frac{d}{dt} \|h\|^2 = 2 \langle h, \mathcal{T}h \rangle = 0$. ■

Collecting the results of Section 2.1 and using $r_0 = \langle r \rangle = 0$, we get:

**Lemma 3.3.** Consider the function $h$ as in Proposition 3.1. With the above notations, let $A := \langle \nabla_X^{\text{skew}} m \rangle$, $b(t) := \langle m \rangle$ and $c(t) := \langle e \rangle$. Then we have

$$ r(t, x) = -x \cdot b'(t) + \frac{\xi_2(x)}{2\sqrt{2d}} c''(t) + \sqrt{\frac{2}{d}} \xi_{\phi} c(t), \quad (3.2a) $$

$$ m(t, x) = Ax + b(t) - \frac{x}{\sqrt{2d}} c'(t), \quad (3.2b) $$

$$ e(t, x) = c(t), \quad (3.2c) $$

where $A$ is a constant skew-symmetric matrix, while $b$ and $c$ are respectively vector valued and scalar functions of $t$ related by (2.12).

We recall that the functions $\xi_2$ and $\xi_\phi$ are defined respectively by (2.10) and (2.11). Equation (2.12) in Proposition (2.1) provides us with various estimates on $A$, $b$ and $c$ which are collected in Sections 3.2, 3.3 and 3.4 in order to prove Proposition 3.1 in Section 3.5.

## 3.2. Control of A

For any $A \in \mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$, let us define

$$ |A|^2 := \int_{\mathbb{R}^d} |Ax|^2 \rho(x) \, dx. $$

The vector space $\mathfrak{M}_{d \times d}^{\text{skew}}(\mathbb{R})$ is of finite dimension: all norms are equivalent to $| \cdot |$. This is also why the rigidity constant $c_K$ given by (1.14) is positive, which implies the Korn-type inequality

$$ \forall A \in \mathcal{R}_{\phi}^{\perp}, \quad \|\nabla_x \phi \cdot A x\|^2 \ge c_K |A|^2. \quad (3.3) $$

By multiplying (2.12) by $x_k$ for $k = 1, \dots, d$, then integrating against $\rho(x)$ and performing some integrations by part, using that $\rho(x) \, dx$ is centred and that the terms involving $\nabla \phi$ vanish, we obtain

$$ \frac{1}{\sqrt{2d}} \langle 2\phi x \rangle c' + \frac{1}{2\sqrt{2d}} \langle |x|^2 x \rangle c''' - \langle x \otimes x \rangle b'' = b. \quad (3.4) $$

With the notation of (3.2), let us define $X$ and $Y$ by

$$ X := \frac{1}{\sqrt{2d}} (2\xi_{\phi} + \nabla_x \phi \cdot x - d) c + \frac{\xi_2}{2\sqrt{2d}} c'' - x \cdot b' \quad (3.5) $$

and

$$ Y := \sqrt{\frac{2}{d}} \langle \phi x \rangle c + \frac{1}{2\sqrt{2d}} \langle |x|^2 x \rangle c'' - \langle x \otimes x \rangle b'. \quad (3.6) $$
Identities (2.12) and (3.4) yield

$$ \frac{d}{dt} (X - Y \cdot \nabla_x \phi) = \nabla_x \phi \cdot A x $$

where, according to Lemma 3.3, the r.h.s. is independent of $t$. As a consequence, we have the following estimate.

**Lemma 3.4.** Consider the function $h$ as in Proposition 3.1. The infinitesimal rotation matrix $A$ of Lemma 3.3 satisfies

$$ -\frac{d}{dt} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle = - \|\nabla_x \phi \cdot A x\|^2 \le -c_K |A|^2. $$

*Proof.* By the conservation law (2.27), we know that $A x = P(m) \in \mathbb{R}_\phi^\perp$, so that (3.3) applies. ■

### 3.3. Control of $b, b'', c'$ and $c'''$

**Lemma 3.5.** Consider the function $h$ as in Proposition 3.1. The functions $b$ and $c$ as defined in Lemma 3.3 are such that

$$ |b| + |b''| + |c'| + |c'''| \le |A|. \quad (3.7) $$

*Proof of Lemma 3.5.* Multiplying (2.12) by $\nabla_x \phi$ and integrating against $\rho(x)$, after integration by parts, using that $\rho$ is centred and observing that the terms involving $2\xi_\phi - d$ and $c'''$ vanish, it follows that

$$ b'' = - \langle \nabla_x^2 \phi \rangle b + \frac{1}{\sqrt{2d}} \langle \nabla_x^2 \phi x \rangle c' + R_1 = -b + \frac{1}{\sqrt{2d}} \langle \nabla_x^2 \phi x \rangle c' + R_1 \quad (3.8) $$

with $R_0 := -\nabla_x \phi \cdot A x$ and $R_1 := \langle R_0 \nabla_x \phi \rangle = O(A)$. By inserting (3.8) into (2.12), one gets

$$ \Psi_1(x) c' + \Psi_2(x) c''' - \Phi(x) \cdot b = R_2 \quad (3.9) $$

with

$$ \begin{gathered} \Phi(x) := \nabla_x \phi - \langle \nabla_x^2 \phi \rangle x = \nabla_x \phi - x, \\ \Psi_1(x) := \frac{2 \xi_\phi(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} - x \cdot \frac{\langle \nabla_x^2 \phi x \rangle}{\sqrt{2d}}, \quad \Psi_2(x) := \frac{\xi_\phi(x)}{2\sqrt{2d}}, \end{gathered} $$

and $R_2 := R_1 \cdot x - R_0$. Let

$$ M_\phi := \langle \Phi \otimes \Phi \rangle \in \mathfrak{M}_{d \times d}^{\text{sym}}(\mathbb{R}), \quad \alpha_i := \langle \Psi_i \Phi \rangle \in \mathbb{R}^d \quad (3.10) $$

with $i = 1, 2$. A multiplication of (3.9) by $\Phi$ and an integration against $\rho$ yields

$$ M_\phi b = \alpha_1 c' + \alpha_2 c''' + R_3 \quad (3.11) $$

where $R_3 := -\langle R_2 \Phi \rangle = O(A)$ thanks to the moment bounds on $\phi$ deduced from (H6). Inverting the matrix $M_\phi$ allows to control $b$ by $c$ and $c''$, and rewrite (3.9) as an ordinary differential equation on $c$, up to an error term of the order of $O(A)$. If $M_\phi$ is not invertible,
Identities (2.12) and (3.4) yield

$$ \frac{d}{dt} (X - Y \cdot \nabla_x \phi) = \nabla_x \phi \cdot A x $$

where, according to Lemma 3.3, the r.h.s. is independent of $t$. As a consequence, we have the following estimate.

**Lemma 3.4.** Consider the function $h$ as in Proposition 3.1. The infinitesimal rotation matrix $A$ of Lemma 3.3 satisfies

$$ -\frac{d}{dt} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle = - \| \nabla_x \phi \cdot A x \|^2 \le -c_K |A|^2. $$

*Proof.* By the conservation law (2.27), we know that $Ax = P(m) \in \mathcal{R}_{\phi}^{\perp}$, so that (3.3) applies. ■

### 3.3. Control of $b$, $b''$, $c'$ and $c'''$

**Lemma 3.5.** Consider the function $h$ as in Proposition 3.1. The functions $b$ and $c$ as defined in Lemma 3.3 are such that

$$ |b| + |b''| + |c'| + |c'''| \le |A|. \quad (3.7) $$

*Proof of Lemma 3.5.* Multiplying (2.12) by $\nabla_x \phi$ and integrating against $\rho(x)$, after integration by parts, using that $\rho$ is centred and observing that the terms involving $2\xi_\phi - d$ and $c'''$ vanish, it follows that

$$ b'' = - \langle \nabla_x^2 \phi \rangle b + \frac{1}{\sqrt{2d}} \langle \nabla_x^2 \phi x \rangle c' + R_1 = -b + \frac{1}{\sqrt{2d}} \langle \nabla_x^2 \phi x \rangle c' + R_1 \quad (3.8) $$

with $R_0 := -\nabla_x \phi \cdot Ax$ and $R_1 := \langle R_0 \nabla_x \phi \rangle = O(A)$. By inserting (3.8) into (2.12), one gets

$$ \Psi_1(x) c' + \Psi_2(x) c''' - \Phi(x) \cdot b = R_2 \quad (3.9) $$

with

$$ \begin{gathered} \Phi(x) := \nabla_x \phi - \langle \nabla_x^2 \phi \rangle x = \nabla_x \phi - x, \\ \Psi_1(x) := \frac{2 \xi_\phi(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} - x \cdot \frac{\langle \nabla_x^2 \phi x \rangle}{\sqrt{2d}}, \quad \Psi_2(x) := \frac{\xi_\phi(x)}{2\sqrt{2d}}, \end{gathered} $$

and $R_2 := R_1 \cdot x - R_0$. Let

$$ M_{\phi} := \langle \Phi \otimes \Phi \rangle \in \mathfrak{M}_{d \times d}^{\text{sym}}(\mathbb{R}), \quad \alpha_i := \langle \Psi_i \Phi \rangle \in \mathbb{R}^d \quad (3.10) $$

with $i = 1, 2$. A multiplication of (3.9) by $\Phi$ and an integration against $\rho$ yields

$$ M_{\phi} b = \alpha_1 c' + \alpha_2 c''' + R_3 \quad (3.11) $$

where $R_3 := -\langle R_2 \Phi \rangle = O(A)$ thanks to the moment bounds on $\phi$ deduced from (H6). Inverting the matrix $M_\phi$ allows to control $b$ by $c$ and $c''$, and rewrite (3.9) as an ordinary differential equation on $c$, up to an error term of the order of $O(A)$. If $M_\phi$ is not invertible,
a similar estimate can still be done after taking into account the global conservation laws of Corollary 2.4.

We recall that $E_\phi$ is defined by (1.9). We distinguish three cases.

▷ Fully non-harmonic case ($d_\phi = d$). The matrix $M_\phi$ is invertible (see Lemma A.2 in Appendix A) and (3.11) yields

$$b = M_{\phi}^{-1} (\alpha_1 c' + \alpha_2 c'' + R_3) \quad (3.12)$$

and hence, together with (3.9), it follows that

$$\tilde{\Psi}_1(x) c' + \tilde{\Psi}_2(x) c''' = R_4, \quad (3.13)$$

with $R_4 := R_2 + \Phi(x) \cdot M_\phi^{-1} R_3$ and

$$\tilde{\Psi}_1(x) := \Psi_1(x) - \Phi(x) \cdot M_\phi^{-1} \alpha_1, \quad \tilde{\Psi}_2(x) := \Psi_2(x) - \Phi(x) \cdot M_\phi^{-1} \alpha_2. \quad (3.14)$$

From Lemma A.3 we know that Rank($\tilde{\Psi}_1, \tilde{\Psi}_2$) = 2 and deduce from (3.13) that $c' = O(A)$ and $c''' = O(A)$. Using then (3.12) and (3.8), we also deduce $b = O(A)$ and $b'' = O(A)$, and the proof is complete in this case.

▷ Partially harmonic case ($1 \le d_\phi \le d-1$). Let $\{e_1, \dots, e_{d_\phi}, e_{d_\phi+1}, \dots, e_d\}$ be a basis of $\mathbb{R}^d$ such that $\{e_1, \dots, e_{d_\phi}\}$ generates $E_\phi$. For any vector $x \in \mathbb{R}^d$, we shall write $x = (\hat{x}, \check{x})$ with $\hat{x} \in \mathbb{R}^{d_\phi}$ and $\check{x} \in \mathbb{R}^{d-d_\phi}$. Similarly, we use the notation $\xi(x) = (\hat{\xi}(x), \check{\xi}(x))$ for a vector-field $\xi: \mathbb{R}^d \to \mathbb{R}^d$. In particular one has $\Phi = (\hat{\Phi}, 0)$ and also $\check{b} = 0$ so that $b = (\hat{b}, 0)$ as a consequence of (2.28). Hence (3.9) becomes

$$\Psi_1 c' + \Psi_2 c''' - \hat{\Phi} \cdot \hat{b} = R_2. \quad (3.15)$$

The matrix $M_\phi$ defined in (3.10) is given by

$$M_\phi = \begin{pmatrix} \hat{M}_\phi & 0 \\ 0 & 0 \end{pmatrix}$$

where

$$\hat{M}_{\phi} := \langle \hat{\Phi} \otimes \hat{\Phi} \rangle \in \mathfrak{M}_{d_{\phi} \times d_{\phi}}^{\text{sym}}(\mathbb{R}). \quad (3.16)$$

Following the same procedure as in the fully non-harmonic case, we obtain after multiplication by $\hat{\Phi}$ and integration in $L^2(\rho)$ that

$$\hat{M}_{\phi} \hat{b} = \hat{\alpha}_1 c' + \hat{\alpha}_2 c''' + \hat{R}_3, \quad (3.17)$$

with $\hat{R}_3 := -\langle R_2 \hat{\Phi} \rangle = O(A)$, $\hat{\alpha}_1 := \langle \Psi_1 \hat{\Phi} \rangle$ and $\hat{\alpha}_2 := \langle \Psi_2 \hat{\Phi} \rangle$. The matrix $\hat{M}_\phi$ is invertible (see Lemma A.2 in Appendix A) and (3.17) yields

$$\hat{b} = \hat{M}_{\phi}^{-1} (\hat{\alpha}_1 c' + \hat{\alpha}_2 c''' + \hat{R}_3). \quad (3.18)$$

Hence, together with (3.15), it follows that

$$\tilde{\Psi}_1(x) c' + \tilde{\Psi}_2(x) c''' = \hat{R}_4$$
a similar estimate can still be done after taking into account the global conservation laws of Corollary 2.4.

We recall that $E_\phi$ is defined by (1.9). We distinguish three cases.

▷ Fully non-harmonic case ($d_\phi = d$). The matrix $M_\phi$ is invertible (see Lemma A.2 in Appendix A) and (3.11) yields

$$b = M_{\phi}^{-1} (\alpha_1 c' + \alpha_2 c'' + R_3) \qquad (3.12)$$

and hence, together with (3.9), it follows that

$$\tilde{\Psi}_1(x) c' + \tilde{\Psi}_2(x) c''' = R_4, \qquad (3.13)$$

with $R_4 := R_2 + \Phi(x) \cdot M_\phi^{-1} R_3$ and

$$\tilde{\Psi}_1(x) := \Psi_1(x) - \Phi(x) \cdot M_\phi^{-1} \alpha_1, \quad \tilde{\Psi}_2(x) := \Psi_2(x) - \Phi(x) \cdot M_\phi^{-1} \alpha_2. \qquad (3.14)$$

From Lemma A.3 we know that Rank($\tilde{\Psi}_1, \tilde{\Psi}_2$) = 2 and deduce from (3.13) that $c' = O(A)$ and $c''' = O(A)$. Using then (3.12) and (3.8), we also deduce $b = O(A)$ and $b'' = O(A)$, and the proof is complete in this case.

▷ Partially harmonic case ($1 \le d_\phi \le d-1$). Let $\{e_1, \dots, e_{d_\phi}, e_{d_\phi+1}, \dots, e_d\}$ be a basis of $\mathbb{R}^d$ such that $\{e_1, \dots, e_{d_\phi}\}$ generates $E_\phi$. For any vector $x \in \mathbb{R}^d$, we shall write $x = (\hat{x}, \check{x})$ with $\hat{x} \in \mathbb{R}^{d_\phi}$ and $\check{x} \in \mathbb{R}^{d-d_\phi}$. Similarly, we use the notation $\xi(x) = (\hat{\xi}(x), \check{\xi}(x))$ for a vector-field $\xi: \mathbb{R}^d \to \mathbb{R}^d$. In particular one has $\Phi = (\hat{\Phi}, 0)$ and also $\check{b} = 0$ so that $b = (\hat{b}, 0)$ as a consequence of (2.28). Hence (3.9) becomes

$$\Psi_1 c' + \Psi_2 c''' - \hat{\Phi} \cdot \hat{b} = R_2. \qquad (3.15)$$

The matrix $M_\phi$ defined in (3.10) is given by

$$M_\phi = \begin{pmatrix} \hat{M}_\phi & 0 \\ 0 & 0 \end{pmatrix}$$

where

$$\hat{M}_{\phi} := \langle \hat{\Phi} \otimes \hat{\Phi} \rangle \in \mathfrak{M}_{d_{\phi} \times d_{\phi}}^{\text{sym}}(\mathbb{R}). \qquad (3.16)$$

Following the same procedure as in the fully non-harmonic case, we obtain after multiplication by $\hat{\Phi}$ and integration against $\rho$ that

$$\hat{M}_{\phi} \hat{b} = \hat{\alpha}_1 c' + \hat{\alpha}_2 c''' + \hat{R}_3, \qquad (3.17)$$

with $\hat{R}_3 := -\langle R_2 \hat{\Phi} \rangle = O(A)$, $\hat{\alpha}_1 := \langle \Psi_1 \hat{\Phi} \rangle$ and $\hat{\alpha}_2 := \langle \Psi_2 \hat{\Phi} \rangle$. The matrix $\hat{M}_\phi$ is invertible (see Lemma A.2 in Appendix A) and (3.17) yields

$$\hat{b} = \hat{M}_{\phi}^{-1} (\hat{\alpha}_1 c' + \hat{\alpha}_2 c''' + \hat{R}_3). \qquad (3.18)$$

Hence, together with (3.15), it follows that

$$\tilde{\Psi}_1(x) c' + \tilde{\Psi}_2(x) c''' = \hat{R}_4$$
with $\hat{R}_4 := R_2 + \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{R}_3 = O(A)$ and

$$ \hat{\Psi}_1(x) := \Psi_1(x) - \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{\alpha}_1, \quad \hat{\Psi}_2(x) := \Psi_2(x) - \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{\alpha}_2. \quad (3.19) $$

As in the full rank case, Rank($\hat{\Psi}_1, \hat{\Psi}_2$) = 2 according to Lemma A.3 and we deduce that $c' = O(A)$ and $c''' = O(A)$. From (3.18) and (3.8), we also get $\hat{b} = O(A)$ and $\hat{b}'' = O(A)$, and since $\check{b} = 0$ we eventually get $b = O(A)$ and $b'' = O(A)$, which completes the proof of the case partially harmonic case.

▷ Fully harmonic case ($d_{\phi} = 0$). We read from (3.2c) and (2.29) that $\langle e \rangle = c = 0 = c' = c''$ and from (3.2b), (2.28) and (H3) that $\langle m \rangle = b = 0 = b' = b''$, which completes the proof of the case fully harmonic case. ■

### 3.4. Control of $b'$, $c''$ and $c$

**Lemma 3.6.** Consider the function $h$ as in Proposition 3.1. The functions $b$ and $c$ as defined in Lemma 3.3 obey the two differential inequalities

$$ \frac{d}{dt} \langle -b, b' \rangle \le -|b'|^2 + O(|A|^2) \quad \text{and} \quad \frac{d}{dt} \langle -c', c'' \rangle \le -|c''|^2 + O(|A|^2). $$

*Proof of Lemma 3.6.* We write

$$ \frac{d}{dt} \langle -b, b' \rangle = \langle -b', b' \rangle + \langle -b, b'' \rangle \quad \text{and} \quad \frac{d}{dt} \langle -c', c'' \rangle = \langle -c'', c'' \rangle + \langle -c', c''' \rangle $$

and notice that $\langle -b, b'' \rangle = O(|A|^2)$ and $\langle -c', c''' \rangle = O(|A|^2)$ by (3.7).

**Lemma 3.7.** The function $c$ as defined in Lemma 3.3 is such that

$$ |c| \le |b'| + |c''| \quad \text{and} \quad |c''| \le |b'| + |c|. \quad (3.20) $$

*Proof of Lemma 3.7.* Multiplying (3.2a) by $\xi_{\phi}$ and integrating against $\rho$, we obtain

$$ c \left( \sqrt{\frac{2}{d}} \langle \xi_{\phi}^2 \rangle + \sqrt{\frac{d}{2}} \right) = \langle x \xi_{\phi} \rangle \cdot b' - \frac{\langle \xi_2 \xi_{\phi} \rangle}{2\sqrt{2d}} c'' $$

using $\langle r \phi \rangle = -\sqrt{\frac{d}{2}} c$ and $\langle r \rangle = 0$ by (2.26), which completes the proof. ■

### 3.5. A Lyapunov function method

We define the Lyapunov function

$$ \mathcal{F}[h] := \|h\|^2 - \varepsilon_A \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle - \varepsilon_b \langle b, b' \rangle - \varepsilon_c \langle c', c'' \rangle, $$

for some positive constants $\varepsilon_A$, $\varepsilon_b$ and $\varepsilon_c$ to be chosen later.
with $\hat{R}_4 := R_2 + \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{R}_3 = O(A)$ and

$$ \hat{\Psi}_1(x) := \Psi_1(x) - \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{\alpha}_1, \quad \hat{\Psi}_2(x) := \Psi_2(x) - \hat{\Phi}(x) \cdot \hat{M}_{\phi}^{-1} \hat{\alpha}_2. \quad (3.19) $$

As in the full rank case, Rank($\hat{\Psi}_1, \hat{\Psi}_2$) = 2 according to Lemma A.3 and we deduce that $c' = O(A)$ and $c''' = O(A)$. From (3.18) and (3.8), we also get $\hat{b} = O(A)$ and $\hat{b}'' = O(A)$, and since $\check{b} = 0$ we eventually get $b = O(A)$ and $b'' = O(A)$, which completes the proof of the case partially harmonic case.

▷ Fully harmonic case ($d_{\phi} = 0$). We read from (3.2c) and (2.29) that $\langle e \rangle = c = 0 = c' = c''$ and from (3.2b), (2.28) and (H3) that $\langle m \rangle = b = 0 = b' = b''$, which completes the proof of the case fully harmonic case. ■

### 3.4. Control of $b'$, $c''$ and $c$

**Lemma 3.6.** Consider the function $h$ as in Proposition 3.1. The functions $b$ and $c$ as defined in Lemma 3.3 obey the two differential inequalities

$$ \frac{d}{dt} \langle -b, b' \rangle \le -|b'|^2 + O(|A|^2) \quad \text{and} \quad \frac{d}{dt} \langle -c', c'' \rangle \le -|c''|^2 + O(|A|^2). $$

*Proof of Lemma 3.6.* We write

$$ \frac{d}{dt} \langle -b, b' \rangle = \langle -b', b' \rangle + \langle -b, b'' \rangle \quad \text{and} \quad \frac{d}{dt} \langle -c', c'' \rangle = \langle -c'', c'' \rangle + \langle -c', c''' \rangle $$

and notice that $\langle -b, b'' \rangle = O(|A|^2)$ and $\langle -c', c''' \rangle = O(|A|^2)$ by (3.7).

**Lemma 3.7.** The function $c$ as defined in Lemma 3.3 is such that

$$ |c| \le |b'| + |c''| \quad \text{and} \quad |c''| \le |b'| + |c|. \quad (3.20) $$

*Proof of Lemma 3.7.* Multiplying (3.2a) by $\xi_{\phi}$ and integrating against $\rho$, we obtain

$$ c \left( \sqrt{\frac{2}{d}} \langle \xi_{\phi}^2 \rangle + \sqrt{\frac{d}{2}} \right) = \langle x \xi_{\phi} \rangle \cdot b' - \frac{\langle \xi_2 \xi_{\phi} \rangle}{2\sqrt{2d}} c'' $$

using $\langle r \phi \rangle = -\sqrt{\frac{d}{2}} c$ and $\langle r \rangle = 0$ by (2.26), which completes the proof. ■

### 3.5. A Lyapunov function method

We define the Lyapunov function

$$ \mathcal{F}[h] := \|h\|^2 - \varepsilon_A \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle - \varepsilon_b \langle b, b' \rangle - \varepsilon_c \langle c', c'' \rangle, $$

for some positive constants $\varepsilon_A$, $\varepsilon_b$ and $\varepsilon_c$ to be chosen later.
**Lemma 3.8.** With the above notations, if $h$ is defined as in Proposition 3.1, then

$$ \mathcal{F}[h] \approx \|h\|^2 \qquad (3.21) $$

for $\varepsilon_A, \varepsilon_b$ and $\varepsilon_c$ small enough.

*Proof of Lemma 3.8.* From (3.1), we know that $\|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2$ and it follows from (3.2a), (3.2b) and (3.2c) that

$$ \|r\|^2 \lesssim |b'|^2 + |c|^2 + |c''|^2, \quad \|m\|^2 \approx |b|^2 + |A|^2 + |c'|^2 \quad \text{and} \quad \|e\|^2 = |c|^2. \qquad (3.22) $$

By Lemma 3.5, we obtain

$$ \|m\|^2 \lesssim |A|^2, \quad |\langle b, b' \rangle| \lesssim |A|^2 + |b'|^2, \quad |\langle c', c'' \rangle| \lesssim |A|^2 + |c''|^2. $$

By Lemma 3.7, we know that $|c|^2 \lesssim |b'|^2 + |c''|^2$ and obtain

$$ \|r\|^2 \lesssim |b'|^2 + |c''|^2, \quad \|c\|^2 \lesssim |b'|^2 + |c''|^2, \quad \|e\|^2 \lesssim |b'|^2 + |c''|^2, \\ |\langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot Ax \rangle| \lesssim |A|^2 + |b'|^2 + |c''|^2. $$

Altogether we have the upper estimate

$$ \mathcal{F}[h] \lesssim |A|^2 + |b'|^2 + |c''|^2 \qquad (3.23) $$

and, using (3.20),

$$ \mathcal{F}[h] \lesssim |A|^2 + |b'|^2 + |c|^2. $$

Using (2.1b), we notice that

$$ b' = \langle \partial_t m \rangle = - \langle \nabla_x r \rangle = - \langle r \nabla_x \phi \rangle \le \|r\| \| \nabla_x \phi \| \le \|r\| \qquad (3.24) $$

performing one integration by parts and using Cauchy-Schwarz inequality. It is then clear that $|A|^2 \lesssim \|m\|^2$ and $|c|^2 = \|e\|^2$, so that by (3.1),

$$ \mathcal{F}[h] \lesssim \|r\|^2 + \|m\|^2 + \|e\|^2 = \|h\|^2. $$

Then, using (3.22) again, we have the lower bound estimate

$$ 2 \mathcal{F}[h] - \|h\|^2 \ge \|r\|^2 + (|b|^2 + |A|^2 + |c'|^2) + |c|^2 \\ - 2 \varepsilon_A (|A|^2 + |b'|^2 + |c''|^2) - 2 \varepsilon_b (|A|^2 + |b'|^2) - 2 \varepsilon_c (|A|^2 + |c''|^2). $$

We know from (3.24) that $|b'|^2 \lesssim \|r\|^2$ and, using (3.20), we also have that $|c''|^2 \lesssim |b'|^2 + |c|^2 \lesssim \|r\|^2 + |A|^2$. As a consequence, we have that

$$ 2 \mathcal{F}[h] - \|h\|^2 \ge 0 $$

if $\varepsilon_A, \varepsilon_b$ and $\varepsilon_c$ are chosen small enough, which completes the proof.

<figure><img src="image_2.png" /></figure>
**Lemma 3.8.** With the above notations, if $h$ is defined as in Proposition 3.1, then

$$ \mathcal{F}[h] \approx \|h\|^2 \qquad (3.21) $$

for $\epsilon_A, \epsilon_b$ and $\epsilon_c$ small enough.

*Proof of Lemma 3.8.* From (3.1), we know that $\|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2$ and it follows from (3.2a), (3.2b) and (3.2c) that

$$ \|r\|^2 \lesssim |b'|^2 + |c|^2 + |c''|^2, \quad \|m\|^2 \approx |b|^2 + |A|^2 + |c'|^2 \quad \text{and} \quad \|e\|^2 = |c|^2. \qquad (3.22) $$

By Lemma 3.5, we obtain

$$ \|m\|^2 \lesssim |A|^2, \quad |\langle b, b' \rangle| \lesssim |A|^2 + |b'|^2, \quad |\langle c, c' \rangle| \lesssim |A|^2 + |c''|^2. $$

By Lemma 3.7, we know that $|c|^2 \lesssim |b'|^2 + |c''|^2$ and obtain

$$ \|r\|^2 \lesssim |b'|^2 + |c''|^2, \quad \|c\|^2 \lesssim |b'|^2 + |c''|^2, \quad \|e\|^2 \lesssim |b'|^2 + |c''|^2, \\ |\langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle| \lesssim |A|^2 + |b'|^2 + |c''|^2. $$

Altogether we have the upper estimate

$$ \mathcal{F}[h] \lesssim |A|^2 + |b'|^2 + |c''|^2 \qquad (3.23) $$

and, using (3.20),

$$ \mathcal{F}[h] \lesssim |A|^2 + |b'|^2 + |c|^2. $$

Using (2.1b), we notice that

$$ b' = \langle \partial_t m \rangle = - \langle \nabla_x r \rangle = - \langle r \nabla_x \phi \rangle \le \|r\| \|\nabla_x \phi\| \le \|r\| \qquad (3.24) $$

performing one integration by parts and using Cauchy-Schwarz inequality. It is then clear that $|A|^2 \lesssim \|m\|^2$ and $|c|^2 = \|e\|^2$, so that by (3.1),

$$ \mathcal{F}[h] \lesssim \|r\|^2 + \|m\|^2 + \|e\|^2 = \|h\|^2. $$

Then, using (3.22) again, we have the lower bound estimate

$$ 2 \mathcal{F}[h] - \|h\|^2 \ge \|r\|^2 + (|b|^2 + |A|^2 + |c'|^2) + |c|^2 \\ - 2 \epsilon_A (|A|^2 + |b'|^2 + |c''|^2) - 2 \epsilon_b (|A|^2 + |b'|^2) - 2 \epsilon_c (|A|^2 + |c''|^2). $$

We know from (3.24) that $|b'|^2 \lesssim \|r\|^2$ and, using (3.20), we also have that $|c''|^2 \lesssim |b'|^2 + |c|^2 \lesssim \|r\|^2 + |A|^2$. As a consequence, we have that

$$ 2 \mathcal{F}[h] - \|h\|^2 \ge 0 $$

if $\epsilon_A, \epsilon_b$ and $\epsilon_c$ are chosen small enough, which completes the proof. ■

<figure><img src="image_3.png" /></figure>
**Lemma 3.9.** With the above notations, if $h$ is defined as in Proposition 3.1, then for some $\varepsilon_A, \varepsilon_b$ and $\varepsilon_c$ small enough, there is a positive constant $\lambda$ such that

$$ \frac{d}{dt} \mathcal{F}[h] \leq -\lambda \mathcal{F}[h]. $$

*Proof of Lemma 3.9.* Using Lemma 3.4, Lemma 3.6 and (3.3), we have

$$ \begin{aligned} -\frac{d}{dt}\mathcal{F}[h] = & -\varepsilon_A \|\nabla_x\phi \cdot Ax\|^2 + \varepsilon_b |b'|^2 + \varepsilon_c |c''|^2 - (\varepsilon_b + \varepsilon_c) O(|A|^2) \\ & \geq \varepsilon'_A |A|^2 + \varepsilon_b |b'|^2 + \varepsilon_c |c''|^2 \geq \mathcal{F}[h], \end{aligned} $$

by choosing $\varepsilon_b$ and $\varepsilon_c$ small enough compared to $\varepsilon_A$ and using (3.23) in the last inequality.

*Proof of Proposition 3.1.* Let $h_0 = h(t=0)$. Thanks to Grönwall's lemma and the equivalence (3.21), we deduce

$$ \forall t \geq 0, \quad \|h(t)\|^2 \leq \mathcal{F}[h(t)] \leq e^{-\lambda t} \mathcal{F}[h_0] \leq e^{-\lambda t} \|h_0\|^2. $$

By Lemma 3.3, we know that $A$ is constant in time. Using for instance (3.22), we deduce from

$$ |A|^2 \lesssim \lim_{t \to +\infty} \|h(t)\|^2 = 0 $$

that $A=0$. By Lemma 3.5, we get that $b=0$ and $c'=0$ for any $t \geq 0$ so that $c$ is independent of $t$. Taking for instance (3.23) into account, we conclude that $h=0$.

Completing the proof of Proposition 3.1 means that Part (1) of Theorem 1.1 is established.

# 4. Proof of hypocoercivity by the micro-macro method

In this section we prove Part (2) of Theorem 1.1 on hypocoercivity using the micro-macro decomposition of the solution as in Section 2.3. The proof of Proposition 3.1 is our a guideline for a new cascade of estimates, but the analysis is however more complex due to the presence of microscopic terms.

## 4.1. Statement

Theorem 1.1, Part (2) can be rewritten as follows.

**Proposition 4.1.** Consider a solution $h$ to (2.22)–(2.23) in $L^2(M)$ with initial datum $h_0$. There exist two positive constants $C$ and $\kappa$ such that

$$ \forall t \geq 0, \quad \|h(t)\| \leq C e^{-\kappa t} \|h_0\|. $$
**Lemma 3.9.** With the above notations, if $h$ is defined as in Proposition 3.1, then for some $\varepsilon_A, \varepsilon_b$ and $\varepsilon_c$ small enough, there is a positive constant $\lambda$ such that

$$ \frac{d}{dt} \mathcal{F}[h] \leq -\lambda \mathcal{F}[h]. $$

*Proof of Lemma 3.9.* Using Lemma 3.4, Lemma 3.6 and (3.3), we have

$$ \begin{aligned} -\frac{d}{dt}\mathcal{F}[h] = & -\varepsilon_A \|\nabla_x\phi \cdot Ax\|^2 + \varepsilon_b |b'|^2 + \varepsilon_c |c''|^2 - (\varepsilon_b + \varepsilon_c) O(|A|^2) \\ & \geq \varepsilon'_A |A|^2 + \varepsilon_b |b'|^2 + \varepsilon_c |c''|^2 \geq \mathcal{F}[h], \end{aligned} $$

by choosing $\varepsilon_b$ and $\varepsilon_c$ small enough compared to $\varepsilon_A$ and using (3.23) in the last inequality.

*Proof of Proposition 3.1.* Let $h_0 = h(t=0)$. Thanks to Grönwall's lemma and the equivalence (3.21), we deduce

$$ \forall t \geq 0, \quad \|h(t)\|^2 \leq \mathcal{F}[h(t)] \leq e^{-\lambda t} \mathcal{F}[h_0] \leq e^{-\lambda t} \|h_0\|^2. $$

By Lemma 3.3, we know that $A$ is constant in time. Using for instance (3.22), we deduce from

$$ |A|^2 \lesssim \lim_{t \to +\infty} \|h(t)\|^2 = 0 $$

that $A=0$. By Lemma 3.5, we get that $b=0$ and $c'=0$ for any $t \geq 0$ so that $c$ is independent of $t$. Taking for instance (3.23) into account, we conclude that $h=0$.

Completing the proof of Proposition 3.1 means that Part (1) of Theorem 1.1 is established.

# 4. Proof of hypocoercivity by the micro-macro method

In this section we prove Part (2) of Theorem 1.1 on hypocoercivity using the micro-macro decomposition of the solution as in Section 2.3. The proof of Proposition 3.1 is our a guideline for a new cascade of estimates, but the analysis is however more complex due to the presence of microscopic terms.

## 4.1. Statement

Theorem 1.1, Part (2) can be rewritten as follows.

**Proposition 4.1.** Consider a solution $h$ to (2.22)–(2.23) in $L^2(M)$ with initial datum $h_0$. There exist two positive constants $C$ and $\kappa$ such that

$$ \forall t \geq 0, \quad \|h(t)\| \leq C e^{-\kappa t} \|h_0\|. $$
Here $C$ and $\kappa$ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to $\phi$ such as the rigidity constant defined in (1.14). We split $h$ into a microscopic part $h^{\perp}$ and a macroscopic part $h^{\|}$ such that

$$h = h^{\|} + h^{\perp} = r + m \cdot v + e \mathfrak{E}(v) + h^{\perp}$$

where $r, m$ and $e$ defined by (2.25) evolve according to the macroscopic equations (2.30) involving the matrix valued function $E[h]$ and the vector valued function $\Theta[h]$ defined by (2.31a) and (2.31b). By construction, we have

$$\|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2 + \|h^{\perp}\|^2. \quad (4.1)$$

Let deviations from averages, or space inhomogeneous, terms be defined by

$$r_s := r - \langle \nabla_x r \rangle \cdot x - \frac{1}{2d} \langle \Delta_x r \rangle \xi_2, \quad (4.2a)$$

$$m_s := m - \langle \nabla_x^{\text{skew}} m \rangle x - \frac{1}{d} \langle \nabla_x \cdot m \rangle x - \langle m \rangle, \quad (4.2b)$$

$$e_s := e - \langle e \rangle, \quad (4.2c)$$

$$w := r - \sqrt{\frac{2}{d}} \langle e \rangle \phi, \quad (4.2d)$$

$$w_s := r_s - \sqrt{\frac{2}{d}} \langle e \rangle \phi_s \quad \text{with} \quad \phi_s := \xi_\phi - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2. \quad (4.2e)$$

We recall that $\xi_2(x) := |x|^2 - \langle |x|^2 \rangle$ and $\xi_\phi(x) := \phi - \langle \phi \rangle$ were already defined in (2.10) and (2.11) while $\nabla_x^{\text{skew}} m$ refers to (2.6). In particular

$$w_s = w - \langle \nabla_x w \rangle x - \frac{1}{2d} \langle \Delta w \rangle \xi_2 + \sqrt{\frac{2}{d}} \langle e \rangle \langle \phi \rangle. \quad (4.3)$$

After introducing some geometric tools in Section 4.2, we split the proof of Proposition 4.1 by considering infinite-dimensional quantities in Section 4.3 and finite-dimensional quantities in Section 4.4; in the latter the analysis closely follows the strategy of Section 3. From now on, we assume that $h$ is as in Proposition 4.1.

## 4.2. Witten-Hodge operator and Korn inequality: a toolbox

Here we collect several classical and less classical estimates that will be used to control the macroscopic quantities. We refer to [9] for references and details of constructive proofs. Assumptions (H3)–(H4)–(H5) coincide with the hypotheses of [9, Section 1.2]. Let

$$[\nabla \phi] := \sqrt{1 + |\nabla \phi|^2}.$$

▷ The strong Poincaré inequality

$$\forall \varphi \in H^1(\rho), \quad \int_{\mathbb{R}^d} |\varphi - \langle \varphi \rangle|^2 [\nabla_x \phi]^2 \rho \, dx \lesssim \int_{\mathbb{R}^d} |\nabla_x \varphi|^2 \rho \, dx \quad (4.4)$$

is proven in [9, Proposition 5].

▷ In order to work in $L^2(\rho)$, we shall use the operator $\Omega$ introduced in (1.15) and considered as an operator acting either on scalar or vector-valued functions. As a consequence
Here $C$ and $\kappa$ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to $\phi$ such as the rigidity constant defined in (1.14). We split $h$ into a microscopic part $h^{\perp}$ and a macroscopic part $h^{\|}$ such that

$$h = h^{\|} + h^{\perp} = r + m \cdot v + e \mathfrak{E}(v) + h^{\perp}$$

where $r, m$ and $e$ defined by (2.25) evolve according to the macroscopic equations (2.30) involving the matrix valued function $E[h]$ and the vector valued function $\Theta[h]$ defined by (2.31a) and (2.31b). By construction, we have

$$\|h\|^2 = \|r\|^2 + \|m\|^2 + \|e\|^2 + \|h^{\perp}\|^2. \quad (4.1)$$

Let deviations from averages, or space inhomogeneous, terms be defined by

$$r_s := r - \langle \nabla_x r \rangle \cdot x - \frac{1}{2d} \langle \Delta_x r \rangle \xi_2, \quad (4.2a)$$

$$m_s := m - \langle \nabla_x^{\text{skew}} m \rangle x - \frac{1}{d} \langle \nabla_x \cdot m \rangle x - \langle m \rangle, \quad (4.2b)$$

$$e_s := e - \langle e \rangle, \quad (4.2c)$$

$$w := r - \sqrt{\frac{2}{d}} \langle e \rangle \phi, \quad (4.2d)$$

$$w_s := r_s - \sqrt{\frac{2}{d}} \langle e \rangle \phi_s \quad \text{with} \quad \phi_s := \xi_\phi - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2. \quad (4.2e)$$

We recall that $\xi_2(x) := |x|^2 - \langle |x|^2 \rangle$ and $\xi_\phi(x) := \phi - \langle \phi \rangle$ were already defined in (2.10) and (2.11) while $\nabla_x^{\text{skew}} m$ refers to (2.6). In particular

$$w_s = w - \langle \nabla_x w \rangle x - \frac{1}{2d} \langle \Delta w \rangle \xi_2 + \sqrt{\frac{2}{d}} \langle e \rangle \langle \phi \rangle. \quad (4.3)$$

After introducing some geometric tools in Section 4.2, we split the proof of Proposition 4.1 by considering infinite-dimensional quantities in Section 4.3 and finite-dimensional quantities in Section 4.4; in the latter the analysis closely follows the strategy of Section 3. From now on, we assume that $h$ is as in Proposition 4.1.

## 4.2. Witten-Hodge operator and Korn inequality: a toolbox

Here we collect several classical and less classical estimates that will be used to control the macroscopic quantities. We refer to [9] for references and details of constructive proofs. Assumptions (H3)–(H4)–(H5) coincide with the hypotheses of [9, Section 1.2]. Let

$$[\nabla \phi] := \sqrt{1 + |\nabla \phi|^2}.$$

▷ The strong Poincaré inequality

$$\forall \varphi \in H^1(\rho), \quad \int_{\mathbb{R}^d} |\varphi - \langle \varphi \rangle|^2 [\nabla_x \phi]^2 \rho \, dx \lesssim \int_{\mathbb{R}^d} |\nabla_x \varphi|^2 \rho \, dx \quad (4.4)$$

is proven in [9, Proposition 5].

▷ In order to work in $L^2(\rho)$, we shall use the operator $\Omega$ introduced in (1.15) and considered as an operator acting either on scalar or vector-valued functions. As a consequence
of (4.4), we have the zeroth-order strong Poincaré inequality (see [9, Proposition 8])
according to which, for any $\varphi \in L^2(\rho)$,

$$
\| \Omega^{-1} \nabla_x^2 \varphi \| + \| \Omega^{-1} (\nabla_x \phi | \nabla_x \varphi) \| + \| \Omega^{-1} (\nabla_x \phi|^2 \varphi) \| \lesssim \| \varphi \|. \quad (4.5)
$$

▷ The following zeroth order Poincaré inequality, sometimes called the *Poincaré-Lions inequality*,

$$
\forall \varphi \in L^2(\rho), \quad \|\varphi - \langle\varphi\rangle\| \lesssim \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\| \lesssim \|\varphi - \langle\varphi\rangle\|, \quad (4.6)
$$

is proven in [9, Proposition 5].

▷ The $(-1)^{th}$ order Poincaré-Lions inequality

$$
\forall \varphi \in H^{-1}(\rho), \quad \| \Omega^{-\frac{1}{2}} (\varphi - \langle \varphi \rangle) \| \lesssim \| \Omega^{-1} \nabla_x \varphi \| \lesssim \| \Omega^{-\frac{1}{2}} (\varphi - \langle \varphi \rangle) \|, \quad (4.7)
$$

is proven in [9, Lemma 10] as well as its variant

$$
\forall \varphi \in L^2(\rho), \quad \|\varphi - \langle\varphi\rangle\| \lesssim \|\nabla_x \Omega^{-\frac{1}{2}} \varphi\| + \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\| \lesssim \|\varphi - \langle\varphi\rangle\|. \quad (4.8)
$$

▷ Another key estimate is the zeroth-order Korn-Poincaré inequality: for any vector field $u: \mathbb{R}^d \to \mathbb{R}^d$ such that $\langle u \rangle = 0$ and $\langle \nabla_x^{\text{skew}} u \rangle = 0$,

$$
\|u\| \lesssim \|\Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} u\|, \quad (4.9)
$$

which is established in [9, Theorem 4] using (2.5).

4.3. Control of infinite-dimensional quantities

We build an entropy function by assembling dissipative functionals for $h^{\perp}$ and the space inhomogeneous terms defined in (4.2).

4.3.1. *Control of h*<sup>*⊥*</sup>. We first control the dissipation of the microscopic part.

**Lemma 4.2.** If *h* is a solution to (2.23) in *L*²(*M*), then

$$
\frac{d}{dt} \|h\|^2 \le -2 c_{\mathcal{C}} \|h^{\perp}\|^2 . \quad (4.10)
$$

*Proof of Lemma 4.2.* Since $C^* = C$ and $\mathcal{T}^* = -\mathcal{T}$, there holds

$$
\frac{1}{2} \frac{d}{dt} \|h\|^2 = \langle Ch, h \rangle .
$$

We conclude that (4.10) holds by the spectral gap assumption (H1) on $\mathcal{C}$.

<figure><img src="image_3.png" /></figure>
of (4.4), we have the zeroth-order strong Poincaré inequality (see [9, Proposition 8])
according to which, for any $\varphi \in L^2(\rho)$,

$$
\|\Omega^{-1} \nabla_x^2 \varphi\| + \|\Omega^{-1} (\nabla_x \phi | \nabla_x \varphi)\| + \|\Omega^{-1} (\nabla_x \phi|^2 \varphi)\| \lesssim \|\varphi\|. \quad (4.5)
$$

▷ The following zeroth order Poincaré inequality, sometimes called the *Poincaré-Lions*
*inequality,*

$$
\forall \varphi \in L^2(\rho), \quad \|\varphi - \langle \varphi \rangle\| \lesssim \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\| \lesssim \|\varphi - \langle \varphi \rangle\|, \quad (4.6)
$$

is proven in [9, Proposition 5].

▷ The $(-1)^{th}$ order Poincaré-Lions inequality

$$
\forall \varphi \in H^{-1}(\rho), \quad \| \Omega^{-\frac{1}{2}} (\varphi - \langle \varphi \rangle) \| \lesssim \| \Omega^{-1} \nabla_x \varphi \| \lesssim \| \Omega^{-\frac{1}{2}} (\varphi - \langle \varphi \rangle) \|, \quad (4.7)
$$

is proven in [9, Lemma 10] as well as its variant

$$
\forall \varphi \in L^2(\rho), \quad \|\varphi - \langle\varphi\rangle\| \lesssim \|\nabla_x \Omega^{-\frac{1}{2}} \varphi\| + \|\Omega^{-\frac{1}{2}} \nabla_x \varphi\| \lesssim \|\varphi - \langle\varphi\rangle\|. \quad (4.8)
$$

▷ Another key estimate is the zeroth-order Korn-Poincaré inequality: for any vector field $u : \mathbb{R}^d \to \mathbb{R}^d$ such that $\langle u \rangle = 0$ and $\langle \nabla_x^{\text{skew}} u \rangle = 0$,

$$
\|u\| \lesssim \|\Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} u\|, \quad (4.9)
$$

which is established in [9, Theorem 4] using (2.5).

4.3. *Control of infinite-dimensional quantities*

We build an entropy function by assembling dissipative functionals for $h^{\perp}$ and the space inhomogeneous terms defined in (4.2).

4.3.1. *Control of h*<sup>*⊥*</sup>. We first control the dissipation of the microscopic part.

**Lemma 4.2.** If *h* is a solution to (2.23) in *L*<sup>2</sup>(*M*), then

$$
\frac{d}{dt} \|h\|^2 \leq -2 c_{\mathcal{C}} \|h^{\perp}\|^2. \qquad (4.10)
$$

*Proof of Lemma 4.2.* Since $C^* = C$ and $\mathcal{T}^* = -\mathcal{T}$, there holds

$$
\frac{1}{2} \frac{d}{dt} \|h\|^2 = \langle Ch, h \rangle .
$$

We conclude that (4.10) holds by the spectral gap assumption (H1) on $\mathcal{C}$.

<figure><img src="image_4.png" /></figure>
### 4.3.2. Control of $e_s$. Let us consider $e_s$ as defined in (4.2c).

**Lemma 4.3.** There are some positive constants $\kappa_1$ and $C$ such that

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle \le -\kappa_1 \|e_s\|^2 + C \|h^{\perp}\| \|h\|. \quad (4.11) $$

*Proof of Lemma 4.3.* Recall that $\Theta[h] = \Theta[h^{\perp}]$ from (2.31b). We compute

$$ \begin{aligned} \frac{d}{dt} \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle &= \langle \Omega^{-1} \nabla_x e, -(1+\frac{2}{d}) \nabla_x e + \Theta[\mathcal{L} h^{\perp}] \rangle + \langle \Omega^{-1} \nabla_x (\partial_t e), \Theta[h] \rangle \\ &\le -\frac{1}{2} (1+\frac{2}{d}) \| \Omega^{-\frac{1}{2}} \nabla_x e \|^2 + C \| \Omega^{-\frac{1}{2}} \Theta[\mathcal{L} h^{\perp}] \|^2 + C \| \Omega^{-1} \nabla_x (\partial_t e) \| \| h^{\perp} \|, \end{aligned} $$

by using Cauchy-Schwarz and Young inequalities. We read from (4.6) that

$$ \| \Omega^{-1/2} \nabla_x e \|^2 \gtrsim \|e_s\|^2. $$

According to (4.6), (H2) and (H6), we have

$$ \| \Omega^{-\frac{1}{2}} \Theta[\mathcal{L} h^{\perp}] \| \le \| h^{\perp} \|. $$

It follows from (2.30c) that

$$ \Omega^{-1} \nabla_x (\partial_t e) = - \sqrt{\frac{2}{d}} \Omega^{-1} \nabla_x (\nabla_x \cdot m) + \Omega^{-1} \nabla_x (\nabla_x^* \cdot \Theta[h^{\perp}]), $$

so that $\| \Omega^{-1} \nabla_x (\partial_t e) \| \lesssim \|h\|$ by (4.5). This completes the proof of (4.11).

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

### 4.3.3. Control of $m_s$. Let us consider $\nabla_x^{\text{sym}} m_s$ as defined by (2.6) and (4.2b).

**Lemma 4.4.** There are some positive constants $\kappa_2$ and $C$ such that

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \le -\kappa_2 \|m_s\|^2 + C (\|e_s\| + \|h^{\perp}\|) \|h\|. \quad (4.12) $$

*Proof of Lemma 4.4.* Let us remark that from (4.2b) one has

$$ \nabla_x^{\text{sym}} m = \nabla_x^{\text{sym}} m_s + \frac{1}{d} \langle \nabla_x \cdot m \rangle \text{Id}_{d \times d}, $$

and from (2.31a),

$$ E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} = \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}]. $$

Moreover, from (2.30c), one gets

$$ \frac{d}{dt} \langle e \rangle = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot m \rangle. $$

As a consequence, from (2.30d), one obtains

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle $$
### 4.3.2. Control of $e_s$. Let us consider $e_s$ as defined in (4.2c).

**Lemma 4.3.** There are some positive constants $\kappa_1$ and $C$ such that

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle \le -\kappa_1 \|e_s\|^2 + C \|h^{\perp}\| \|h\|. \quad (4.11) $$

*Proof of Lemma 4.3.* Recall that $\Theta[h] = \Theta[h^{\perp}]$ from (2.31b). We compute

$$ \begin{aligned} \frac{d}{dt} \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle &= \langle \Omega^{-1} \nabla_x e, -\left(1 + \frac{2}{d}\right) \nabla_x e + \Theta[\mathcal{L} h^{\perp}] \rangle + \langle \Omega^{-1} \nabla_x (\partial_t e), \Theta[h] \rangle \\ &\le -\frac{1}{2} \left(1 + \frac{2}{d}\right) \|\Omega^{-\frac{1}{2}} \nabla_x e\|^2 + C \|\Omega^{-\frac{1}{2}} \Theta[\mathcal{L} h^{\perp}]\|^2 + C \|\Omega^{-1} \nabla_x (\partial_t e)\| \|h^{\perp}\|, \end{aligned} $$

by using Cauchy-Schwarz and Young inequalities. We read from (4.6) that

$$ \| \Omega^{-1/2} \nabla_x e \|^2 \gtrsim \|e_s\|^2. $$

According to (4.6), (H2) and (H6), we have

$$ \|\Omega^{-\frac{1}{2}} \Theta[\mathcal{L} h^{\perp}]\| \le \|h^{\perp}\|. $$

It follows from (2.30c) that

$$ \Omega^{-1} \nabla_x (\partial_t e) = -\sqrt{\frac{2}{d}} \Omega^{-1} \nabla_x (\nabla_x \cdot m) + \Omega^{-1} \nabla_x (\nabla_x^* \cdot \Theta[h^{\perp}]), $$

so that $\|\Omega^{-1} \nabla_x (\partial_t e)\| \lesssim \|h\|$ by (4.5). This completes the proof of (4.11).

<figure><img src="image_5.png" /></figure>

### 4.3.3. Control of $m_s$. Let us consider $\nabla_x^{\text{sym}} m_s$ as defined by (2.6) and (4.2b).

**Lemma 4.4.** There are some positive constants $\kappa_2$ and $C$ such that

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \le -\kappa_2 \|m_s\|^2 + C (\|e_s\| + \|h^{\perp}\|) \|h\|. \quad (4.12) $$

*Proof of Lemma 4.4.* Let us remark that from (4.2b) one has

$$ \nabla_x^{\text{sym}} m = \nabla_x^{\text{sym}} m_s + \frac{1}{d} \langle \nabla_x \cdot m \rangle \text{Id}_{d \times d}, $$

and from (2.31a),

$$ E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} = \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}]. $$

Moreover, from (2.30c), one gets

$$ \frac{d}{dt} \langle e \rangle = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot m \rangle. $$

As a consequence, from (2.30d), one obtains

$$ \frac{d}{dt} \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle $$
$$
\begin{align*}
&= \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^{\perp}] + \frac{2}{d} \langle \nabla_x \cdot m \rangle \text{Id}_{d \times d} \rangle \\
&\quad + \langle \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s), E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \\
&= -2 \left\| \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s \right\|^2 + \left\langle \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s, \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \right\rangle \\
&\quad + \left\langle \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s), \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}] \right\rangle.
\end{align*}
$$

Using the Cauchy-Schwarz inequality, we deduce

$$
\begin{aligned}
& \frac{d}{dt} \left\langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \right\rangle \\
& \le - \left\| \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s \right\|^2 + C \left\| \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \right\|^2 \\
& \qquad + C \left\| \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s) \right\| \left\| \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}] \right\|.
\end{aligned}
$$

Using Korn's inequality (4.9) and observing by (2.30b) that

$$
\| \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s) \| = O(\|h\|) \quad \text{and} \quad \| \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \| = O(\|h^{\perp}\|)
$$

from (4.5) and (4.8) as in the proof of Lemma 4.3, we prove (4.12).

<figure><img src="image_5.png" /></figure>

4.3.4. Control of $w_s$. Let us consider $w_s$ as defined in (4.2e).

**Lemma 4.5.** *There are some positive constants $\kappa_3$ and $C$ such that*

$$
\frac{d}{dt} \langle \Omega^{-1} \nabla_x w_s, m_s \rangle \le - \kappa_3 \|w_s\|^2 + C \|e_s\|^2 + C \|h^\perp\|^2 + C \|m_s\| \|h\| \quad (4.13)
$$

and

$$
\frac{d}{dt} \langle -\Omega^{-1} \partial_t w_s, w_s \rangle \le - \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + C \|w_s\| \|h\|. \quad (4.14)
$$

*Proof of Lemma 4.5.* Observe that (2.30b), (4.2c) and (4.3) imply

$$
\begin{align*}
\partial_t m &= -\nabla_x w + \sqrt{\frac{2}{d}} \nabla_x^* e_s + \nabla_x^* \cdot E[h^\perp] \\
&= -\nabla_x w_s - \langle \nabla_x w \rangle - \frac{1}{d} \langle \Delta_x w \rangle x + \sqrt{\frac{2}{d}} \nabla_x^* e_s + \nabla_x^* \cdot E[h^\perp].
\end{align*}
$$

Integrating (2.30b) and using (4.2d), one gets

$$
\frac{d}{dt} \langle m \rangle = - \langle \nabla_x r \rangle = - \langle \nabla_x w \rangle
$$

and

$$
\begin{align*}
\frac{d}{dt} \langle \nabla_x \cdot m \rangle &= - \langle \Delta_x r \rangle + \sqrt{\frac{2}{d}} \langle e \Delta_x \phi \rangle + \langle \operatorname{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle \\
&= - \langle \Delta_x w \rangle + \sqrt{\frac{2}{d}} \langle e_s \Delta_x \phi \rangle + \langle \operatorname{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle.
\end{align*}
$$
$$
\begin{align*}
&= \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^{\perp}] + \frac{2}{d} \langle \nabla_x \cdot m \rangle \text{Id}_{d \times d} \rangle \\
&\quad + \langle \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s), E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \\
&= -2 \left\| \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s \right\|^2 + \left\langle \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s, \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \right\rangle \\
&\quad + \left\langle \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s), \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}] \right\rangle.
\end{align*}
$$

Using the Cauchy-Schwarz inequality, we deduce

$$
\begin{align*}
& \frac{d}{dt} \left\langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \right\rangle \\
& \le - \left\| \Omega^{-\frac{1}{2}} \nabla_x^{\text{sym}} m_s \right\|^2 + C \left\| \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \right\|^2 \\
& \phantom{\le} + C \left\| \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s) \right\| \left\| \sqrt{\frac{2}{d}} e_s \text{Id}_{d \times d} + E[h^{\perp}] \right\|.
\end{align*}
$$

Using Korn's inequality (4.9) and observing by (2.30b) that

$$
\| \Omega^{-1} \nabla_x^{\text{sym}} (\partial_t m_s) \| = O(\|h\|) \quad \text{and} \quad \| \Omega^{-\frac{1}{2}} E[\mathcal{L} h^{\perp}] \| = O(\|h^{\perp}\|)
$$

from (4.5) and (4.8) as in the proof of Lemma 4.3, we prove (4.12).

<figure><img src="image_6.png" /></figure>

4.3.4. Control of $w_s$. Let us consider $w_s$ as defined in (4.2e).

**Lemma 4.5.** *There are some positive constants $\kappa_3$ and $C$ such that*

$$
\frac{d}{dt} \langle \Omega^{-1} \nabla_x w_s, m_s \rangle \le - \kappa_3 \|w_s\|^2 + C \|e_s\|^2 + C \|h^\perp\|^2 + C \|m_s\| \|h\| \quad (4.13)
$$

and

$$
\frac{d}{dt} \langle -\Omega^{-1} \partial_t w_s, w_s \rangle \le - \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + C \|w_s\| \|h\|. \quad (4.14)
$$

*Proof of Lemma 4.5.* Observe that (2.30b), (4.2c) and (4.3) imply

$$
\begin{align*}
\partial_t m &= -\nabla_x w + \sqrt{\frac{2}{d}} \nabla_x^* e_s + \nabla_x^* \cdot E[h^\perp] \\
&= -\nabla_x w_s - \langle \nabla_x w \rangle - \frac{1}{d} \langle \Delta_x w \rangle x + \sqrt{\frac{2}{d}} \nabla_x^* e_s + \nabla_x^* \cdot E[h^\perp].
\end{align*}
$$

Integrating (2.30b) and using (4.2d), one gets

$$
\frac{d}{dt} \langle m \rangle = - \langle \nabla_x r \rangle = - \langle \nabla_x w \rangle
$$

and

$$
\begin{align*}
\frac{d}{dt} \langle \nabla_x \cdot m \rangle &= - \langle \Delta_x r \rangle + \sqrt{\frac{2}{d}} \langle e \cdot \Delta_x \phi \rangle + \langle \operatorname{Tr}(E[h^{\perp}] \nabla_x^2 \phi) \rangle \\
&= - \langle \Delta_x w \rangle + \sqrt{\frac{2}{d}} \langle e_s \cdot \Delta_x \phi \rangle + \langle \operatorname{Tr}(E[h^{\perp}] \nabla_x^2 \phi) \rangle.
\end{align*}
$$
Finally, by differentiating (2.30b), one has

$$\partial_t \nabla_x m = - \nabla_x^2 r + \sqrt{\frac{2}{d}} \nabla_x^* \nabla_x e + \sqrt{\frac{2}{d}} \nabla_x^2 \phi e + \nabla_x^* \cdot (\nabla_x \otimes E[h^\perp]) + E[h^\perp] \nabla_x^2 \phi$$

and the integration of the skew-symmetric part yields

$$\frac{d}{dt} \langle \nabla_x^{\text{skew}} m \rangle = \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle. \quad (4.15)$$

As a consequence of these identities and (4.2b), one gets

$$\partial_t m_s = - \nabla_x w_s + \sqrt{\frac{2}{d}} \left( \nabla_x^* e_s - \frac{1}{d} \langle e_s \Delta_x \phi \rangle x \right) + m_E$$

where $m_E := \nabla_x^* \cdot E[h^\perp] - \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x - \frac{1}{d} \langle \text{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle x$. Hence

$$\langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle = - \|\Omega^{-\frac{1}{2}} \nabla_x w_s\|^2 + \|\Omega^{-\frac{1}{2}} \nabla_x w_s\| \|\Omega^{-\frac{1}{2}} m_E\| \\ + \sqrt{\frac{2}{d}} \|\Omega^{-\frac{1}{2}} \nabla_x w_s\| \|\Omega^{-\frac{1}{2}} (\nabla_x^* e_s - \frac{1}{d} \langle e_s \Delta_x \phi \rangle x)\|$$

Using the zeroth order Poincaré inequality (4.6) and (4.2c), we can estimate $\|\Omega^{-1/2} \nabla_x^* e_s\|$ by $\|e_s\|$. Up to a few integrations by parts, using (H4), (H6) and (H7), we end up for some constant $C > 0$ with

$$\langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle \le -\frac{1}{2} \|\Omega^{-\frac{1}{2}} \nabla_x w_s\|^2 + C \left( \|e_s\|^2 + \|h^\perp\|^2 \right). \quad (4.16)$$

From the definitions (4.2a) and (4.2e), we have

$$\partial_t w_s = \partial_t r - \langle \nabla_x \partial_t r \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \partial_t r \rangle \xi_2, -\sqrt{\frac{2}{d}} \langle \partial_t e \rangle \phi_s,$$

so that, by (2.30a) and (2.30c),

$$\partial_t w_s = \nabla_x^* m - \langle \nabla_x \nabla_x^* m \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* m \rangle \xi_2 - \frac{2}{d} \langle \nabla_x \cdot m \rangle \phi_s. \quad (4.17)$$

Using (4.5) in order to estimate the first term, and performing several integration by parts and using the boundedness assumption (H6) on $\phi$ in order to estimate the three last terms, we obtain

$$\|\Omega^{-1} \nabla_x \partial_t w_s\| \lesssim \|m\| \lesssim \|h\|. \quad (4.18)$$

Inserting (4.16) and (4.18) in

$$\frac{d}{dt} \langle \Omega^{-1} \nabla_x w_s, m_s \rangle = \langle \Omega^{-1} \nabla_x \partial_t w_s, m_s \rangle + \langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle$$

completes the proof of (4.13).

In order to control the time-derivative of $w_s$, we write

$$\frac{d}{dt} \langle -\Omega^{-1} \partial_t w_s, w_s \rangle = - \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 - \langle \Omega^{-1} \partial_t^2 w_s, w_s \rangle. \quad (4.19)$$
Finally, by differentiating (2.30b), one has

$$\partial_t \nabla_x m = - \nabla_x^2 r + \sqrt{\frac{2}{d}} \nabla_x^* \nabla_x e + \sqrt{\frac{2}{d}} \nabla_x^2 \phi e + \nabla_x^* \cdot (\nabla_x \otimes E[h^\perp]) + E[h^\perp] \nabla_x^2 \phi$$

and the integration of the skew-symmetric part yields

$$\frac{d}{dt} \langle \nabla_x^{\text{skew}} m \rangle = \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle. \quad (4.15)$$

As a consequence of these identities and (4.2b), one gets

$$\partial_t m_s = - \nabla_x w_s + \sqrt{\frac{2}{d}} \left( \nabla_x^* e_s - \frac{1}{d} \langle e_s \Delta_x \phi \rangle x \right) + m_E$$

where $m_E := \nabla_x^* \cdot E[h^\perp] - \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x - \frac{1}{d} \langle \text{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle x$. Hence

$$\langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle = - \|\Omega^{-\frac{1}{2}} \nabla_x w_s\|^2 + \|\Omega^{-\frac{1}{2}} \nabla_x w_s\| \|\Omega^{-\frac{1}{2}} m_E\| \\ + \sqrt{\frac{2}{d}} \|\Omega^{-\frac{1}{2}} \nabla_x w_s\| \|\Omega^{-\frac{1}{2}} (\nabla_x^* e_s - \frac{1}{d} \langle e_s \Delta_x \phi \rangle x)\|$$

Using the zeroth order Poincaré inequality (4.6) and (4.2c), we can estimate $\|\Omega^{-1/2} \nabla_x^* e_s\|$ by $\|e_s\|$. Up to a few integrations by parts, using (H4), (H6) and (H7), we end up for some constant $C > 0$ with

$$\langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle \le -\frac{1}{2} \|\Omega^{-\frac{1}{2}} \nabla_x w_s\|^2 + C \left( \|e_s\|^2 + \|h^\perp\|^2 \right). \quad (4.16)$$

From the definitions (4.2a) and (4.2e), we have

$$\partial_t w_s = \partial_t r - \langle \nabla_x \partial_t r \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \partial_t r \rangle \xi_2, -\sqrt{\frac{2}{d}} \langle \partial_t e \rangle \phi_s$$

so that, by (2.30a) and (2.30c),

$$\partial_t w_s = \nabla_x^* m - \langle \nabla_x \nabla_x^* m \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* m \rangle \xi_2 - \frac{2}{d} \langle \nabla_x \cdot m \rangle \phi_s. \quad (4.17)$$

Using (4.5) in order to estimate the first term, and performing several integration by parts and using the boundedness assumption (H6) on $\phi$ in order to estimate the three last terms, we obtain

$$\|\Omega^{-1} \nabla_x \partial_t w_s\| \lesssim \|m\| \lesssim \|h\|. \quad (4.18)$$

Inserting (4.16) and (4.18) in

$$\frac{d}{dt} \langle \Omega^{-1} \nabla_x w_s, m_s \rangle = \langle \Omega^{-1} \nabla_x \partial_t w_s, m_s \rangle + \langle \Omega^{-1} \nabla_x w_s, \partial_t m_s \rangle$$

completes the proof of (4.13).

In order to control the time-derivative of $w_s$, we write

$$\frac{d}{dt} \langle -\Omega^{-1} \partial_t w_s, w_s \rangle = - \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 - \langle \Omega^{-1} \partial_{tt}^2 w_s, w_s \rangle. \quad (4.19)$$
Differentiating (4.17) with respect to $t$, we have

$$ \partial_{tt}^2 w_s = \nabla_x^* \cdot (\partial_t m) - \langle \nabla_x \nabla_x^* \cdot (\partial_t m) \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* \cdot (\partial_t m) \rangle \xi_2 - \frac{2}{d} \langle \nabla_x \cdot (\partial_t m) \rangle \phi_s, $$

where the first term is obtained by differentiating (2.30b) and amounts to

$$ \nabla_x^*(\partial_t m) = -\nabla_x^* \cdot \nabla_x w + \sqrt{\frac{2}{d}} \nabla_x^* \cdot \nabla_x^* e_s + \nabla_x^* \cdot \nabla_x^* E[h^{\perp}], $$

using (4.2c) and (4.2d). Similar expressions hold for the three next terms. Arguing similarly as for (4.18), we have

$$ \Omega^{-1} \partial_{tt}^2 w_s = O(\|h\|). $$

Together with (4.19) this proves (4.14). ■

### 4.3.5. First Lyapunov functional.
We end this section by introducing a first, partial Lyapunov functional

$$ \mathcal{F}_1[h] := \|h\|^2 + \varepsilon_1 \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle + \varepsilon_2 \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \\ + \varepsilon_3 \langle \Omega^{-1} \nabla_x w_s, m_s \rangle + \varepsilon_4 \langle -\Omega^{-1} \partial_t w_s, w_s \rangle \quad (4.20) $$

where

$$ \varepsilon_1 = \varepsilon, \quad \varepsilon_2 = \varepsilon^{3/2}, \quad \varepsilon_3 = \varepsilon^{7/4}, \quad \varepsilon_4 = \varepsilon^{15/8}. \quad (4.21) $$

Let us define the dissipation functional

$$ \mathcal{D}_1[h] := \|h^{\perp}\|^2 + \|e_s\|^2 + \|m_s\|^2 + \|w_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2. \quad (4.22) $$

**Lemma 4.6.** There are some positive constants $\kappa_0$, $C_0$ and $\kappa$ such that for any $\varepsilon > 0$ small enough, we have

$$ \frac{d}{dt} \mathcal{F}_1[h] \le -\kappa_0 \|h^{\perp}\|^2 - \varepsilon^{15/8} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2. $$

*Proof of Lemma 4.6.* By collecting the results of Lemmata 4.2, 4.3, 4.4 and 4.5, we obtain

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_1[h] \le & -2 c_{\mathcal{C}} \|h^{\perp}\|^2 - \varepsilon_1 \kappa_1 \|e_s\|^2 + \varepsilon_1 C \|h^{\perp}\| \|h\| \\ & - \varepsilon_2 \kappa_2 \|m_s\|^2 + \varepsilon_2 C (\|h^{\perp}\| + \|e_s\|) \|h\| \\ & - \varepsilon_3 \kappa_3 \|w_s\|^2 + \varepsilon_3 C (\|e_s\|^2 + \|h^{\perp}\|^2 + \|m_s\| \|h\|) \\ & - \varepsilon_4 \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \varepsilon_4 C \|w_s\| \|h\| \end{aligned} $$

for any $\varepsilon_i \in (0, 1)$, $i = 1, 2, 3, 4$, up to a renaming of the generic constant $C > 0$. Using repeatedly Young's inequality, we have

$$ \varepsilon_1 C \|h^{\perp}\| \|h\| \le \frac{1}{2} c_{\mathcal{C}} \|h^{\perp}\|^2 + \varepsilon_1^2 \frac{C^2}{2c_{\mathcal{C}}} \|h\|^2, $$

$$ \varepsilon_2 C \|h^{\perp}\| \|h\| \le \frac{1}{2} c_{\mathcal{C}} \|h^{\perp}\|^2 + \varepsilon_2^2 \frac{C^2}{2c_{\mathcal{C}}} \|h\|^2, $$
Differentiating (4.17) with respect to $t$, we have

$$ \partial_{tt}^2 w_s = \nabla_x^* \cdot (\partial_t m) - \langle \nabla_x \nabla_x^* \cdot (\partial_t m) \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* \cdot (\partial_t m) \rangle \xi_2 - \frac{2}{d} \langle \nabla_x \cdot (\partial_t m) \rangle \phi_s, $$

where the first term is obtained by differentiating (2.30b) and amounts to

$$ \nabla_x^*(\partial_t m) = -\nabla_x^* \cdot \nabla_x w + \sqrt{\frac{2}{d}} \nabla_x^* \cdot \nabla_x^* e_s + \nabla_x^* \cdot \nabla_x^* E[h^{\perp}], $$

using (4.2c) and (4.2d). Similar expressions hold for the three next terms. Arguing similarly as for (4.18), we have

$$ \Omega^{-1} \partial_{tt}^2 w_s = O(\|h\|). $$

Together with (4.19) this proves (4.14).

### 4.3.5. First Lyapunov functional.
We end this section by introducing a first, partial Lyapunov functional

$$ \mathcal{F}_1[h] := \|h\|^2 + \varepsilon_1 \langle \Omega^{-1} \nabla_x e, \Theta[h] \rangle + \varepsilon_2 \langle \Omega^{-1} \nabla_x^{\text{sym}} m_s, E[h] - \sqrt{\frac{2}{d}} \langle e \rangle \text{Id}_{d \times d} \rangle \\ + \varepsilon_3 \langle \Omega^{-1} \nabla_x w_s, m_s \rangle + \varepsilon_4 \langle -\Omega^{-1} \partial_t w_s, w_s \rangle \quad (4.20) $$

where

$$ \varepsilon_1 = \varepsilon, \quad \varepsilon_2 = \varepsilon^{3/2}, \quad \varepsilon_3 = \varepsilon^{7/4}, \quad \varepsilon_4 = \varepsilon^{15/8}. \quad (4.21) $$

Let us define the dissipation functional

$$ \mathcal{D}_1[h] := \|h^{\perp}\|^2 + \|e_s\|^2 + \|m_s\|^2 + \|w_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2. \quad (4.22) $$

**Lemma 4.6.** There are some positive constants $\kappa_0$, $C_0$ and $\kappa$ such that for any $\varepsilon > 0$ small enough, we have

$$ \frac{d}{dt} \mathcal{F}_1[h] \le -\kappa_0 \|h^{\perp}\|^2 - \varepsilon^{\frac{15}{8}} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2. $$

*Proof of Lemma 4.6.* By collecting the results of Lemmata 4.2, 4.3, 4.4 and 4.5, we obtain

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_1[h] \le & -2 c_{\mathcal{C}} \|h^{\perp}\|^2 - \varepsilon_1 \kappa_1 \|e_s\|^2 + \varepsilon_1 C \|h^{\perp}\| \|h\| \\ & - \varepsilon_2 \kappa_2 \|m_s\|^2 + \varepsilon_2 C (\|h^{\perp}\| + \|e_s\|) \|h\| \\ & - \varepsilon_3 \kappa_3 \|w_s\|^2 + \varepsilon_3 C (\|e_s\|^2 + \|h^{\perp}\|^2 + \|m_s\| \|h\|) \\ & - \varepsilon_4 \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \varepsilon_4 C \|w_s\| \|h\| \end{aligned} $$

for any $\varepsilon_i \in (0, 1)$, $i = 1, 2, 3, 4$, up to a renaming of the generic constant $C > 0$. Using repeatedly Young's inequality, we have

$$ \varepsilon_1 C \|h^{\perp}\| \|h\| \le \frac{1}{2} c_{\mathcal{C}} \|h^{\perp}\|^2 + \varepsilon_1^2 \frac{C^2}{2c_{\mathcal{C}}} \|h\|^2, $$

$$ \varepsilon_2 C \|h^{\perp}\| \|h\| \le \frac{1}{2} c_{\mathcal{C}} \|h^{\perp}\|^2 + \varepsilon_2^2 \frac{C^2}{2c_{\mathcal{C}}} \|h\|^2, $$
$$ \varepsilon_2 C \|e_s\| \|h\| \le \frac{1}{2} \varepsilon_1 \kappa_1 \|e_s\|^2 + \frac{\varepsilon_2^2}{\varepsilon_1} \frac{C^2}{2\kappa_1} \|h\|^2, $$

$$ \varepsilon_3 C \|m_s\| \|h\| \le \frac{1}{2} \varepsilon_2 \kappa_2 \|m_s\|^2 + \frac{\varepsilon_3^2}{\varepsilon_2} \frac{C^2}{2\kappa_2} \|h\|^2, $$

$$ \varepsilon_4 C \|w_s\| \|h\| \le \frac{1}{2} \varepsilon_3 \kappa_3 \|w_s\|^2 + \frac{\varepsilon_4^2}{\varepsilon_3} \frac{C^2}{2\kappa_3} \|h\|^2, $$

and therefore

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_1[h] \le -(c_{\mathcal{C}} - \varepsilon_3 C) \|h^{\perp}\|^2 - \varepsilon_1 \kappa_1 \left(\frac{1}{2} - \frac{\varepsilon_3}{\kappa_1 \varepsilon_1} C\right) \|e_s\|^2 - \frac{1}{2} \varepsilon_2 \kappa_2 \|m_s\|^2 \\ - \frac{1}{2} \varepsilon_3 \kappa_3 \|w_s\|^2 - \varepsilon_4 \left\|\Omega^{-\frac{1}{2}} \partial_t w_s\right\|^2 \\ + \frac{1}{2} C^2 \left( \frac{\varepsilon_1^2}{c_{\mathcal{C}}} + \frac{\varepsilon_2^2}{c_{\mathcal{C}}} + \frac{\varepsilon_1^2}{\kappa_1 \varepsilon_1} + \frac{\varepsilon_3^2}{\kappa_2 \varepsilon_2} + \frac{\varepsilon_4^2}{\kappa_3 \varepsilon_3} \right) \|h\|^2. \end{aligned} $$

The choice $\kappa_0 = \frac{1}{4} c_{\mathcal{C}}$, $\kappa = \min \{\frac{c_{\mathcal{C}}}{4}, \frac{\kappa_1}{4}, \frac{\kappa_2}{2}, \frac{\kappa_3}{2}, 1\}$ and $C_0 = \frac{1}{2} C^2 (\frac{2}{c_{\mathcal{C}}} + \frac{1}{\kappa_1} + \frac{1}{\kappa_2} + \frac{1}{\kappa_3})$ with $\varepsilon_i, i = 1, \dots, 4$, given by (4.21) and

$$ 0 < \varepsilon < \min \left\{ 1, \left(\frac{4C}{\kappa_1}\right)^{-4/3}, \left(\frac{2C}{c_{\mathcal{C}}}\right)^{-4/7} \right\} $$

completes the proof.

<figure><img src="image_6.png" /></figure>

### 4.4. Control of finite-dimensional quantities

After estimating the decay of the *deviations from averages terms* defined by (4.2), let us consider the time-dependent *global scalar quantities* $\langle e \rangle$, $\langle \nabla_x^{\text{skew}} m \rangle$, $\langle \nabla_x \cdot m \rangle$, $\langle m \rangle$, $\langle \nabla_x r \rangle$, $\langle \Delta_x r \rangle$. We proceed as in the proof of Proposition 3.1. Let

$$ A(t) := \langle \nabla_x^{\text{skew}} m \rangle, \quad b(t) := \langle m \rangle, \quad c(t) := \langle e \rangle, \qquad (4.23\text{a}) $$

$$ z(t,x) := r(t,x) + b'(t) \cdot x - c''(t) \frac{\xi_2(x)}{2\sqrt{2d}} - c(t) \sqrt{\frac{2}{d}} \xi_{\phi}(x). \qquad (4.23\text{b}) $$

By comparison with (3.2a), we know that $z = 0$ if $h$ corresponds to a special macroscopic mode. We observe here that there is no reason for $A(t)$ to be neither the orthogonal projection of $m$ onto infinitesimal rotation matrices nor independent of $t$. We shall have to take this fact into account later and remember that, according to (4.15),

$$ A'(t) = \langle (E[h^{\perp}] \nabla_x^2 \phi)^{\text{skew}} \rangle. \qquad (4.24) $$

#### 4.4.1. The macroscopic equations.
As defined by (2.25), the functions $r, m$ and $e$ can be rewritten in the new variables as follows.

**Lemma 4.7.** With previous notations, if $h$ solves (2.22)–(2.23) in $L^2(M)$, then

$$ r(t,x) = -b'(t) \cdot x + c''(t) \frac{\xi_2(x)}{2\sqrt{2d}} + c(t) \sqrt{\frac{2}{d}} \xi_{\phi}(x) + z(t,x), \qquad (4.25\text{a}) $$

$$ m(t,x) = A(t)x + b(t) - c'(t) \frac{1}{\sqrt{2d}} x + m_s(t,x), \qquad (4.25\text{b}) $$

$$ e(t,x) = c(t) + e_s(t,x), \qquad (4.25\text{c}) $$
$$ \varepsilon_2 C \|e_s\| \|h\| \le \frac{1}{2} \varepsilon_1 \kappa_1 \|e_s\|^2 + \frac{\varepsilon_2^2}{\varepsilon_1} \frac{C^2}{2\kappa_1} \|h\|^2, $$

$$ \varepsilon_3 C \|m_s\| \|h\| \le \frac{1}{2} \varepsilon_2 \kappa_2 \|m_s\|^2 + \frac{\varepsilon_3^2}{\varepsilon_2} \frac{C^2}{2\kappa_2} \|h\|^2, $$

$$ \varepsilon_4 C \|w_s\| \|h\| \le \frac{1}{2} \varepsilon_3 \kappa_3 \|w_s\|^2 + \frac{\varepsilon_4^2}{\varepsilon_3} \frac{C^2}{2\kappa_3} \|h\|^2, $$

and therefore

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_1[h] \le -(c_{\mathcal{C}} - \varepsilon_3 C) \|h^{\perp}\|^2 - \varepsilon_1 \kappa_1 \left(\frac{1}{2} - \frac{\varepsilon_3}{\kappa_1 \varepsilon_1} C\right) \|e_s\|^2 - \frac{1}{2} \varepsilon_2 \kappa_2 \|m_s\|^2 \\ - \frac{1}{2} \varepsilon_3 \kappa_3 \|w_s\|^2 - \varepsilon_4 \left\| \Omega^{-\frac{1}{2}} \partial_t w_s \right\|^2 \\ + \frac{1}{2} C^2 \left( \frac{\varepsilon_1^2}{c_{\mathcal{C}}} + \frac{\varepsilon_2^2}{c_{\mathcal{C}}} + \frac{\varepsilon_1^2}{\kappa_1 \varepsilon_1} + \frac{\varepsilon_3^2}{\kappa_2 \varepsilon_2} + \frac{\varepsilon_4^2}{\kappa_3 \varepsilon_3} \right) \|h\|^2. \end{aligned} $$

The choice $\kappa_0 = \frac{1}{4} c_{\mathcal{C}}$, $\kappa = \min \{\frac{c_{\mathcal{C}}}{4}, \frac{\kappa_1}{4}, \frac{\kappa_2}{2}, \frac{\kappa_3}{2}, 1\}$ and $C_0 = \frac{1}{2} C^2 (\frac{2}{c_{\mathcal{C}}} + \frac{1}{\kappa_1} + \frac{1}{\kappa_2} + \frac{1}{\kappa_3})$ with $\varepsilon_i, i = 1, \dots, 4$, given by (4.21) and

$$ 0 < \varepsilon < \min \left\{ 1, \left(\frac{4C}{\kappa_1}\right)^{-4/3}, \left(\frac{2C}{c_{\mathcal{C}}}\right)^{-4/7} \right\} $$

completes the proof.

<figure><img src="image_7.png" /></figure>

### 4.4. Control of finite-dimensional quantities

After estimating the decay of the *deviations from averages terms* defined by (4.2), let us consider the time-dependent *global scalar quantities* $\langle e \rangle$, $\langle \nabla_x^{\text{skew}} m \rangle$, $\langle \nabla_x \cdot m \rangle$, $\langle m \rangle$, $\langle \nabla_x r \rangle$, $\langle \Delta_x r \rangle$. We proceed as in the proof of Proposition 3.1. Let

$$ A(t) := \langle \nabla_x^{\text{skew}} m \rangle, \quad b(t) := \langle m \rangle, \quad c(t) := \langle e \rangle, \qquad (4.23\text{a}) $$

$$ z(t,x) := r(t,x) + b'(t) \cdot x - c''(t) \frac{\xi_2(x)}{2\sqrt{2d}} - c(t) \sqrt{\frac{2}{d}} \xi_{\phi}(x). \qquad (4.23\text{b}) $$

By comparison with (3.2a), we know that $z = 0$ if $h$ corresponds to a special macroscopic mode. We observe here that there is no reason for $A(t)$ to be neither the orthogonal projection of $m$ onto infinitesimal rotation matrices nor independent of $t$. We shall have to take this fact into account later and remember that, according to (4.15),

$$ A'(t) = \langle (E[h^{\perp}] \nabla_x^2 \phi)^{\text{skew}} \rangle. \qquad (4.24) $$

#### 4.4.1. The macroscopic equations.
As defined by (2.25), the functions $r, m$ and $e$ can be rewritten in the new variables as follows.

**Lemma 4.7.** With previous notations, if $h$ solves (2.22)–(2.23) in $L^2(M)$, then

$$ r(t,x) = -b'(t) \cdot x + c''(t) \frac{\xi_2(x)}{2\sqrt{2d}} + c(t) \sqrt{\frac{2}{d}} \xi_{\phi}(x) + z(t,x), \qquad (4.25\text{a}) $$

$$ m(t,x) = A(t)x + b(t) - c'(t) \frac{1}{\sqrt{2d}} x + m_s(t,x), \qquad (4.25\text{b}) $$

$$ e(t,x) = c(t) + e_s(t,x), \qquad (4.25\text{c}) $$
where $z$ obeys the bounds

$$ \|z\|^2 \le \|w_s\|^2 + \|e_s\|^2 + \|h^\perp\|^2, \quad (4.26a) $$

$$ \|\Omega^{-\frac{1}{2}} \partial_t z\|^2 \le \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \|m_s\|^2 + \|h^\perp\|^2. \quad (4.26b) $$

*Proof of Lemma 4.7.* The expression (4.25a) follows from the definition of $z$ in (4.23b), while (4.25c) is no more than a rewriting of (4.2c). From (2.30c) one observes that

$$ c' = \frac{d}{dt} \langle e \rangle = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot m \rangle, \quad (4.27) $$

so that (4.25b) follows from the definition (4.2b) of $m_s$.

From (2.30b) we have

$$ b' = \frac{d}{dt} \langle m \rangle = - \langle \nabla_x r \rangle. \quad (4.28) $$

Using (4.2a) and (4.2e), we write

$$ \begin{aligned} r &= w_s + \sqrt{\frac{2}{d}} \langle e \rangle \phi_s + \langle \nabla_x r \rangle \cdot x + \frac{1}{2d} \langle \Delta_x r \rangle \xi_2 \\ &= w_s + \sqrt{\frac{2}{d}} c \left( \xi_\phi - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2 \right) - b' \cdot x + \frac{1}{2d} \langle \Delta_x r \rangle \xi_2. \end{aligned} $$

From (4.25a), we deduce

$$ z = w_s + \frac{1}{\sqrt{2d}} \left( \frac{1}{\sqrt{2d}} \langle \Delta_x r \rangle - \frac{1}{2} c'' - \frac{1}{d} \langle \Delta_x \phi \rangle c \right) \xi_2. \quad (4.29) $$

Finally, thanks to (4.27) and (2.30b), we compute

$$ c'' = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot \partial_t m \rangle = \sqrt{\frac{2}{d}} \langle \Delta_x r \rangle - \frac{2}{d} \langle \nabla_x \cdot \nabla_x^* e \rangle - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot (\nabla_x^* E[h^\perp]) \rangle, $$

and thus obtain

$$ c'' = \sqrt{\frac{2}{d}} \langle \Delta_x r \rangle - \frac{2}{d} \langle e \Delta \phi \rangle - \sqrt{\frac{2}{d}} \langle \operatorname{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle. \quad (4.30) $$

By inserting (4.30) in (4.29), we obtain

$$ z = w_s + \frac{1}{2d} \langle E[h^\perp] : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{d\sqrt{2d}} \langle e_s \Delta_x \phi \rangle \xi_2, $$

from which (4.26a) follows. By differentiating $z$ with respect to $t$, we get

$$ \partial_t z = \partial_t w_s + \frac{1}{2d} \langle \partial_t E[h^\perp] : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{d\sqrt{2d}} \langle \partial_t e_s \Delta_x \phi \rangle \xi_2. $$

By (2.30d) and (2.31a), we know that

$$ \partial_t E[h] = -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^\perp] = \sqrt{\frac{2}{d}} \partial_t e \operatorname{Id}_{d \times d} + \partial_t E[h^\perp]. $$

Besides, we learn from (4.2b) and (4.27) that

$$ \nabla_x^{\text{sym}} m = \nabla_x^{\text{sym}} m_s + \frac{1}{d} \langle \nabla_x \cdot m \rangle \operatorname{Id}_{d \times d} = \nabla_x^{\text{sym}} m_s - \frac{1}{\sqrt{2d}} \langle \partial_t e \rangle \operatorname{Id}_{d \times d} $$
where $z$ obeys the bounds

$$ \|z\|^2 \le \|w_s\|^2 + \|e_s\|^2 + \|h^\perp\|^2, \quad (4.26a) $$

$$ \|\Omega^{-\frac{1}{2}} \partial_t z\|^2 \le \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \|m_s\|^2 + \|h^\perp\|^2. \quad (4.26b) $$

*Proof of Lemma 4.7.* The expression (4.25a) follows from the definition of $z$ in (4.23b), while (4.25c) is no more than a rewriting of (4.2c). From (2.30c) one observes that

$$ c' = \frac{d}{dt} \langle e \rangle = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot m \rangle, \quad (4.27) $$

so that (4.25b) follows from the definition (4.2b) of $m_s$.

From (2.30b) we have

$$ b' = \frac{d}{dt} \langle m \rangle = - \langle \nabla_x r \rangle. \quad (4.28) $$

Using (4.2a) and (4.2e), we write

$$ \begin{aligned} r &= w_s + \sqrt{\frac{2}{d}} \langle e \rangle \phi_s + \langle \nabla_x r \rangle \cdot x + \frac{1}{2d} \langle \Delta_x r \rangle \xi_2 \\ &= w_s + \sqrt{\frac{2}{d}} c \left( \xi_\phi - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2 \right) - b' \cdot x + \frac{1}{2d} \langle \Delta_x r \rangle \xi_2. \end{aligned} $$

From (4.25a), we deduce

$$ z = w_s + \frac{1}{\sqrt{2d}} \left( \frac{1}{\sqrt{2d}} \langle \Delta_x r \rangle - \frac{1}{2} c'' - \frac{1}{d} \langle \Delta_x \phi \rangle c \right) \xi_2. \quad (4.29) $$

Finally, thanks to (4.27) and (2.30b), we compute

$$ c'' = - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot \partial_t m \rangle = \sqrt{\frac{2}{d}} \langle \Delta_x r \rangle - \frac{2}{d} \langle \nabla_x \cdot \nabla_x^* e \rangle - \sqrt{\frac{2}{d}} \langle \nabla_x \cdot (\nabla_x^* E[h^\perp]) \rangle, $$

and thus obtain

$$ c'' = \sqrt{\frac{2}{d}} \langle \Delta_x r \rangle - \frac{2}{d} \langle e \Delta \phi \rangle - \sqrt{\frac{2}{d}} \langle \operatorname{Tr}(E[h^\perp] \nabla_x^2 \phi) \rangle. \quad (4.30) $$

By inserting (4.30) in (4.29), we obtain

$$ z = w_s + \frac{1}{2d} \langle E[h^\perp] : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{d\sqrt{2d}} \langle e_s \Delta_x \phi \rangle \xi_2, $$

from which (4.26a) follows. By differentiating $z$ with respect to $t$, we get

$$ \partial_t z = \partial_t w_s + \frac{1}{2d} \langle \partial_t E[h^\perp] : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{d\sqrt{2d}} \langle \partial_t e_s \Delta_x \phi \rangle \xi_2. $$

By (2.30d) and (2.31a), we know that

$$ \partial_t E[h] = -2 \nabla_x^{\text{sym}} m + E[\mathcal{L} h^\perp] = \sqrt{\frac{2}{d}} \partial_t e \operatorname{Id}_{d \times d} + \partial_t E[h^\perp]. $$

Besides, we learn from (4.2b) and (4.27) that

$$ \nabla_x^{\text{sym}} m = \nabla_x^{\text{sym}} m_s + \frac{1}{d} \langle \nabla_x \cdot m \rangle \operatorname{Id}_{d \times d} = \nabla_x^{\text{sym}} m_s - \frac{1}{\sqrt{2d}} \langle \partial_t e \rangle \operatorname{Id}_{d \times d} $$
and, as a consequence,

$$ \partial_t E[h^\perp] = -2 \nabla_x^{\text{sym}} m_s + E[\mathcal{L} h^\perp] - \sqrt{\frac{2}{d}} \partial_t e_s \text{Id}_{d \times d}. $$

Hence

$$ \partial_t z = \partial_t w_s - \frac{1}{d} \langle \nabla_x^{\text{sym}} m_s : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{2d} \langle E[\mathcal{L} h^\perp] : \nabla_x^2 \phi \rangle \xi_2 $$

and Estimate (4.26b) follows using an integration by parts and (H6). ■

Using (4.25a) on the one hand, and (2.30a) combined with (4.25b) on the other hand, we write

$$ \begin{aligned} \partial_t r &= \sqrt{\frac{2}{d}} \xi_{\phi} c' - x \cdot b'' + \frac{\xi_2}{2\sqrt{2d}} c''' + \partial_t z \\ &= \nabla_x^* m_s + \nabla_x \phi \cdot A x - \frac{1}{\sqrt{2d}} (\nabla_x \phi \cdot x - d) c' + \nabla_x \phi \cdot b. \end{aligned} $$

We deduce a differential equation which is very similar to (2.12) up to additional terms involving $m_s$ and $z$, namely

**Proposition 4.8.** The functions A, b, c, z and $m_s$ defined by (4.23) and (4.2b) solve

$$ \frac{2 \xi_{\phi}(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - x \cdot b'' - \nabla_x \phi \cdot Ax \\ = \nabla_x^* m_s - \partial_t z. \quad (4.31) $$

### 4.4.2. Control of A. The counterpart of Section 3.2 goes as follows.

**Lemma 4.9.** There are some positive constants $\kappa_4$ and $C$ such that

$$ -\frac{d}{dt} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle \le -\kappa_4 |A|^2 + C (\mathcal{D}_1[h] + C \|h^\perp\| \|h\|), \quad (4.32) $$

where $X = X(b', c, c'')$, $Y = Y(b', c, c'')$ and $\mathcal{D}_1[h]$ are defined respectively in (3.5), (3.6) and (4.22).

**Proof of Lemma 4.9.** We argue as for Lemma 3.4. We multiply (4.31) by $x_k$ for $k = 1, \dots, d$ and after integration, we get

$$ \sqrt{\frac{2}{d}} \langle \xi_{\phi} x \rangle c' + \frac{1}{2\sqrt{2d}} \langle \xi_2 x \rangle c''' - b - \langle x \otimes x \rangle b'' = \langle m_s \rangle - \langle x \partial_t z \rangle. \quad (4.33) $$

Using the definitions of X and Y, (4.31) and (4.33) yield

$$ \frac{d}{dt} (X - Y \cdot \nabla_x \phi) = \nabla_x \phi \cdot A x + \nabla_x^* m_s - \langle m_s \rangle \cdot \nabla_x \phi - \partial_t z + \langle x \partial_t z \rangle \cdot \nabla_x \phi. $$

Using (4.23a) and (4.24), we obtain

$$ \begin{aligned} \frac{d}{dt} \langle -(X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle \\ = - \| \nabla_x \phi \cdot A x \|^2 - \langle \nabla_x^* m_s, \nabla_x \phi \cdot A x \rangle + \langle \langle m_s \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot A x \rangle \end{aligned} $$
and, as a consequence,

$$ \partial_t E[h^\perp] = -2 \nabla_x^{\text{sym}} m_s + E[\mathcal{L} h^\perp] - \sqrt{\frac{2}{d}} \partial_t e_s \text{Id}_{d \times d}. $$

Hence

$$ \partial_t z = \partial_t w_s - \frac{1}{d} \langle \nabla_x^{\text{sym}} m_s : \nabla_x^2 \phi \rangle \xi_2 + \frac{1}{2d} \langle E[\mathcal{L} h^\perp] : \nabla_x^2 \phi \rangle \xi_2 $$

and Estimate (4.26b) follows using an integration by parts and (H6). ■

Using (4.25a) on the one hand, and (2.30a) combined with (4.25b) on the other hand, we write

$$ \begin{aligned} \partial_t r &= \sqrt{\frac{2}{d}} \xi_\phi c' - x \cdot b'' + \frac{\xi_2}{2\sqrt{2d}} c''' + \partial_t z \\ &= \nabla_x^* m_s + \nabla_x \phi \cdot A x - \frac{1}{\sqrt{2d}} (\nabla_x \phi \cdot x - d) c' + \nabla_x \phi \cdot b. \end{aligned} $$

We deduce a differential equation which is very similar to (2.12) up to additional terms involving $m_s$ and $z$, namely

**Proposition 4.8.** The functions A, b, c, z and $m_s$ defined by (4.23) and (4.2b) solve

$$ \frac{2 \xi_{\phi}(x) + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - x \cdot b'' - \nabla_x \phi \cdot Ax \\ = \nabla_x^* m_s - \partial_t z. \quad (4.31) $$

### 4.4.2. Control of A. The counterpart of Section 3.2 goes as follows.

**Lemma 4.9.** There are some positive constants $\kappa_4$ and $C$ such that

$$ -\frac{d}{dt} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle \le -\kappa_4 |A|^2 + C (\mathcal{D}_1[h] + C \|h^\perp\| \|h\|), \quad (4.32) $$

where $X = X(b', c, c'')$, $Y = Y(b', c, c'')$ and $\mathcal{D}_1[h]$ are defined respectively in (3.5), (3.6) and (4.22).

**Proof of Lemma 4.9.** We argue as for Lemma 3.4. We multiply (4.31) by $x_k$ for $k = 1, \dots, d$ and after integration, we get

$$ \sqrt{\frac{2}{d}} \langle \xi_{\phi} x \rangle c' + \frac{1}{2\sqrt{2d}} \langle \xi_2 x \rangle c''' - b - \langle x \otimes x \rangle b'' = \langle m_s \rangle - \langle x \partial_t z \rangle. \quad (4.33) $$

Using the definitions of $X$ and $Y$, (4.31) and (4.33) yield

$$ \frac{d}{dt} (X - Y \cdot \nabla_x \phi) = \nabla_x \phi \cdot A x + \nabla_x^* m_s - \langle m_s \rangle \cdot \nabla_x \phi - \partial_t z + \langle x \partial_t z \rangle \cdot \nabla_x \phi. $$

Using (4.23a) and (4.24), we obtain

$$ \begin{aligned} \frac{d}{dt} \langle -(X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle \\ = - \| \nabla_x \phi \cdot A x \|^2 - \langle \nabla_x^* m_s, \nabla_x \phi \cdot A x \rangle + \langle \langle m_s \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot A x \rangle \end{aligned} $$
$$ + \langle \partial_t z, \nabla_x \phi \cdot A x \rangle - \langle \langle x \partial_t z \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot A x \rangle \\ - \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x \rangle . $$

For the first term and thanks to the conservation law (2.27), we note that

$$ \mathcal{R}_{\phi}^{\perp} \ni \mathbb{P}(m) = Ax + \mathbb{P}(m_s), $$

so that we can apply inequality (3.3) to $x \mapsto Ax + \mathbb{P}(m_s)$ to get

$$ c_K \|Ax + \mathbb{P}(m_s)\|^2 \le \|\nabla_x \phi \cdot Ax + \nabla_x \phi \cdot \mathbb{P}(m_s)\|^2, $$

which yields

$$ c_K |A|^2 = c_K \|Ax\|^2 \le 4 \|\nabla_x \phi \cdot Ax\|^2 + C \|m_s\|^2 \quad (4.34) $$

for any $C > 4 + c_K$. In order to estimate the other terms, we use

$$ \begin{aligned} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x \rangle & \le \|\Omega^{-1} (X - Y \cdot \nabla_x \phi)\| \|\Omega (\nabla_x \phi x)\| \|\langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle\| \\ & \le (|b'| + |c'| + |c'''|) \|h^\perp\|, \end{aligned} $$

and

$$ \langle \langle m_s \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot Ax \rangle \le \|m_s\| \|\nabla_x \phi \cdot Ax\| \le \|m_s\| |A|. $$

Thanks to the zeroth order Poincaré inequality (4.6), we also have

$$ \langle \nabla_x^* m_s, \nabla_x \phi \cdot Ax \rangle = \langle \Omega^{-\frac{1}{2}} (\nabla_x^* m_s), \Omega^{\frac{1}{2}} (\nabla_x \phi \cdot Ax) \rangle \le \|m_s\| |A| $$

as well as similar estimates for the terms in $\partial_t z$ and $\langle x \partial_t z \rangle \cdot \nabla_x \phi$. Collecting these estimates with (4.34), we get

$$ \begin{aligned} \frac{d}{dt} \langle - (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot Ax \rangle & \le - \frac{1}{4} c_K |A|^2 + C \left( \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t z\| \right) |A| + \left( |b'| + |c'| + |c'''| \right) \|h^\perp\|, \end{aligned} $$

for some $\kappa_4 < \frac{1}{4} c_K$ and $C > 0$ large enough. Young's inequality and (4.26b) conclude the proof of (4.32). ■

### 4.4.3. Control of $b, b', b''$ and $c, c', c''$ and $c'''$. It follows from three lemmata.

**Lemma 4.10.** The following estimate holds

$$ |b| + |b''| + |c'| + |c'''| \le |A| + \|\Omega^{-\frac{1}{2}} \partial_t w_s\| + \|m_s\| + \|h^\perp\|. \quad (4.35) $$

*Proof.* Using (4.31), we can write

$$ \frac{2 \xi_{\phi} + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - x \cdot b'' = R_0 $$
$$ + \langle \partial_t z, \nabla_x \phi \cdot A x \rangle - \langle \langle x \partial_t z \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot A x \rangle \\ - \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x \rangle. $$

For the first term and thanks to the conservation law (2.27), we note that

$$ \mathcal{R}_{\phi}^{\perp} \ni \mathbb{P}(m) = A x + \mathbb{P}(m_s), $$

so that we can apply inequality (3.3) to $x \mapsto Ax + \mathbb{P}(m_s)$ to get

$$ c_K \|Ax + \mathbb{P}(m_s)\|^2 \le \|\nabla_x \phi \cdot Ax + \nabla_x \phi \cdot \mathbb{P}(m_s)\|^2, $$

which yields

$$ c_K |A|^2 = c_K \|Ax\|^2 \le 4 \|\nabla_x \phi \cdot Ax\|^2 + C \|m_s\|^2 \quad (4.34) $$

for any $C > 4 + c_K$. In order to estimate the other terms, we use

$$ \begin{aligned} \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot \langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle x \rangle & \le \|\Omega^{-1} (X - Y \cdot \nabla_x \phi)\| \|\Omega (\nabla_x \phi x)\| \|\langle (E[h^\perp] \nabla_x^2 \phi)^{\text{skew}} \rangle\| \\ & \le (|b'| + |c'| + |c'''|) \|h^\perp\|, \end{aligned} $$

and

$$ \langle \langle m_s \rangle \cdot \nabla_x \phi, \nabla_x \phi \cdot A x \rangle \le \|m_s\| \|\nabla_x \phi \cdot A x\| \le \|m_s\| |A|. $$

Thanks to the zeroth order Poincaré inequality (4.6), we also have

$$ \langle \nabla_x^* m_s, \nabla_x \phi \cdot A x \rangle = \langle \Omega^{-\frac{1}{2}} (\nabla_x^* m_s), \Omega^{\frac{1}{2}} (\nabla_x \phi \cdot A x) \rangle \le \|m_s\| |A| $$

as well as similar estimates for the terms in $\partial_t z$ and $\langle x \partial_t z \rangle \cdot \nabla_x \phi$. Collecting these estimates with (4.34), we get

$$ \begin{aligned} & \frac{d}{dt} \langle - (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle \\ & \le - \frac{1}{4} c_K |A|^2 + C \left( \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t z\| \right) |A| + \left( |b'| + |c'| + |c'''| \right) \|h^\perp\|, \end{aligned} $$

for some $\kappa_4 < \frac{1}{4} c_K$ and $C > 0$ large enough. Young's inequality and (4.26b) conclude the proof of (4.32). ■

### 4.4.3. Control of $b, b', b''$ and $c, c', c''$ and $c''''$. It follows from three lemmata.

**Lemma 4.10.** The following estimate holds

$$ |b| + |b''| + |c'| + |c'''| \le |A| + \|\Omega^{-\frac{1}{2}} \partial_t w_s\| + \|m_s\| + \|h^\perp\|. \quad (4.35) $$

*Proof.* Using (4.31), we can write

$$ \frac{2 \xi_{\phi} + \nabla_x \phi \cdot x - d}{\sqrt{2d}} c' + \frac{\xi_2}{2\sqrt{2d}} c''' - \nabla_x \phi \cdot b - x \cdot b'' = R_0 $$
with $R_0 := \nabla_x \phi \cdot A x + \nabla_x^* m_s - \partial_t z$. Arguing as in the proof of Lemma 3.5 with this new definition of $R_0$, we obtain that (3.11) holds with

$$R_3 := - \langle R_0 \tilde{\Phi} \rangle, \quad \tilde{\Phi} := \nabla \phi - x - \langle (\nabla \phi - x) \otimes x \rangle \nabla \phi.$$

Observing that

$$R_3 = - \langle \nabla_x \phi \cdot A x \tilde{\Phi} \rangle - \langle T(D \tilde{\Phi}) m_s \rangle + \langle \Omega^{\frac{1}{2}} \tilde{\Phi}, \Omega^{-\frac{1}{2}} \partial_t z \rangle \\ = O(|A| + \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t z\|)$$

and using (4.26b), the proof of (4.35) follows for the same reasons as in the proof of Lemma 3.5. ■

With $D_1[h]$ defined in (4.22), we obtain the counterpart of Lemma 3.6.

**Lemma 4.11.** There exists a constant $C > 0$ so that

$$\begin{aligned} \frac{d}{dt} \langle -b, b' \rangle &\le -|b'|^2 + C|A|^2 + C D_1[h], \\ \frac{d}{dt} \langle -c', c'' \rangle &\le -|c''|^2 + C|A|^2 + C D_1[h]. \end{aligned}$$

**Lemma 4.12.** The following estimates hold

$$|c| \lesssim |b'| + |c''| + \|w_s\|^2 + \|e_s\|^2 + \|h^{\perp}\|^2, \quad (4.36)$$

$$\|r\| \lesssim |b'| + |c''| + \|w_s\|^2 + \|e_s\|^2 + \|h^{\perp}\|^2. \quad (4.37)$$

*Proof.* According to (4.25a), we can write $r$ as

$$r = \sqrt{\frac{2}{d}} \xi_{\phi} c + R_5 \quad (4.38)$$

where $R_5 := z - x \cdot b' + \frac{\xi_2}{2\sqrt{2d}} c'' = O(|b'| + |c''| + D_1[h])$ because of (4.26a). Using this expression in (2.26) and recalling that $\langle e \rangle = c$ yields

$$\sqrt{\frac{d}{2}} c \left( 1 + \frac{2}{d} \langle \xi_{\phi}^2 \rangle \right) = - \langle \xi_{\phi} R_5 \rangle,$$

from which (4.36) follows. Coming back to (4.38), we establish (4.37). ■

**4.4.4. Second Lyapunov functional.** Let us introduce the Lyapunov function

$$F_2[h] := F_1[h] - \varepsilon_5 \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle - \varepsilon_6 \langle b, b' \rangle - \varepsilon_6 \langle c', c'' \rangle \quad (4.39)$$

for some additional small parameters $\varepsilon_5$ and $\varepsilon_6$, and the associated dissipation functional

$$D_2[h] := D_1[h] + |A|^2 + |b'|^2 + |c''|^2. \quad (4.40)$$

**Lemma 4.13.** For any $0 < \varepsilon_6 < \varepsilon_5 < \varepsilon_4 < \varepsilon_3 < \varepsilon_2 < \varepsilon_1$ with $\varepsilon_1$ small enough, there holds

$$\|h\|^2 \lesssim F_2[h] \lesssim D_2[h] \lesssim \|h\|^2. \quad (4.41)$$