Special macroscopic modes and hypocoercivity

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.


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 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 [43,10]. 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 [27,45].
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.

Equation and assumptions. Consider the kinetic equation
for the unknown distribution function f = f (t, x, v) depending on the time variable t ≥ 0, the position variable x ∈ R d , and the velocity variable v ∈ R d , where d ≥ 1 is an arbitrary dimension. The transport operator T is given by with a stationary, position dependent potential φ : R d → R. We assume that the linear collision operator C is acting only along the velocity variable v ∈ R d , is self-adjoint in L 2 (µ −1 ), with weight given by the local Maxwellian function (1.2) ∀ v ∈ R d , µ(v) := e −|v| 2 /2 (2 π) d/2 , and has the (d + 2)-dimensional kernel of collision invariants given by (H0) Ker C = Span µ, v 1 µ, . . . , v d µ, |v| 2 µ , corresponding to the local conservation of mass, momentum and energy.
Here L 2 (µ −1 ) is the subspace of L 2 loc (R d , dv) of the functions f such that is finite.
We assume that C satisfies the following spectral gap property (which is a quantitative version of the spatially homogeneous linearized H-theorem) for some constant c C > 0 and all f in the domain of C , where Π denotes the L 2 (µ −1 )-orthogonal projection onto Ker C . Moreover, we suppose that for any polynomial function p(v) : R d → R of degree at most 4, the function p µ is in the domain of C and We provide examples of collision operators satisfying these conditions in Appendix B.1, including the linearized Boltzmann and Landau operators.
Throughout the paper, we assume that the potential φ : R d → R is such that ρ(x) := e −φ(x) is a centred probability density, i.e.,  We also assume that φ is of class C 2 (R d ; R), and for all ε > 0, there exists a constant C ε such that where ∇ 2 x φ denotes the Hessian matrix of φ. We further assume that the measure ρ(x) dx satisfies the Poincaré inequality with a constant c P > 0, for all ϕ ∈ L 2 (ρ), where is the average of ϕ. Here L 2 (ρ) is the subspace of L 2 loc (R d , dx) of the functions ϕ such that ϕ 2 L 2 (ρ) = R d |ϕ| 2 ρ dx is finite. We assume moment bounds on ρ, namely (H6) for some constant C φ > 0. We also introduce the normalization (H7) where Id d×d the identity matrix of size d. The assumption that ∇ 2 x φ 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 B.3.
The potential φ(x) := (1 + |a γ x| 2 ) γ/2 − Z γ , with γ > 1 and real normalization constants a γ , Z γ , satisfies (H3)-(H4)-(H5)-(H6)-(H7). See Appendix B.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 B.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 C and φ 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 (H8) t → e tL is a strongly continuous semi-group on the space L 2 (M −1 ), where M is the global Maxwellian equilibrium function given by (2 π) d/2 , and the space L 2 (M −1 ) is the subspace of L 2 loc (R d × R d , dx dv) of the functions f such that is finite.
1.2. The main result. From here on, we assume the normalization conditions (H3)-(H7). The function 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 Of course we read from (H0) that F = α M, α ∈ R, is a special macroscopic mode but we also look for solutions to (1.4) that can be written as (1.5) for some functions r, m and e with values respectively in R, R d and R, with The energy mode F = β H M, β ∈ R, is another stationary solution to (1.4) where H defined by (1.7) H(x, v) := 1 2 |v| 2 − d + φ(x) − φ is the Hamiltonian energy associated with the characteristics of the transport equation ∂ t f = 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 φ as observed by L. Boltzmann in [5]. These modes deserve some explanations.
The set of infinitesimal rotations compatible with φ defined as is identified with a subset of the space of skew-symmetric matrices In other words, we write A ∈ R φ if and only if φ is invariant by the rotation group θ → e θA , i.e., The set R φ gives rise to the corresponding set of rotation modes compatible with φ defined by Functions in R φ are stationary solutions of (1.4) associated to all invariances of φ under rotation.
There are also some time-periodic special macroscopic modes when φ has harmonic directions. Let us define Notice that E φ is a subspace of R d and d φ ≤ d. Alternatively, we can characterize d φ by We face three different cases. The case d φ = d is called fully non-harmonic: E φ = R d and φ has no harmonic direction and it that case, as we shall see below, there are no timeperiodic solutions.
In the case 1 ≤ d φ ≤ d − 1, the potential is called partially harmonic. In the harmonic coordinates x d φ +1 , . . . , x d , we have ∂ x i φ = x i and define harmonic directional modes by Harmonic directional modes are also defined if d φ = 0. By convention, we set D φ := {0} if d φ = d. All functions in D φ 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 ⊥ φ . These modes are independent of each other.
In the case d φ = 0, the potential is called fully harmonic and φ(x) = 1 2 |x| 2 + d 2 log(2 π). In addition to the harmonic directional modes, the set of harmonic pulsating modes (1.11) P φ := Span is also made of solutions to (1.4). By convention, we set P φ = {0} if d φ ≥ 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 With these definitions at hand, we can now state the main result of this paper.
(2) There are explicit constants C > 0 and κ > 0 such that, for any solu- there exists a unique special macroscopic mode F such that The constants in the decay estimate being explicit means that the proof is constructive and provides a finite algorithm for computing C and κ.
In the following, the norm and scalar product without subscript, · and ·, · , refer to the space L 2 (M), so that Note that when considering functions of x only, the L 2 (ρ)-norm coincides with the L 2 (M)-norm. · stands for the average with respect to ρ dx.
In Theorem 1.1, the constants C and κ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to φ 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 [20,21,22] 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 and 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 Hilbertian 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 memoir [45] by C. Villani for an overview of the initial developments of this theory and to [27,35,26,13,14] 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 arguments. 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 perturbation 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 [3,44] and to [23,28,7] in polynomially weighted spaces.
In this article, we focus on an important and old problem. We study kinetic equations 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 assumption 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 contribution is precisely the study of these consequences, which requires new methods, by classifying all special macroscopic modes and proving hypocoercivity results with constructive 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), A x·v is a stationary solution of the transport equation where A x · v is known as the angular momentum generated by A ∈ M d×d (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 [10,43,6,19]. 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 B.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 φ 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 where R ⊥ φ is the orthogonal complement in L 2 (ρ) of the set R φ defined in (1.8). The time-periodic special macroscopic modes in D φ and P φ , defined in (1.10) and (1.11) respectively, are related to the (partial) harmonicity of φ and another difficulty is to quantify "how far" the potential φ is from being (partially) harmonic. As for Korn type inequalities, the analysis relies on the finite dimension of the space E φ 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 ∇ * x the adjoint of ∇ x in L 2 (ρ) acting on vector fields The Witten-Laplace operator ∇ * x · ∇ x is self-adjoint in L 2 (ρ), with kernel spanned by constant functions and its first non-zero eigenvalue determines the optimal Poincaré constant c P . The operator (1.14) is used in the 0th-order Poincaré inequality 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 micromacro decomposition as in [20,21,22] 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 [24,38,39] 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 C is bounded and φ has bounded derivatives of order two and more. The method is based on commutator estimates as in [27,35,45] in the spirit of the hypoellipticity theory of [29]. In practice, an elegant triple cascade of commutators based on the equality [∇ v , v · ∇ x ] = ∇ 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. Appendix B 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.

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 By (1.4), we know that ∂ t F = T F . By integrating in v the evolution equation against 1, v, v ⊗ v and v E, we obtain macroscopic equations on the macroscopic quantities r = r(t, x), m = m(t, x) and e = e(t, x): where the symmetric gradient is defined by From (2.1e), we deduce that e does not depend on the space variable, and therefore is a function t → c(t) depending on t only. We recall that the average is defined as ϕ := R d ϕ ρ dx. Then we read from (2.1d) that (2.4) By the Schwarz Lemma applied to m = (m k ) d k=1 , we have for any i, j, k = 1, . . . , d. By differentiating (2.4) with respect to x i , x j and x k , we get that ∇ 2 x m = 0, so that in particular the skew-symmetric gradient, defined by is constant in the x-variable and equal to its average. Together with (2.4), 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 x . 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 An integration against ρ shows that r 0 = r(t, ·) 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): Equation (2.12) suggests, on the one hand, that (partial) harmonicity of the potential φ allows for non-trivial choices of b and c, as we shall indeed see later. On the other hand, for a generic potential φ in the sense that the for two constants r 0 and c ∈ R. In other words, we have if φ is not a (partially) harmonic potential: any special macroscopic mode is then a linear combination of a Maxwellian function and an energy mode.

Global conservation laws. Consider a solution
. Associated with the symmetries of the equation, there are local conservations which, after integration on the phase space R d × R d , give rise to global conservation laws. These laws allow us to identify the special macroscopic modes compatible with φ which, as we shall see later, attract the solutions to the Cauchy problem.
The conservation of mass writes Hence α M is a solution to (1.4) with same mass , the conservation of energy amounts to is a solution to (1.4) with same energy as the conserved energy of f . With Moreover, the global conservations of mass and energy write The transport operator can be written as 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 H is conserved along the characteristics of Newton's equationsẋ = v andv = − ∇ x φ. This is reflected by the Poisson brackets: a stationary solution F satisfies 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 T invariant, we can associate an infinitesimal transformation G(x, v) such that {G, H} = 0 and as a consequence, if f solves the transport equation is in the kernel of the collision operator C for any x ∈ R d , then γ G M is a solution of (1.4), i.e., a special macroscopic mode, for any γ ∈ R. Then (2.17) holds for any solution f of (1.1) with initial datum f 0 and there is a unique γ ∈ R such that More considerations on symmetries, local and global conservation laws, and Noether's theorem can be found in textbooks on classical mechanics like, for instance, [31,41]. The case of rotational symmetries enters this framework: When φ is invariant under a rotation, stationary rotation modes appear. Let A ∈ R φ as defined in (1.8) and consider the rotation group (R θ ) θ∈R defined by R θ := e θA and a point for any θ ∈ R. By differentiation with respect to θ, we get Integrating the above identity against (u · x) ρ yields u = 0 after an integration by parts because ρ(x) dx is centred according to Assumption (H3). Rotation modes compatible with φ 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 and the conservation of the total angular momentum associated with this rotation writes Given f 0 , let us identify the corresponding special macroscopic mode.
Associated with f 0 , we introduce the initial momentum m 0 (x) := We can then check that x → A x belongs to R φ and thus the function (rotational mode compatible with φ) the momentum of f , the associated conservation law then reads φ is the orthogonal of R φ , seen as a subspace of R, for the scalar product induced by L 2 (ρ). We refer to Lemma A.1 in Appendix A.1 for a precise statement and a short proof.
, then the conservation of the total angular momentum implies for all j ∈ J φ : Now let us turn our attention to the time-periodic special macroscopic modes and start with the harmonic directional modes, which appear when i depends only on (x 1 , . . . , x d φ ) and for any i ∈ I φ the harmonic directional modes, defined for all (t, which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1. For any i ∈ I φ , let us define The function solves (1.4) and belongs to D φ as defined by (1.10). Moreover (f − F dir ) satisfies the following two global conservation laws: for any t ≥ 0, When all coordinates are harmonic (d φ = 0), then φ(x) = 1 2 |x| 2 + d 2 log(2 π) due to the normalization (H7) and the harmonic pulsating modes, defined for all (t, which implies that these two global quantities evolve as an harmonic oscillator with period equal to 1/2. With solves (1.4) and belongs to P φ as defined by (1.11). Moreover (f − F pul ) satisfies the following two global conservation laws: for any t ≥ 0, Let us consider the set of the generators of all above special macroscopic modes We have the following orthogonality property. 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 with initial datum f 0 ∈ L 2 (M −1 ). With the above notations, for any f ∈ S and any t ≥ 0, we have

2.3.
A micro-macro decomposition. Let us consider a solution f ∈ C R + ; L 2 (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 (M) in terms of The operator C acts only on the velocity variable, is self-adjoint in L 2 (M) (when integrating in x, v) and L 2 (µ) (when integrating in v) and Let us consider the micro-macro decomposition where h is the L 2 (µ)-orthogonal projection of h on Ker C and E defined by (1.6) is a normalized Hermite polynomial of degree 2. In other words, are the macroscopic quantities corresponding to the spatial density, the local flux and thermal energy, while h ⊥ is the microscopic part of the decomposition. 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 According to Corollary 2.3, h has multiple global conservation laws. Global conservation laws associated to rotational symmetries of φ This also means P(m) ∈ R ⊥ φ as in (2.19). Global conservation laws corresponding to the harmonic directional modes In the fully harmonic case d φ = 0, global conservation laws corresponding to the harmonic pulsating modes (2.29) m · x = 0 and d 2 e − φ r = 0 .

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 : R d → R considered as a function of the velocity variable, we compute J p [h] = R d p h µ dv using standard properties of Hermite functions: and, using (2.23), we also get If f is a special macroscopic mode, then (2.30) is reduced to (2.1) because, in that case, h ⊥ = 0 and Θ[h ] = 0.

Classification of the special macroscopic modes
In this section, we prove Part (1) of Theorem 1.
is also a special macroscopic mode and (1.5) implies that h defined in (2.22) (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 E(v) are orthonormal Hermite polynomials, we have Collecting the results of Section 2.1 and using r 0 = r = 0, we get: 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 ξ 2 and ξ φ 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.

Control of A.
For any A ∈ M skew d×d (R), let us define The vector space M skew d×d (R) is of finite dimension: all norms are equivalent to | · |. This is also why the rigidity constant c K given by (1.13) is positive, which implies the Korn-type inequality By multiplying (2.12) by x k for k = 1, . . . , d, then integrating against ρ(x) and performing some integrations by part, using that ρ(x) dx is centred and that the terms involving ∇φ vanish, we obtain With the notation of (3.2), let us define X and Y by where, according to Lemma 3.3, the r.h.s. is independent of t. As a consequence, we have the following estimate.
Proof of Lemma 3.5. Multiplying (2.12) by ∇ x φ and integrating against ρ(x), after integration by parts, using that ρ is centred and observing that the terms involving 2 ξ φ − d and c vanish, it follows that . By inserting (3.8) into (2.12), one gets A multiplication of (3.9) by Φ and an integration against ρ yields We recall that E φ is defined by (1.9). We distinguish three cases. and hence, together with (3.9), it follows that Similarly, we use the notation ξ(x) = (ξ(x),ξ(x)) for a vector-field ξ : R d → R d . In particular one has Φ = ( Φ, 0) and alsob = 0 so that b = (b, 0) as a consequence of (2.28). Hence (3.9) becomes The matrix M φ defined in (3.10) is given by . Following the same procedure as in the fully non-harmonic case, we obtain after multiplication by Φ and integration in L 2 (ρ) that (3.17) M φb =α 1 c +α 2 c +R 3 , Hence, together with (3.15), it follows that As in the full rank case, Rank( Ψ 1 , Ψ 2 ) = 2 according to Lemma A. 3 Proof of Lemma 3.6. We write using r φ = − d 2 c and r = 0 by (2.26), which completes the proof.

3.5.
A Lyapunov function method. We define the Lyapunov function for some positive constants ε A , ε b and ε c to be chosen later.
Lemma 3.8. With the above notations, if h is defined as in Proposition 3.1, then for ε A , ε b and ε c small enough.
By Lemma 3.5, we obtain By Lemma 3.7, we know that |c| 2 |b | 2 + |c | 2 and obtain performing one integration by parts and using Cauchy-Schwarz inequality. It is then clear that |A| 2 m 2 and |c| 2 = e 2 , so that by (3.1), Then, using (3.22) again, we have the lower bound estimate We know from (3.24) that |b | 2 r 2 and, using (3.20), we also have that |c | 2 |b | 2 + |c| 2 r 2 + |A| 2 . As a consequence, we have that and ε c are chosen small enough, which completes the proof.
by choosing ε b and ε c small enough compared to ε 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 By Lemma 3.3, we know that A is constant in time. Using for instance (3.22), we deduce from |A| 2 lim t→+∞ h(t) 2 = 0 that A = 0. By Lemma 3.5, we get that b = 0 and c = 0 for any t ≥ 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.

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.
Here C and κ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to φ such as the rigidity constant defined in (1.13). We split h into a microscopic part h ⊥ and a macroscopic part h such that Let deviations from averages, or space inhomogeneous, terms be defined by  We recall that ξ 2 (x) := |x| 2 − |x| 2 and ξ φ (x) := φ− φ were already defined in (2.10) and (2.11) while ∇ skew x m refers to (2.6). In particular 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.

Control of infinite-dimensional quantities.
We build an entropy function by assembling dissipative functionals for h ⊥ and the space inhomogeneous terms defined in (4.2).

4.3.1.
Control of h ⊥ . We first control the dissipation of the microscopic part.  We conclude that (4.10) holds by the spectral gap assumption (H1) on C .

4.3.2.
Control of e s . Let us consider e s as defined in (4.2c).
Using the Cauchy-Schwarz inequality, we deduce . Using Korn's inequality (4.9) and observing by (2.30b) that Proof of Lemma 4.5. Observe that (2.30b), (4.2c) and (4.3) imply . Integrating (2.30b) and using (4.2d), one gets x φ and the integration of the skew-symmetric part yields As a consequence of these identities and (4.2b), one gets Hence x Using the zeroth order Poincaré inequality (4.6) and (4.2c), we can estimate Ω −1/2 ∇ * 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 (4. 16) From the definitions (4.2a) and (4.2e), we have  (4.5) in order to estimate the first term, and performing several integration by parts and using the boundedness assumption (H6) on φ in order to estimate the three last terms, we obtain (4. 18) Inserting (4.16) and (4.18) in completes the proof of (4.13).
In order to control the time-derivative of w s , we write Differentiating (4.17) with respect to t, we have where the first term is obtained by differentiating (2.30b) and amounts to , using (4.2c) and (4.2d). Similar expressions hold for the three next terms. Arguing similarly as for (4.18), we have . 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 (4.20) Let us define the dissipation functional Lemma 4.6. There are some positive constants κ 0 , C 0 and κ such that for any ε > 0 small enough, we have Proof of Lemma 4.6. By collecting the results of Lemmata 4.2, 4.3, 4.4 and 4.5, we obtain for any ε i ∈ (0, 1), i = 1, 2, 3, 4, up to a renaming of the generic constant C > 0. Using repeatedly Young's inequality, we have
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), where z obeys the bounds 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 so that (4.25b) follows from the definition (4.2b) of m s .
Using (4.2a) and (4.2e), we write Finally, thanks to (4.27) and (2.30b), we compute , and thus obtain 2 d e s ∆ x φ ξ 2 , from which (4.26a) follows. By differentiating z with respect to t, we get By (2.30d) and (2.31a), we know that . Besides, we learn from (4.2b) and (4.27) that Id d×d and, as a consequence, x φ ξ 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 We deduce a differential equation which is very similar to (2.12) up to additional terms involving m s and z, namely

4.4.2.
Control of A. The counterpart of Section 3.2 goes as follows.
Lemma 4.9. There are some positive constants κ 4 and C such that Proof of Lemma 4.9. We argue as for Lemma 3.4. We multiply (4.31) by x k for k = 1, . . . , d and after integration, we get Using the definitions of X and Y , (4.31) and (4.33) yield Using (4.23a) and (4.24), we obtain For the first term and thanks to the conservation law (2.27), we note that so that we can apply inequality (3.3) to x → A x + P(m s ) to get which yields for any C > 4 + c K . In order to estimate the other terms, we use Thanks to the zeroth order Poincaré inequality (4.6), we also have as well as similar estimates for the terms in ∂ t z and x ∂ t z · ∇ x φ. Collecting these estimates with (4.34), we get for some κ 4 < 1 4 c K and C > 0 large enough. Young's inequality and (4.26b) conclude the proof of (4.32). Control of b, b , b and c, c , c and c . It follows from three lemmata.  |b|

4.4.3.
Proof. Using (4.31), we can write Arguing as in the proof of Lemma 3.5 with this new definition of R 0 , we obtain that (3.11) holds with Observing that 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 Lemma 4.12. The following estimates hold Proof. According to (4.25a), we can write r as ) because of (4.26a). Using this expression in (2.26) and recalling that e = c yields from which (4.36) follows. Coming back to (4.38), we establish (4.37).

Second Lyapunov functional. Let us introduce the Lyapunov function
for some additional small parameters ε 5 and ε 6 , and the associated dissipation functional Lemma 4.13. For any 0 < ε 6 < ε 5 < ε 4 < ε 3 < ε 2 < ε 1 with ε 1 small enough, there holds Proof of Lemma 4.13. We can control all quantities involved in the definitions of F 2 and D 2 by h 2 . Indeed, from (4.1) and (4.23a), we have (4.41) r + m + e + h ⊥ + |b| + |c| h and thus also e s e + |c| h from (4.25c). Next, we observe from (4.27) that and thus also m s m + |A| + |b| + |c | h from (4.25b). Similarly, we observe from (4.28) that Coming back to the definition of w s and using (4.28), we get and deduce w s r + |c| + |b | h . Similarly, from (4.30), we also have |c | r + e + h ⊥ ≤ h . Summing up, we have proved (4.42) e s + m s + w s + |A| + |c | + |b | + |c | h .
We finally have to control the terms Ω −1/2 ∂ t w s . From (4.2e), (4.2a) and (2.30), we have x · m ξ 2 − 2 d c φ s and, after performing several integration by parts, As a consequence of the estimates (4.41), (4.42), (4.43) and of the definition (4.39) of F 2 (also see (4.20)), we have This completes the proof of the first equivalence in (4.40  Proof of Proposition 4.1. We differentiate with respect to t the Lyapunov function F 2 [h] and use Lemmata 4.6, 4.9 and 4.11 to get d dt Using Young's inequality we have As in the proof of Lemma 4.6, we choose appropriately the small parameters ε i such that the quantities ε 5 , ε 6 , ε 5 /ε 4 and ε 6 /ε 5 are small enough in terms of ε. With ε 5 := ε 61/32 and ε 6 := ε 62/32 , we obtain Choosing ε > 0 small enough, the differential inequality simplifies into Because of the equivalences established in Lemma 4.13, there are two con- Choosing ε > 0 smaller if necessary, we obtain . This completes the proof of Proposition 4.1, that is, of Part (2) of Theorem 1.1, by using once again the equivalences of Lemma 4.13.

Proof of hypocoercivity by the commutator method
In this section we give an alternative proof of our main result in Theorem 1.1 using a commutator method, under the additional hypotheses that (H9) The linear collision operator C is bounded in L 2 (µ −1 ) , Assumption (H9) means that the operator C as defined in (2.24) is bounded on L 2 (µ) while (H10) means that the potential φ has bounded second derivatives and is superlinear at infinity. These assumptions are added merely in order to simplify the computations but the bound (H9) on the collision operator can be relaxed into just (H2) by usingÃ i := Π * A i Π instead of the A i 's defined in (5.2) in Proposition 5.2 (where Πh = h is the orthogonal projection on the macroscopic part), and the bound in (H10) can be relaxed into simply |∇ 2 x φ| 1 + |∇ x φ| thanks to the additional weight ∇ x φ in the Poincaré inequality (H5). We include the commutator method because of its interesting algebraic properties and potential applications to a larger class of equations. Then part (2)  where C and κ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to φ such as the rigidity constant defined in (1.13).
While ∇ x and ∇ v map scalar functions to vector-valued functions, their L 2 (M)-adjoints are ∇ * x = − ∇ x · + ∇ x φ and ∇ * v = − ∇ v · + v · and map vector-valued functions back to scalar functions. The operators ∇ x and ∇ v commute but each does not commute with its adjoint. We have These scalar operators also act, coordinate by coordinate, on tensors.

5.1.
Cascade of infinite-dimensional correctors. The three following operators play the role of correctors: From [25,27] or by standard pseudo-differential calculus arguments (see Lemma A.4), A 0 , A 1 , A 2 are bounded operators in L 2 (M). Let R n [V ] be the space of real polynomials of v with degree less or equal than n ∈ N, then This means for instance that A 2 can access the local energy without seeing the local density and local momentum, in a descending cascade. Note that the simplest "order 1" corrector A 0 is inspired by the corrector ∇ * x Λ −1/2 : Λ −1/2 ∇ v introduced in [27] for Fokker-Planck type equations. We define macroscopic deviations from averages quantities that are slighlty different from (4.2a)-(4.2c): The core commutator estimates are: where the B i 's are bounded operators that satisfy and where the Λ i 's are nonnegative self-adjoint operators that satisfy, for Proof of Proposition 5.2. Since C is self-adjoint and T is skew-adjoint, d dt We can therefore write [A i , T ]+A i C +C A i =: B i −Λ i where, by using (5.1), we have the following explicit formulas since the three derivatives in velocity on the right hand side cancel all macroscopic quantities. The other lines are dealt with similarly, using the boundedness of H φ , and we deduce | Now we deal with B 1 . We consider the second line B 1,2 . Then Λ Γ 1/2 [Λ −1 Γ −1/2 , T ] is bounded by standard pseudo-differential calculus and so b 1,2 : The macroscopic quantities r and m are canceled since two derivatives in velocity are involved. We then use ∇ v ⊗ ∇ v ⊗ ∇ x e E = ∇ x e ⊗ Id d×d and Appendix A.4 shows (using Ω ≥ 1 for the first inequality) The other lines are similar and yield the same estimate.
We then deal with B 0 , and focus on the second line B 0,2 again. The T is bounded arguing as before, and (5.4) Indeed a direct computation gives withΩ := (∇ * x ∇ x + 2). The factor 2 comes from the fact that for all α ∈ R, since v is an eigenfunction of ∇ * v ∇ v with eigenvalue 1. To complete the proof of (5.4), it is sufficient to notice that The other lines in B 0 are treated similarly again. Now we deal with the main nonnegative terms Λ i h, h in (5.3). We first compute the contribution of Λ 0 , Λ 1 and Λ 2 acting respectively on r, m and e. Due to the number of derivatives in velocity in the right hand side of the expressions giving the Λ i 's, we have Λ Next we use the cascade of Poincaré inequalities of Lemma A.6. For the density r, this implies that there isa constant λ 0 > 0 such that (5.8) Regarding the momentum m, one first observes that thanks to the Schwarz lemma written as Applied to (5.5)-(5.7), the cascade of Poincaré inequalities at order 2 stated in Lemma A.6 implies that there exists a constant λ 1 such that Regarding the energy e, we use the standard Poincaré inequality in L 2 (ρ) (the order 1 inequality of Lemma A.6) to get λ 2 > 0 such that (5.10) Λ 2 (e E(v)), (e E(v)) = 4 d + 2 Ω − 1 2 ∇ x e 2 ≥ 2 λ 2 e − e 2 = 2 λ 2 ẽ 2 .
With the above estimates in hand, we can investigate all terms appearing in Λ i h, h . According to the number of velocity and space gradients in Λ 2 , we get that from which one obtains with (5.10) that Similarly for m, using in addition that Λ 1 is self-adjoint and Λ 1 (e E(v)) = Λ 1 (ẽ E(v)), one has Finally, regarding the local density r, we get similarly and it follows from (5.8) that By direct computation, we have x where the last line follows from an integration by parts. The first term in the right hand side vanishes because ∇ * x is symmetric and B is skew-symmetric, the third term also vanishes. This proves that d dt m(t) − m 0 , B x = 0 and completes the proof.
A.2. Special macroscopic modes: the invertibility and rank. We state and prove two results used in Section 3.3 and implicitely in Section 4.5.
The first result deals with the invertibility of the matrices M φ and M φ defined respectively in (3.10) and (3.16).
Proof of Lemma A.2. Assume that d φ = d in (1.9). Let u ∈ R d be such that M φ u = 0. Then M φ u · u = |Φ · u| 2 = 0, which implies that Φ(x) · u = 0 for any x ∈ R d , hence u = 0. This means that Ker M φ = {0}. The proof in the case d φ ≤ d − 1 follows exactly the same scheme.
The second result deals with the linear independence of the two functions Ψ 1 and Ψ 2 defined in (3.14), and similarly for Ψ 1 and Ψ 2 defined in (3.19).
A.3. Some computations for the commutator method. Here we prove technical claims used in Section 5. The first result is concerned with boundedness of the operators defined in Section 5.1 under Assumptions (H9) and (H10). Proof of Lemma A.4. As a typical example, we focus on Λ 1 for which it is sufficient to show that Λ ForΓ −1/2 (∂ v k + v k /2) this is due to the fact thatΓ −1/2 is of order −1 and ∂ v k + v k /2 of order 1 in the pseudo-differential calculus associated to the metric (dv 2 + dη 2 )/(1 + |v| 2 + |η| 2 ), η being the dual variable of v.
The composition is then of order 0 and the Calderón-Vaillancourt Theorem (see [8]) implies the boundedness.
is of order 2 in the pseudo-differential calculus associated to the metric (dx 2 + dv 2 + dξ 2 + dη 2 )/(1 + |η| 2 + |v| 2 + |∇φ| 2 + |ξ| 2 ), ξ being the dual variable of v. This implies the desired boundedness. Such calculus with two levels (involving Λ in all variable and Γ only in velocity variables) is also at the core of the boundedness of terms like Λ Γ 1/2 Λ −1 Γ −1/2 , T where we use that Λ and Γ commute and that the commutation decrease the order by 1, so that this operator is of order 0 and therefore bounded by the Calderón-Vaillancourt Theorem. Note in addition that H φ (which appears, e.g., in the B i 's) is of order 0 which greatly simplifies the proofs. For all other terms Λ i , A i and B i , similar computations give the result.
The second result deals with the symmetry and nonnegativity of Λ 1 .
Lemma A.5. The operator Λ 1 is symmetric and nonnegative.
Proof of Lemma A.5. First we check that x is symmetric, since for the other part of Λ 1 this is obvious: Next check that Λ 1 is indeed a nonnegative operator: This completes the proof.
A.4. A cascade of Poincaré-Lions inequalities. Under Assumptions (H9) and (H10), we prove several inequalities used in Section 5.1. Let ϕ a smooth function in L 2 (ρ) with compact support and P 0 (ϕ) := ϕ , with parameter γ ∈ [−2, 1]. This operator is non-local, of order 2 in velocity (of diffusive type) and therefore not bounded. It satisfies the spectral gap condition (H1) when γ ∈ [−2, 1] (see, e.g., [2,36] for constructive estimates) and again all polynomials in velocity multiplied by µ are in its domain and it satisfies the boundedness property (H2). Note that the most important physical case, the linearisation of the so-called Landau-Coulomb collision operator (describing statistically collisions for a gas of electrons with Coulomb interactions) corresponds to γ = −3 in dimension d = 3 and is not covered by our assumption. It is however likely that our method could be extended for weakened versions of the condition (H1), leading also to a weaker conclusion, with a stretched exponential relaxation rate, rather than an exponential one, in the spirit of [42]. one gets an additional term ρ(x) in front of the collision operator that goes to zero at infinity. We have not considered this degeneracy in the present paper: it is likely to create significant difficulties since there is then no uniform-in-x spectral gap for ρ C .

B.2.
Examples of potentials. Let us discuss and illustrate the hypotheses (H5) and (H6) on the potential φ. The bounded moment hypothesis (H6) is not restrictive. Functions like φ(x) = d+5 2 ln(1 + |x| 2 ) − Z φ which are very slowly increasing at infinity satisfy this hypothesis, as well as fast-increasing ones like φ(x) = e |x| 4 − Z φ (here Z φ is the constant of normalization of e −φ in L 1 ). Regarding the Poincaré inequality (H5), many works have been devoted to the study of sufficient conditions in order to guarantee the existence of a spectral gap. Here are some examples.
Example B.5. The harmonic potential φ(x) = 1 2 |x| 2 + d 2 log(2 π) satisfies the Poincaré inequality with constant c P = 1. The inequality is equivalent to the spectral gap inequality for the operator Ω defined in (1.14). In the flat L 2 space, the change of unknown u = v e −φ/2 shows that the Poincaré inequality is also equivalent to the spectral gap inequality for the quantum harmonic oscillator operator −∆ x + 1 4 |x| 2 − d 2 .
Example B.6. For a general φ, the change of unknown u = v e −φ/2 yields the following Schrödinger-type operator According to the so-called Bakry-Emery theory (see for instance to [1]), there is a spectral gap as soon as the Hessian ∇ 2 x φ is uniformly strictly positive at infinity. When it is uniformly strictly positive everywhere the following estimate is available on the spectral gap c P : where λ 1 (∇ 2 x φ) > 0 is the lowest eigenvalue of ∇ 2 x φ. Example B.7. All potentials φ such that P φ has compact resolvent satisfy the Poincaré inequality (H5). This happens in particular when which is implied for instance by the stronger assumption This is a standard result on Schrödinger operators, see for instance [37, Theorem XIII.67 p. 249], and 0 is then a simple discrete eigenvalue. The argument in the latter reference is not constructive, and for a simpler constructive argument we refer for instance to [45,Theorem A.1] or the IMS truncation method in [40].
Example B.8. Here is an exotic example of potential that does not satisfy (B.2) nor the Bakry-Émery criterion (uniform convexity of φ) and for which the Poincaré inequality holds. Consider on R 2 φ(x, y) = x 2 1 + y 2 2 − Z φ where Z φ is the normalization constant so that ρ = e −φ is a probability density. One can check that P φ has a spectral gap, although φ is constant on the unbounded set {x = 0}.
B.3. Change of coordinates. Let us discuss the reduction to the normalization (H7). Note that the formulas for Ker C are invariant by orthonormal change of coordinates in the velocity variable. By orthonormal change of coordinates in both the velocity and space variables, we can then reduce to the case when φ satisfies 0 0 · · · 0 0 p 2 2 0 · · · 0 . . . . . .
where we suppose without loss of generality that all p j 's are positive. The analysis of the present paper can be adapted to this case, including the main Theorem 1.1, with the following changes. We define the set of adapted centred rotational modes compatible with φ as in (1.8): We then choose orthonormal coordinates x = (x 1 , x 2 , . . . , x d ) such that ∂ x j φ = p 2 j x j for some p j > 0 if and only if j ∈ I φ := {d φ + 1, . . . , d}, and x j = 0 for any j ∈ I φ if x ∈ E φ (the linear subspace defined in (1.9)). We define the set of harmonic directional modes by where f − j (t, x, v) := (p j x j cos(p j t) − v j sin(p j t)) M(x, v) , f + j (t, x, v) := (p j x j sin(p j t) + v j cos(p j t)) M(x, v) . If d φ = 0 and for some p > 0, p j = p for all j ∈ {1 , . . . , d}, we define the set of harmonic pulsating modes by where f − (t, x, v) := 1 2 |p x| 2 − |v| 2 cos(2 p t) − p x · v sin(2 p t) M(x, v) , f + (t, x, v) := 1 2 |p x| 2 − |v| 2 sin(2 p t) + p x · v cos(2 p t) M(x, v) . The functions in R φ , D φ and P φ are special macroscopic modes of (1.1). With these definitions, the proof of Theorem 1.1 can be adapted to prove a hypocoercivity result taking into account all special macroscopic modes.
B.4. Spectral interpretation. We have focused so far on real solutions to (1.1), which is natural since physical solutions (densities of probability) are real valued. By considering complex solutions, we can interpret the results in terms of the complex spectrum of the nonnegative operator , the complexification of L 2 (M −1 ). We consider φ as in (B.3). We can then describe precisely the spectrum of −L and obtain resolvent estimates in a half-plane that includes the imaginary axis. Notice first that 0 is in the spectrum of −L with associated eigenspace where R φ,C is the set of rotation modes as defined in (B.4) but extended to the corresponding C-vector space. This set is then of (complex) dimension 2 + dim(R φ ). Depending on the harmonicity of φ we have three cases which are summarized in Figure 1.   where I φ := {d φ + 1, . . . , d}, to which we can associate the eigenfunctions of (−L ) corresponding to the eigenvalues ∓ i p j and given by (c) Case with harmonic directional and pulsating modes (d φ = 0). In this last case necessarily all p j 's are equal to a common value p > 0 and φ(x) = 1 2 |p x| 2 + d 2 log(2 π) − d log(p). All possible harmonic directional modes exist, as well as all possible infinitesimal rotational modes R φ,C with R φ = M skew d×d (C). The complexification of the set P φ defined in (B.5) is P φ,C = Span e 2 i p t f + (x, v) , e − 2 i p t f − (x, v) where f ± (x, v) := p x · v ± i 2 |p x| 2 − |v| 2 M(x, v) are eigenfunctions of (−L ) with eigenvalues ± 2 i p.
The analysis of the paper can be extended to the complex Hilbertian space L 2 C (M −1 ) with a set of special macroscopic modes defined by S := Span where the f j 's and the f ± 's are defined above (when φ has the relevant harmonicity). Let S ⊥ be the orthogonal of S in L 2 C (M −1 ). We note that since L is a real operator, both S and S ⊥ are stable by conjugation and therefore stable by L and L * . Using the Laplace transform, we obtain from Theorem 1.1 the following resolvent estimate for −L | S ⊥ : whereC is an explicit constant depending on κ and C in Theorem 1.1 and · B(S ⊥ ) stands for the operator norm on S ⊥ . The provides the resolvent estimates in the left half-planes in Figure 1.
B.5. Special macroscopic modes for the full nonlinear Boltzmann equation. The special macroscopic modes which minimize the entropy for the full nonlinear Boltzmann equation are the nonlinear counterparts to the linearized special macroscopic modes studied in the present paper. They appear for the first time in the literature in Boltzmann's paper [5] as mentioned in the introduction. The full nonlinear inhomogeneous Boltzmann equation is where, with the classical notations F = F (v ), F * = F (v * ) and F * = F (v * ) associated to elastic collisions (v, v * ) → (v , v * ), such that the microscopic conservation of momentum v + v * = v + v * and energy |v | 2 + |v * | 2 = |v| 2 + |v * | 2 hold, the Boltzmann collision operator writes Here B ≥ 0 is the cross-section. We refer to [10] for more details. Let us assume the normalization (H7) on φ. We consider the functions in the space S of special macroscopic modes generated by the set R φ of rotation modes compatible with φ if φ admits any, the set D φ of harmonic directional modes if φ has harmonic directions, the set P φ of harmonic pulsating modes if φ is fully harmonic.