11institutetext: Anton Arnold 22institutetext: Technische Universität Wien,
Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8, A-1040
Wien, Österreich, 22email<EMAIL_ADDRESS>
https://www.asc.tuwien.ac.at/~arnold/ 33institutetext: Jean Dolbeault
44institutetext: CEREMADE (CNRS UMR n∘ 7534), PSL university, Université
Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France, 44email:
<EMAIL_ADDRESS>
https://www.ceremade.dauphine.fr/~dolbeaul/ 55institutetext: Christian
Schmeiser 66institutetext: Fakultät für Mathematik, Universität Wien, Oskar-
Morgenstern-Platz 1, 1090 Wien, Austria, 66email:
<EMAIL_ADDRESS>
https://homepage.univie.ac.at/christian.schmeiser/ 77institutetext: Tobias
Wöhrer 88institutetext: Technische Universität Wien, Institut für Analysis und
Scientific Computing, Wiedner Hauptstr. 8, A-1040 Wien, Österreich, 88email:
<EMAIL_ADDRESS>
https://www.asc.tuwien.ac.at/~arnold/

# Sharpening of decay rates in Fourier based hypocoercivity methods

Anton Arnold  Jean Dolbeault  Christian Schmeiser  and Tobias Wöhrer

###### Abstract

This paper is dealing with two $\mathrm{L}^{2}$ hypocoercivity methods based
on Fourier decomposition and mode-by-mode estimates, with applications to
rates of convergence or decay in kinetic equations on the torus and on the
whole Euclidean space. The main idea is to perturb the standard
$\mathrm{L}^{2}$ norm by a twist obtained either by a nonlocal perturbation
build upon diffusive macroscopic dynamics, or by a change of the scalar
product based on Lyapunov matrix inequalities. We explore various estimates
for equations involving a Fokker–Planck and a linear relaxation operator. We
review existing results in simple cases and focus on the accuracy of the
estimates of the rates. The two methods are compared in the case of the
Goldstein–Taylor model in one-dimension.
_MSC (2020):_ Primary: 82C40. Secondary: 76P05, 35H10, 35K65, 35P15, 35Q84.

###### Keywords:

Hypocoercivity, linear kinetic equations, entropy - entropy production
inequalities, Goldstein–Taylor model, Fokker–Planck operator, linear
relaxation operator, linear BGK operator, transport operator, Fourier modes
decomposition, pseudo-differential operators, Nash’s inequality

__

###### Contents

  1. I.1 An abstract hypocoercivity result based on a twisted $\mathrm{L}^{2}$ norm
  2. I.2 An abstract hypocoercivity result based on Lyapunov matrix inequalities
  3. II.1 A detailed mode-by-mode approach
    1. II.1.1 Introduction
    2. II.1.2 A first optimization in the general setting
    3. II.1.3 Mode-by-mode hypocoercivity
    4. II.1.4 Further observations
  4. II.2 Convergence rates and decay rates
    1. II.2.1 Exponential convergence rate on a small torus
    2. II.2.2 Algebraic decay rate in the whole Euclidean space
  5. III.1 General setting and Fourier decomposition
  6. III.2 Comparison of the two hypocoercivity methods for $\mathcal{X}=\mathbb{T}$
    1. III.2.1 The optimal global Lyapunov functional
    2. III.2.2 The defective cases
    3. III.2.3 Decay results for the case $\mathcal{X}=\mathbb{T}$
  7. III.3 Decay results for the case $\mathcal{X}={\mathbb{R}}$

### Introduction

We consider dynamical systems involving a degenerate dissipative operator and
a conservative operator, such that the combination of both operators implies
the convergence to a uniquely determined equilibrium state. In the typical
case encountered in kinetic theory, the dissipative part is not coercive and
has a kernel which is unstable under the action of the conservative part. Such
dynamical systems are called _hypocoercive_ according to Mem-villani . We are
interested in the decay rate of a natural dissipated functional, the _entropy_
, in spite of the indefiniteness of the entropy dissipation term. In a linear
setting, the functional typically is quadratic and can be interpreted as the
square of a Hilbert space norm. Classical examples are evolutions of
probability densities for Markov processes with positive equilibria. Over the
last 15 years, various hypocoercivity methods have been developed, which rely
either on Fisher type functionals (the $\mathrm{H}^{1}$ approach) or on
entropies which are built upon weighted $\mathrm{L}^{2}$ norms, or even weaker
norms as in armstrong2019variational . In the $\mathrm{L}^{2}$ approach, it is
very natural to introduce spectral decompositions and handle the free
transport operator, for instance in Fourier variables, as a simple
multiplicative operator. In the appropriate functional setting, the problem is
then reduced to the study of a system of ODEs, which might be finite or
infinite. This is the point of view that we adopt here, with the purpose of
comparing several methods and benchmarking them on some simple examples.

Decay rates are usually obtained by adding a twist to the entropy or squared
Hilbert space norm. In hyperbolic systems with dissipation, early attempts can
be traced back to the work of Kawashima and Shizuta shizuta1985 ; MR1057534 ,
where the twist is defined in terms of a _compensating function_. The
similarities between hypocoercivity and hypoellipticity are not only motivated
the creation of the latter terminology, as explained in Mem-villani , but also
serve as a guideline for proofs of hypocoercivity Herau ; Mouhot-Neumann ;
Mem-villani  and in particular for the construction of the twist. Here we
shall focus on two approaches to $\mathrm{L}^{2}$-hypocoercivity.
$\rhd$ In Dolbeault2009511 ; DMS-2part , an abstract method motivated by Herau
and by the compensating function approach has been formulated, which provides
constructive hypocoercivity estimates. The twist is built upon a non-local
term associated with the spectral gap of the diffusion operator obtained in
the _diffusion limit_ and controls the relaxation of the macroscopic part in
the limiting diffusion equation, that is, the projection of the distribution
function on the orthogonal of the kernel of the dissipative part of the
evolution operator. The motivating applications are kinetic transport models
with diffusive macroscopic dynamics, see, _e.g._ , Dolbeault22102012 ; CRS ;
goudon:hal-01421710 ; NeuSch ; addala2019l2hypocoercivity ; FavSch , where the
results yield decay estimates in an $\mathrm{L}^{2}$ setting.
$\rhd$ The goal of the second approach is to find sharp decay estimates in
special situations, where sufficient explicit information about the dynamics
is available. Examples are ODE systems AAC20  as well as problems where a
spectral decomposition into ODE problems exists Arnold2014 ; ArnEinSigWoe ;
ASS20 ; AESW . In these situations, sharp decay estimates can be derived by
employing _Lyapunov matrix inequalities_.

In the standard definitions of hypocoercivity, a spectral gap and an
exponential decay to equilibrium are required. This, however, can be expected
only in sufficiently confined situations, _i.e._ , in bounded domains or for
sufficiently strong confining forces. Problems without or with too weak
confinement have been treated either by regaining spectral gaps pointwise in
frequency after Fourier transformation as in BDMMS ; MR2927622  or by
employing specially adapted functional inequalities in BDMMS ; BDS-very-weak ;
BDLS , with the Nash inequality Nash58  as the most prominent example.

The aim of this work is to present a review and a comparison of the two
approaches mentioned above, executed for both confined and unconfined
situations, where for the former a periodic setting is chosen, such that the
Fourier decomposition method can be used in all cases. A special emphasis is
put on optimizing the procedures with the ultimate goal of proving sharp decay
rates. Attention is restricted to abstract linear hyperbolic systems with
linear relaxation, where ’abstract’ means that infinite systems such as
kinetic transport equations are allowed. Note that in the finite dimensional
case, the setting is as in MR2927622 .

In Part I of this work, both methods are presented in an abstract framework.
Concerning the method of DMS-2part ; BDMMS , the setting is abstract linear
ODEs, where the dynamics is driven by the sum of a dissipative and a
conservative operator such that the dissipation rate is indefinite, but the
conservative operator provides enough mixing to create hypocoercivity. Then
the approach based on Lyapunov matrix inequalities is discussed at the hand of
hyperbolic systems with relaxation. By Fourier decomposition the problem is
reduced to ODE systems and Lyapunov functionals with optimal decay rates are
built. The results of this section can be seen as a sharpening of the abstract
decay estimates in MR2927622 .

Part II is concerned with sharpening the approach of DMS-2part ; BDMMS
applied to linear kinetic equations with centered Maxwellian equilibria (for
sake of simplicity). It contains results on the optimal choice of parameters
in the abstract setting, on the mode-by-mode application of the method after
Fourier transformation, on the convergence of an optimized rate estimate to
the sharp rate in the macroscopic diffusion limit and, finally, on the
derivation of global convergence or decay rates for the cases of small tori
and of the Euclidean space without confinement.

Part III is devoted to a comparison of both approaches for a particular
example, the Goldstein–Taylor model with constant exchange rate, a hyperbolic
system of two equations with an exchange term in one space dimension, which
can be interpreted as a discrete velocity model with two velocities. It has
already been used as a model problem in DMS-2part , and the sharp decay rate
on the one-dimensional torus has been derived by the Lyapunov matrix
inequality approach in ArnEinSigWoe . The challenging problem of finding the
sharp decay rate for a position dependent exchange rate has been treated in BS
; BScorr . It is shown that the mode-by-mode Lyapunov functionals derived by
both methods, the Lyapunov matrix inequality approach and the modal
optimization of the abstract framework outlined in Part II, coincide for the
Goldstein–Taylor model. On the torus the mode-by-mode Lyapunov functionals can
be combined into a global Lyapunov functional which provides the sharp decay
rate (see Theorem III.2.3). On the real line, the modal results combine into a
global estimate with sharp algebraic decay rate. Due to the presence of a
defective eigenvalue in the modal equations, the standard approach requires
modifications to obtain reasonable multiplicative constants.

Part I. Review of two hypocoercivity methods

We consider the abstract evolution equation

$\frac{dF}{dt}+\mathsf{T}F=\mathsf{L}F\,,\quad t>0\,,$ (1)

with initial datum $F(t=0,\cdot)=F_{0}$. Applied to kinetic equations,
$\mathsf{T}$ and $\mathsf{L}$ are respectively the _transport_ and the
_collision_ operators, but the abstract result of this section is not
restricted to such operators. We shall assume that $\mathsf{T}$ and
$\mathsf{L}$ are respectively anti-Hermitian and Hermitian operators defined
on a complex Hilbert space
$\big{(}\mathcal{H},\langle\cdot,\cdot\rangle\big{)}$ with corresponding norm
denoted by $\|\cdot\|$.

##  I.1 An abstract hypocoercivity result based on a twisted $\mathrm{L}^{2}$
norm

Let us start by recalling the basic method of DMS-2part . This technique is
inspired by diffusion limits and we invite the reader to consider DMS-2part
for detailed motivations. We define

$\mathsf{A}:=\Big{(}\mathrm{Id}+(\mathsf{T}\Pi)^{*}\mathsf{T}\Pi\Big{)}^{-1}(\mathsf{T}\Pi)^{*}$
(2)

where ∗ denotes the adjoint with respect to $\langle\cdot,\cdot\rangle$ and
$\Pi$ is the orthogonal projection onto the null space of $\mathsf{L}$. We
assume that positive constants $\lambda_{m}$, $\lambda_{M}$, and $C_{M}$
exist, such that, for any $F\in\mathcal{H}$, the following properties hold:
$\rhd$ _microscopic coercivity_

$-\,\langle\mathsf{L}F,F\rangle\geq\lambda_{m}\,\|(\mathrm{Id}-\Pi)F\|^{2}\,,$
(H1)

$\rhd$ _macroscopic coercivity_

$\|\mathsf{T}\Pi F\|^{2}\geq\lambda_{M}\,\|\Pi F\|^{2}\,,$ (H2)

$\rhd$ _parabolic macroscopic dynamics_

$\Pi\mathsf{T}\Pi\,F=0\,,$ (H3)

$\rhd$ _bounded auxiliary operators_

$\|\mathsf{AT}(\mathrm{Id}-\Pi)F\|+\|\mathsf{AL}F\|\leq
C_{M}\,\|(\mathrm{Id}-\Pi)F\|\,.$ (H4)

A simple computation shows that a solution $F$ of (1) is such that

$\frac{1}{2}\,\frac{d}{dt}\|F\|^{2}=\langle\mathsf{L}F,F\rangle\leq-\,\lambda_{m}\,\|(\mathrm{Id}-\Pi)F\|^{2}\,.$

We assume that (1) has, up to normalization, a unique steady state
$F_{\infty}$. By linearity, we can replace $F_{0}$ by $F_{0}-\langle
F_{0},F_{\infty}\rangle\,F_{\infty}$ or simply $F_{0}-F_{\infty}$, with
$F_{\infty}$ appropriately normalized. With no loss of generality, we can
therefore assume that $F_{\infty}=0$. This is however not enough to conclude
that $\|F(t,\cdot)\|^{2}$ decays exponentially with respect to $t\geq 0$. As
in the _hypocoercivity_ method introduced in DMS-2part  for real valued
operators and extended in BDMMS  to complex Hilbert spaces, we consider the
Lyapunov functional

$\mathsf{H}_{1}[F]:=\frac{1}{2}\,\|F\|^{2}+\delta\,\mathrm{Re}\langle\mathsf{A}F,F\rangle$
(3)

for some $\delta>0$ to be determined later. If $F$ solves (1), then

$\begin{array}[]{rl}-\,\frac{d}{dt}\mathsf{H}_{1}[F]=\mathsf{D}[F]:=&-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi
F,F\rangle\\\\[4.0pt]
&-\,\delta\,\mathrm{Re}\langle\mathsf{TA}F,F\rangle+\delta\,\mathrm{Re}\langle\mathsf{AT}(\mathrm{Id}-\Pi)F,F\rangle-\delta\,\mathrm{Re}\langle\mathsf{AL}F,F\rangle\,.\end{array}$
(4)

The following result has been established in DMS-2part ; BDMMS .

###### Theorem I.1.1

Let $\mathsf{L}$ and $\mathsf{T}$ be closed linear operators in the complex
Hilbert space $\big{(}\mathcal{H},\langle\cdot,\cdot\rangle\big{)}$. We assume
that $\mathsf{L}$ is Hermitian and $\mathsf{T}$ is anti-Hermitian, and that
(H1)–(H4) hold for some positive constants $\lambda_{m}$, $\lambda_{M}$, and
$C_{M}$. Then for some $\delta>0$, there exists $\lambda>0$ and $C>1$ such
that, if $F$ solves (1) with initial datum $F_{0}\in\mathcal{H}$, then

$\mathsf{H}_{1}[F(t,\cdot)]\leq\mathsf{H}_{1}[F_{0}]\,e^{-\lambda\,t}\quad\mbox{and}\quad\|F(t,\cdot)\|^{2}\leq
C\,e^{-\lambda\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq 0\,.$ (5)

Here we assume that the unique steady state is $F_{\infty}=0$ otherwise we
have to replace $F(t,\cdot)$ by $F(t,\cdot)-F_{\infty}$ and $F_{0}$ by
$F_{0}-F_{\infty}$ in (5). The strategy of DMS-2part , later extended in BDMMS
, is to prove that for any $\delta>0$ small enough, we have

$\lambda\,\mathsf{H}_{1}[F]\leq\mathsf{D}[F]$ (6)

for some $\lambda>0$, and

$c_{-}\,\|F\|^{2}\leq\mathsf{H}_{1}[F]\leq c_{+}\,\|F\|^{2}$ (7)

for some constants $c_{-}$ and $c_{+}$ such that $0\leq c_{-}\leq 1/2\leq
c_{+}$. As a consequence, if $c_{-}>0$, we obtain the estimate $C\leq
c_{+}/c_{-}$. We learn from (BDMMS, , Proposition 4) that Theorem I.1.1 holds
with $c_{\pm}=(1\pm\delta)/2$,

$\lambda=\frac{\lambda_{M}}{3\,(1+\lambda_{M})}\min\left\\{1,\lambda_{m},\tfrac{\lambda_{m}\,\lambda_{M}}{(1+\lambda_{M})\,C_{M}^{2}}\right\\}\;\mbox{and}\;\delta=\frac{1}{2}\,\min\left\\{1,\lambda_{m},\tfrac{\lambda_{m}\,\lambda_{M}}{(1+\lambda_{M})\,C_{M}^{2}}\right\\}\,.$
(8)

Our primary goal of Part II is to obtain sharper estimates of $\lambda$,
$c_{\pm}$ and $C$ for an appropriate choice of $\delta$ in specific cases.
Notice that it is convenient to work in an Hilbert space framework because
this allows us to use Fourier transforms.

##  I.2 An abstract hypocoercivity result based on Lyapunov matrix
inequalities

Here we review our second hypocoercivity method, as developed on various
examples in Arnold2014 ; AAC ; Achleitner2018 , before comparing it with the
method of § I.1.

As in § I.1, without loss of generality we assume that (1) has the unique
steady state $F_{\infty}=0$. We are interested in explicit decay rates for
$\|F(t,\cdot)\|^{2}\to 0$ as $t\to+\infty$. To fix the ideas we start with
some prototypical examples:

  1. 1.

Although almost trivial, the _stable ODEs_ with constant-in-$t$ coefficients

$\frac{dF}{dt}=-\,C\,F$ (9)

is at the core of the method. Here $F(t)\in{\mathbb{C}}^{n}$,
$\mathsf{T}:=C_{AH}\in{\mathbb{C}}^{n\times n}$ is an anti-Hermitian matrix,
and $\mathsf{L}:=-\,C_{H}\in{\mathbb{C}}^{n\times n}$ is a Hermitian negative
semi-definite matrix, where $C_{AH}$ and $C_{H}$ denote the anti-Hermitian and
Hermitian parts of $C=C_{AH}+C_{H}$. Several other examples will be reduced to
(9), mostly via Fourier transformation in $x$. We shall use the same index
notation (‘$AH$’ and ‘$H$’) for matrix $B$ in Example 4 and matrix $C$ in
Example 5. The hypocoercivity structure of (9) is discussed in AAC20 .

  2. 2.

_Discrete velocity BGK models_ , _i.e._ transport-relaxation equations (see §
2.1 and § 4.1 in AAC ) can be written in the form of (1) where
$F(t,x)=\big{(}f_{1}(t,x),...,f_{n}(t,x)\big{)}^{\top}$,
$x\in\mathcal{X}\subset{\mathbb{R}}$, $\mathsf{T}:=V\,\partial_{x}$ with the
diagonal matrix $V\in{\mathbb{R}}^{n\times n}$ representing the velocities,
and the collision operator $\mathsf{L}:=\sigma\,B$ with $\sigma>0$. Here, the
matrix $B\in{\mathbb{R}}^{n\times n}$ is in _BGK form_

$B=\begin{pmatrix}b_{1}\\\ \vdots\\\
b_{n}\end{pmatrix}\otimes(1,\,\ldots,\,1)-\mathrm{Id}$

with $b=(b_{1},\ldots,b_{n})^{\top}\in(0,1)^{n}$ such that
$\sum_{j=1}^{n}b_{j}=1$, and $\mathrm{Id}$ denotes the identity matrix. The
collision operator $\mathsf{L}$ is symmetric on the velocity-weighted
$\mathrm{L}^{2}$-space
$\mathcal{H}=\mathrm{L}^{2}(\mathcal{X}\times\\{1,\ldots,n\\};\\{b_{j}^{-1}\\})$.
Due to this structure, $B$ has a simple eigenvalue 0 with corresponding left
eigenvector $l_{1}=(1,\ldots,1)$ associated with the mass conservation of the
system. The corresponding right eigenvector $b$ spans the local-in-$x$ steady
states, which are of the form $\rho(x)\,b$ for some arbitrary scalar function
$\rho(x)$. The case with only two velocities, or _Goldstein–Taylor model_ , is
dealt with in Part III.

  3. 3.

A linear _kinetic BGK model_ is analyzed in AAC , where
$F=f(t,x,v)\in{\mathbb{R}}$, $x\in\mathbb{T}$ (the $1$-dimensional torus of
length $2\pi$), and $v\in{\mathbb{R}}$. The kinetic transport operator is
$\mathsf{T}:=v\,\partial_{x}$, and the BGK operator
$\mathsf{L}f:=\mathcal{M}_{\upvartheta}(v)\,\int_{\mathbb{R}}f\,dv-f$ is
symmetric in the weighted space
$\mathcal{H}=\mathrm{L}^{2}\big{(}\mathbb{T}\times{\mathbb{R}};dx\,dv/(2\pi\,\mathcal{M}_{\upvartheta}(v))\big{)}$,
where $\mathcal{M}_{\upvartheta}(v)$ denotes the centered Maxwellian with
variance (or temperature) $\upvartheta$. The kernel of $\mathsf{L}$ is spanned
by $\mathcal{M}_{\upvartheta}(v)$, which is also the global steady state
$F_{\infty}(v)$, due to the setting on the torus.

  4. 4.

The (degenerate) _reaction-diffusion systems_ of FePrTa17  can also be written
as in (1), with $F(t,x)=(f_{1}(t,x),...,f_{n}(t,x))^{\top}$,
$\mathsf{T}:=-B_{AH}$, $\mathsf{L}:=D\,\Delta+B_{H}$. Here, $0\leq
D\in{\mathbb{R}}^{n\times n}$ is a diagonal matrix, $B\in{\mathbb{R}}^{n\times
n}$ is an essentially non-negative matrix, _i.e._ , $b_{ij}\geq 0$ for any
$i\neq j$, and $b_{ii}=-\sum_{i\neq j}b_{ij}$, and $B_{AH}$ and $B_{H}$ are
its anti-symmetric and symmetric parts.

  5. 5.

As a final example, let us mention (possibly degenerate) _Fokker–Planck
equations_ with linear-in-$x$ drift for $F=f(t,x)$, $x\in{\mathbb{R}}^{d}$.
After normalization (in the sense of ASS20 ), they can be identified with (1),
where
$\mathsf{T}f:=-\,\mbox{div}\big{(}f_{\infty}\,C_{AH}\,\nabla(f/f_{\infty})\big{)}$,
$\mathsf{L}f:=\mbox{div}\big{(}f_{\infty}\,C_{H}\,\nabla(f/f_{\infty})\big{)}$,
with a positive stable drift matrix $C\in{\mathbb{R}}^{d\times d}$ such that
$C_{H}\geq 0$, and $f_{\infty}=(2\pi)^{-d/2}\exp\big{(}-|x|^{2}/2\big{)}$ is
the unique normalized steady state. As shown in ASS20 , these Fokker–Planck
equations are equivalent to (9) and tensorized versions of it.

In the articles cited above for Examples 1–3, an $\mathrm{L}^{2}$-based
hypocoercive entropy method has been used to derive sharp decay estimates for
the solution $F(t)$ towards its steady state $F_{\infty}$, and the same
strategy can also be applied to Example 4. In Arnold2014  an
$\mathrm{H}^{1}$-based hypocoercive entropy method was developed for the
Fokker–Planck equations in Example 5. But in view of its subspace
decomposition given in ASS20 , an $\mathrm{L}^{2}$-analysis is also feasible.

In our second hypocoercive entropy method, we construct a problem adapted
Lypunov functional that is able to reveal the sharp decay behavior as
$t\to+\infty$. We shall illustrate this strategy for Examples 2 and 3, where
the anti-Hermitian operator $\mathsf{T}$ is either $V\partial_{x}$ (for
discrete velocities) or $v\,\partial_{x}$ (for continuous velocities). In
order to establish the _mode-by-mode hypocoercivity_ , we Fourier transform
(1) w.r.t. $x\in\mathcal{X}$, with either
$\mathcal{X}=\mathbb{T}^{1}=:\mathbb{T}$ or $\mathcal{X}={\mathbb{R}}^{d}$. In
the torus case, we assume $d=1$ for simplicity, but the method extends to
higher dimensions (see Achleitner2018 ). With the abuse of notations of
keeping $F$ for the distribution function written in the variables
$(t,\xi,v)$, this yields

$\frac{dF}{dt}=-\,i\,\xi\,VF+\mathsf{L}F=:-\,C(\xi)\,F\,,$ (10)

with a discrete modal variable $\xi\in{\mathbb{Z}}$ for the torus and
$\xi\in{\mathbb{R}}^{d}$ in the whole space case. In (10), $V$ is a diagonal
matrix for Example 2, and for Example 3 it either represents the
multiplication operator by $v$ or, when using a basis in the $v$-variable, a
symmetric, real-valued “infinite matrix” (_cf._ (AAC, , § 4)).

For each fixed mode $\xi$, (10) is now an ODE with constant coefficients (of
dimension $n<\infty$ for Example 2, and infinite dimensional for Example 3).
For finite $n$, we define the _modal spectral gap_ of $C(\xi)$ as

$\mu(\xi):=\min_{0\neq\lambda_{j}\in\sigma(C(\xi))}\mathrm{Re}(\lambda_{j})\,.$
(11)

If no eigenvalue of $C(\xi)$ with $\mathrm{Re}(\lambda_{j})=\mu(\xi)$ is
defective (_i.e._ , all eigenvalues have matching algebraic and geometric
multiplicities), see _e.g._ HJ , then the exponential decay of
$\|F(t,\xi)\|^{2}$ with the sharp rate $2\,\mu(\xi)$ is shown using a Lyapunov
functional obtained as a twisted Euclidean norm on ${\mathbb{C}}^{n}$. To this
end we use the following algebraic result.

######  Lemma 1 ((AAC, , Lemma 2))

For a given matrix $C\in{\mathbb{C}}^{n\times n}$, let $\mu$ be defined as in
(11). Assume that $0\not\in\sigma(C)$ and that $C$ has no defective
eigenvalues with $\mathrm{Re}(\lambda_{j})=\mu$. Then there exists a positive
definite Hermitian matrix $P\in{\mathbb{C}}^{n\times n}$ such that

$\displaystyle C^{*}P+P\,C$ $\displaystyle\geq 2\,\mu\,P\,.$ (12)

Moreover, if all eigenvalues of $C$ are non-defective, any matrix

$\displaystyle P:=\sum\limits_{j=1}^{n}c_{j}\,w_{j}\otimes w_{j}^{*}$ (13)

satisfies (12), where $w_{j}\in{\mathbb{C}}^{n}$ denote the normalized (right)
eigenvectors of $C^{*}$ and, for all $j=1,\dots,n$, the coefficient
$c_{j}\in(0,+\infty)$ is an arbitrary weight.

For the extension of this lemma to the case $0\in\sigma(C)$ we refer to (AAC,
, Lemma 3), but, anyhow, this is typically relevant only for $\xi=0$. The more
technical case when $C$ has defective eigenvalues was analyzed in (Arnold2014,
, Lemma 4.3(i)). In the case $n=\infty$ (occuring in the kinetic BGK models of
Example 3), the eigenfunction construction of the operator (or “infinite
matrix”) $P$ via (13) is, in general, not feasible. A systematic construction
of _approximate_ matrices $P$ with a suboptimal value compared with $\mu$ in
(12) was presented in (AAC, , § 4.3-4.4) and (Achleitner2018, , § 2.3).

Using the deformation matrix $P$, we define the “twisted Euclidean norm” in
${\mathbb{C}}^{n}$ as

$\|F\|_{P}^{2}:=\langle F,P\,F\rangle\,,$

which is equivalent to the Euclidean norm $\|\cdot\|$ through the estimate

$\lambda_{1}^{P}\,\|F\|^{2}\leq\|F\|_{P}^{2}\leq\lambda_{n}^{P}\,\|F\|^{2}\,,$
(14)

where $\lambda_{1}^{P}$ and $\lambda_{n}^{P}$ are the smallest and largest
eigenvalues of $P$, respectively. From (9) and (12) follows that

$\frac{d}{dt}\|F\|_{P}^{2}=-\,\langle
F,(C^{*}P+PC)F\rangle\leq-\,2\,\mu\,\|F\|_{P}^{2}\,.$

This shows that solutions to (9) satisfy

$\|F(t)\|_{P}^{2}\leq e^{-2\,\mu\,t}\,\|F_{0}\|_{P}^{2}\quad\forall\,t\geq
0\,,$

and hence in the Euclidean norm:

$\|F(t)\|^{2}\leq\mbox{cond}(P)\,e^{-2\,\mu\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq
0\,,$ (15)

where $\mbox{cond}(P):=\lambda_{n}^{P}/\lambda_{1}^{P}$ denotes the _condition
number_ of $P$. We recall from AAS19  that $\mbox{cond}(P)$ is in general not
the minimal multiplicative constant for (15). In fact, in general it is
impossible to obtain that optimal constant from a Lyapunov functional, even
for $n=2$, see (AAS19, , Theorem 4.1). We also remark that the matrix $P$ from
(13) is not uniquely determined (even beyond trivial multiples). As a
consequence, $\operatorname{cond}(P)$ may be different for different
admissible choices of $P$. For an example with $n=3$, we refer to (AAS19, , §
3).

Analogous decay estimates hold for solutions $F(t,\xi)$ to the modal ODEs
(10), and they involve the deformation matrices $P(\xi)$ and the modal
spectral gaps $\mu(\xi)$:

$\|F(t,\xi)\|_{P(\xi)}^{2}\leq
e^{-2\,\mu(\xi)\,t}\,\|F_{0}(\xi)\|_{P(\xi)}^{2}\quad\forall\,t\geq 0\,.$ (16)

This motivates the definition of a _modal-based Lyapunov functional_ by
assembling the modal functionals. We present two variants of this approach.

Strategy 1. We consider the global Lyapunov functional

$\mathsf{H}_{2}[F]:=\sum_{\xi\in{\mathbb{Z}}}\|F(\xi)\|^{2}_{P(\xi)}\,,$ (17)

which is written here for the case of discrete modes, _i.e._ ,
$\mathcal{X}=\mathbb{T}$.

We recall that the matrix $P(\xi)$ is not unique. In the kinetic BGK examples
studied so far (_cf._ AAC ; Achleitner2018 ) it was convenient to choose $P$
depending continuously on $\xi$ (for $\xi\in{\mathbb{R}}^{d}$) and such that
$P(\xi)\to\mathrm{Id}$ as $|\xi|\to+\infty$. For kinetic equations with a
local-in-$x$ dissipative operator $\mathsf{L}$, the matrix $C(\xi)$ has the
form given in (10). Under the assumption of a uniform spectral gap
$\overline{\mu}:=\inf_{\xi}\mu(\xi)>0$, the form
$P(\xi)=\mathrm{Id}+O\left(1/|\xi|\right)$ is very natural (see $P^{(1)}(\xi)$
in (53) for an example) in view of the matrix inequality (12).

The modal decay (16) implies the following decay estimate for the solution to
(1):

$\mathsf{H}_{2}[F(t)]\leq
e^{-2\,\bar{\mu}\,t}\,\mathsf{H}_{2}[F_{0}]\quad\forall\,t\geq
0\,,\quad\text{for any }F_{0}\perp F_{\infty}\,.$

Using Parseval’s identity and the norm equivalence from (14), this yields

$\|F(t)\|^{2}\leq\bar{c}_{P}\,e^{-2\,\bar{\mu}\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq
0\,,\quad\text{for any }F_{0}\perp F_{\infty}\,,$ (18)

where $\bar{c}_{P}:=\sup_{\xi}\mbox{cond}(P(\xi))$.

Strategy 2. If all modes $\xi$ have the same spectral gap $\mu(\xi)$, then the
estimate (18) clearly yields the minimal multiplicative constant $\bar{c}_{P}$
(obtainable by Lyapunov methods). This is the case when the relaxation rate
$\sigma<2$ in the Goldstein–Taylor model, which is studied in AESW  and in
Part III below. But faster decaying modes may have a “too large” condition
number $\mbox{cond}(P(\xi))$, as it is the case for $\sigma>2$ in AESW . Then,
the matrices $P(\xi)$ from (12) have to be modified in order to reduce
$\mbox{cond}(P(\xi))$ by lowering $\mu=\mu(\xi)$ in (12). For simplicity we
detail this strategy only for the case that the infimum $\bar{\mu}$ is
actually attained. Main steps are:

  * •

Let $\Xi:=\\{\xi\,:\,\mu(\xi)=\bar{\mu}\\}$ be the set of the modes with
slowest decay. Set $c_{\Xi}:=\sup_{\xi\in\Xi}\mbox{cond}(P(\xi))$, _i.e._ the
worst common multiplicative constant for these slow modes.

  * •

For all modes $\xi\not\in\Xi$, we distinguish several cases:

    * –

If $\mbox{cond}(P(\xi))\leq c_{\Xi}$, set $\tilde{P}(\xi):=P(\xi)$.

    * –

If $\mbox{cond}(P(\xi))>c_{\Xi}$, then replace $P(\xi)$ by
$\tilde{P}(\xi)\in{\mathbb{C}}^{n\times n}$, which is a positive definite
Hermitian solution to the matrix inequality

$C(\xi)^{*}P+P\,C(\xi)\geq 2\,\bar{\mu}\,P\,.$

In particular, $P$ should be either chosen as any such solution that satisfies
$\mbox{cond}(P(\xi))\leq c_{\Xi}$ or, if this is impossible, then by a
solution $P$ having the least condition number.

  * •

Let
$\tilde{c}_{\Xi}:=\sup_{\xi\not\in\Xi}\mbox{cond}\left(\tilde{P}(\xi)\right)$
be the best multiplicative constant for the faster modes.

  * •

Set $\tilde{c}_{P}:=\max\\{c_{\Xi},\,\tilde{c}_{\Xi}\\}$. With this
construction we define a second, refined Lyapunov functional (again written
for the case $\mathcal{X}=\mathbb{T}$) by

$\widetilde{\mathsf{H}}_{2}[F]:=\sum_{\xi\in\Xi}\|F(\xi)\|^{2}_{P(\xi)}+\sum_{\xi\in\Xi^{c}}\|F(\xi)\|^{2}_{\tilde{P}(\xi)}\,,$
(19)

where $\Xi^{c}:={\mathbb{Z}}\setminus\Xi$.

This yields the improved decay estimate (_w.r.t._ the multiplicative
constant):

$\|F(t)\|^{2}\leq\tilde{c}_{P}\,e^{-2\,\bar{\mu}\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq
0\,,\quad\text{for any }F_{0}\perp F_{\infty}\,.$ (20)

Note that, by construction, $\tilde{c}_{P}\leq\bar{c}_{P}$. Altogether, our
estimates on a solution to the evolution equation (1) rewritten as (10) in
Fourier variables can be summarized into the following result.

###### Proposition 1

On $\mathbb{T}$, let us consider an operator $C$ such that, in Fourier
variables, $C(\xi)$ takes values in ${\mathbb{C}}^{n\times n}$ for any
$\xi\in{\mathbb{Z}}$. Assume the existence of a uniform spectral gap
$\overline{\mu}:=\inf_{\xi\in{\mathbb{Z}}}\mu(\xi)>0$ where $\mu(\xi)$ is
defined by (11).

  1. a)

If the corresponding modal deformation matrices $P(\xi)$ satisfy
$\bar{c}_{P}<\infty$, then the solutions of (1) satisfy the decay estimate
(18).

  2. b)

If the modified deformation matrices $\tilde{P}(\xi)$ satisfy
$\tilde{c}_{P}<\infty$, then the solutions of (1) satisfy the decay estimate
(20).

The above procedure was applied in AESW  to the Goldstein–Taylor model, and in
AAC  to Examples 2–3, considered on $\mathbb{T}$.

The hypocoercivity results based on the Lyapunov matrix inequalities (12) and
mode-by-mode estimates as in (16) have the advantage that, in simple cases, it
is possible to identify the optimal decay rates. They are less flexible than
the hypocoercivity results based on the twisted $\mathrm{L}^{2}$ norm inspired
by diffusion limits of § I.1. Our purpose of Part III is to detail several
variants of these methods in simple cases, draw a few consequences and compare
the estimates of the two methods.

Part II. Optimization of twisted $\mathrm{L}^{2}$ norms

This part is devoted to accurate hypocoercivity estimates in Fourier variables
based on our _first abstract method_ , for two simple kinetic equations with
Gaussian local equilibria. It is a refined version of the paper BDMMS  devoted
to a larger class of equilibria, but to the price of weaker bounds. Here we
underline some key ideas of mode-by-mode hypocoercivity and perform more
accurate and explicit computations. New estimates are obtained, which
numerically improve upon known ones. Rates and constants are discussed and
numerically illustrated, with the purpose of establishing benchmarks for the
_$\mathrm{L}^{2}$ -hypocoercivity theory based upon a twist inspired by
diffusion limits_. Exponential rates are obtained on the torus, with a
discussion on high frequency estimates. On the whole space case, low
frequencies are involved in the computation of the asymptotic decay rates. We
also detail how spectral estimates of the mode-by-mode $\mathrm{L}^{2}$
hypocoercivity method can be systematically turned into rates of decay using
the ideas of the original proof of Nash’s inequality.

##  II.1 A detailed mode-by-mode approach

###  II.1.1 Introduction

We consider the Cauchy problem

$\partial_{t}f+v\cdot\nabla_{x}f=\mathsf{L}f\,,\quad f(0,x,v)=f_{0}(x,v)\,,$
(21)

for a distribution function $f(t,x,v)$, where $x\in{\mathbb{R}}^{d}$ denotes
the position variable, $v\in{\mathbb{R}}^{d}$ is the velocity variable, and
$t\geq 0$ is the time. Concerning the collision operator, $\mathsf{L}$ denotes
the _Fokker–Planck operator_ $\mathsf{L}_{1}$ or, as in Dolbeault2009511 , the
_linear BGK operator_ $\mathsf{L}_{2}$, which are defined respectively by

$\mathsf{L}_{1}f:=\Delta_{v}f+\nabla_{v}\cdot(v\,f)\quad\mbox{and}\quad\mathsf{L}_{2}f:=\rho_{f}\,\mathcal{M}-f\,.$

Here $\mathcal{M}$ is the normalized Gaussian function

$\mathcal{M}(v)=\frac{e^{-\,\frac{1}{2}\,|v|^{2}}}{(2\,\pi)^{d/2}}\quad\forall\,v\in{\mathbb{R}}^{d}$

and $\rho_{f}:=\int_{{\mathbb{R}}^{d}}f\,dv$ is the spatial density. Notice
that $\mathcal{M}$ spans the kernel of $\mathsf{L}$. We introduce the _weight_

$d\gamma:=\gamma(v)\,dv\quad\mbox{where}\quad\gamma:=\frac{1}{\mathcal{M}}$

and the weighted norm

$\|f\|_{\mathrm{L}^{2}(dx\,d\gamma)}^{2}:=\iint_{\mathcal{X}\times{\mathbb{R}}^{d}}|f(x,v)|^{2}\,dx\,d\gamma\,,$

where $\mathcal{X}$ denotes either the cube $[0,L)^{d}$ with periodic boundary
conditions or $\mathcal{X}={\mathbb{R}}^{d}$, that is, the whole Euclidean
space.

Let us consider the Fourier transform of $f$ in $x$ defined by

$\hat{f}(t,\xi,v)=\int_{\mathcal{X}}e^{-i\,x\cdot\xi}\,f(t,x,v)\,dx\,,$ (22)

where either $\mathcal{X}=[0,L)^{d}$ (with periodic boundary conditions), or
$\mathcal{X}={\mathbb{R}}^{d}$. We denote by
$\xi\in(2\pi/L)^{d}\,{\mathbb{Z}}^{d}\subset{\mathbb{R}}^{d}$ or
$\xi\in{\mathbb{R}}^{d}$ the Fourier variable. Details will be given in §
II.1.3. Next, we rewrite Equation (21) for $F=\hat{f}$ as

$\partial_{t}F+\mathsf{T}F=\mathsf{L}F\,,\quad
F(0,\xi,v)=\hat{f_{0}}(\xi,v)\,,\quad\mathsf{T}F=i\,(v\cdot\xi)F\,.$ (23)

Here we abusively use the same notation $\mathsf{T}$ for the transport
operator in the original variables and after the Fourier transform, where it
is a simple multiplication operator. We shall also consider $\xi$ as a given,
fixed parameter and omit it whenever possible, so that we shall write that $F$
is a function of $(t,v)$, for sake of simplicity. Let us define

$\mathcal{H}=\mathrm{L}^{2}\left(d\gamma\right)\,,\quad\|F\|^{2}=\int_{{\mathbb{R}}^{d}}|F|^{2}\,d\gamma\,,\quad\Pi
F=\mathcal{M}\,\int_{{\mathbb{R}}^{d}}F\,dv=\mathcal{M}\,\rho_{F}\,.$ (24)

Our goal is to obtain decay estimates of $\|F\|$ parameterized by $\xi$ and
this is why such an approach can be qualified as a _mode-by-mode
hypocoercivity method._

###  II.1.2 A first optimization in the general setting

The estimates of (BDMMS, , Proposition 4) are rough and it is possible to
improve upon the choice for $\delta$ and $\lambda$. On the triangle

$\mathcal{T}_{m}:=\Big{\\{}(\delta,\lambda)\in(0,\lambda_{m})\times(0,2\,\lambda_{m})\,:\,\lambda<2\,(\lambda_{m}-\delta)\Big{\\}}\,,$

let us define

$h_{\star}(\delta,\lambda):=\delta^{2}\left(C_{M}+\frac{\lambda}{2}\right)^{2}-4\left(\lambda_{m}-\,\delta-\frac{\lambda}{2}\right)\left(\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}-\frac{\lambda}{2}\right)\,,$
$\lambda_{\star}(\delta):=\sup\Big{\\{}\lambda\in(0,2\,\lambda_{m})\,:\,h_{\star}(\delta,\lambda)\leq
0\Big{\\}}\quad\mbox{and}\quad
C_{\star}(\delta):=\frac{2+\delta}{2-\delta}\,.$

We will also need later

$K_{M}:=\frac{\lambda_{M}}{1+\lambda_{M}}<1\quad\mbox{and}\quad\delta_{\star}:=\frac{4\,K_{M}\,\lambda_{m}}{4\,K_{M}+C_{M}^{2}}<\lambda_{m}\,.$

Our first result provides us with the following refinement of (5).

###### Proposition 2

Under the assumptions of Theorem I.1.1, we have

$\mathsf{H}_{1}[F(t,\cdot)]\leq\mathsf{H}_{1}[F_{0}]\,e^{-\lambda\,t}\quad\forall\,t\geq
0$

with
$\lambda=\max\Big{\\{}\lambda_{\star}(\delta)\,:\,\delta\in(0,\delta_{\star})\Big{\\}}$.
Moreover, for any $\delta<\min\\{2,\delta_{\star}\\}$, if $F$ solves (1) with
initial datum $F_{0}\in\mathcal{H}$, then

$\|F(t)\|^{2}\leq
C_{\star}(\delta)\,e^{-\,\lambda_{\star}(\delta)\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq
0\,.$

On the boundary of the triangle $\mathcal{T}_{m}$, we notice that

$h_{\star}(0,\lambda)=\lambda\,(2\,\lambda_{m}-\lambda)>0\quad\forall\,\lambda\in(0,2\,\lambda_{m})\,,$
$h_{\star}\Big{(}\delta,2\,(\lambda_{m}-\delta)\Big{)}=(C_{M}+\lambda_{m}-\delta)^{2}\,\delta^{2}>0\quad\forall\,\delta\in(0,\lambda_{m})\,,$

and
$h_{\star}(\delta,0)/\delta=\left(C_{M}^{2}+4\,K_{M}\right)\,\delta-4\,K_{M}\,\lambda_{m}$
is negative if $0<\delta<\delta_{\star}$. As a consequence, the set
$\\{(\delta,\lambda)\in\mathcal{T}_{m}\,:\,h_{\star}(\delta,\lambda)\leq 0\\}$
is non-empty. The functions $\lambda\mapsto h_{\star}(\delta,\lambda)$ for a
fixed $\delta\in(0,\lambda_{m})$ and $\delta\mapsto h_{\star}(\delta,\lambda)$
for a fixed $\lambda\in(0,2\,\lambda_{m})$ are both polynomials of second
degree. The expression of $\lambda_{\star}(\delta)$ is explicitly computed as
the smallest root of $\lambda\mapsto h_{\star}(\delta,\lambda)$ but has no
interest by itself. It is also elementary to check that $h_{\star}$ is
positive if $(\delta,\lambda)\in\mathcal{T}_{m}$ with $\delta>\delta_{\star}$.

###### Proof

The method is the same as in DMS-2part  and (BDMMS, , Proposition 4), except
that we use sharper estimates.

Since $\mathsf{AT}\Pi$ can be interpreted as $z\mapsto(1+z)^{-1}\,z$ applied
to $(\mathsf{T}\Pi)^{*}\mathsf{T}\Pi$, the spectral theorem and conditions
(H1) and (H2) imply that

$-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi
F,F\rangle\geq\lambda_{m}\,\|(\mathrm{Id}-\Pi)F\|^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,\|\Pi
F\|^{2}\,.$ (25)

From that point, one has to prove that
$-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi F,F\rangle$
controls the other terms in the expression of $\mathsf{D}[F]$. By (H4), we
know that

$\left|\mathrm{Re}\langle\mathsf{AT}(\mathrm{Id}-\Pi)F,F\rangle+\,\mathrm{Re}\langle\mathsf{AL}F,F\rangle\right|\leq
C_{M}\,\|\Pi F\|\,\|(\mathrm{Id}-\Pi)F\|\,.$ (26)

As in (DMS-2part, , Lemma 1), if $G=\mathsf{A}F$, _i.e._ , if
$(\mathsf{T}\Pi)^{*}F=G+(\mathsf{T}\Pi)^{*}\,\mathsf{T}\Pi\,G$, then

$\langle\mathsf{TA}F,F\rangle=\langle
G,(\mathsf{T}\Pi)^{*}\,F\rangle=\|G\|^{2}+\|\mathsf{T}\Pi
G\|^{2}=\|\mathsf{A}F\|^{2}+\|\mathsf{TA}F\|^{2}\,.$

By the Cauchy-Schwarz inequality, we know that

$\langle
G,(\mathsf{T}\Pi)^{*}\,F\rangle=\langle\mathsf{TA}F,(\mathrm{Id}-\Pi)F\rangle\\\
\leq\|\mathsf{TA}F\|\,\|(\mathrm{Id}-\Pi)F\|\leq\frac{1}{2\,\mu}\,\|\mathsf{TA}F\|^{2}+\frac{\mu}{2}\,\|(\mathrm{Id}-\Pi)F\|^{2}$

for any $\mu>0$. Hence

$2\,\|\mathsf{A}F\|^{2}+\left(2-\frac{1}{\mu}\right)\|\mathsf{TA}F\|^{2}\leq\mu\,\|(\mathrm{Id}-\Pi)F\|^{2}\,,$

which, by taking either $\mu=1/2$ or $\mu=1$, proves that

$\|\mathsf{A}F\|\leq\frac{1}{2}\,\|(\mathrm{Id}-\Pi)F\|\,,\quad\|\mathsf{TA}F\|\leq\|(\mathrm{Id}-\Pi)F\|$

and establishes (7). Incidentally, this proves that

$\left|\langle\mathsf{TA}F,F\rangle\right|=\left|\langle\mathsf{TA}F,(\mathrm{Id}-\Pi)F\rangle\right|\leq\|(\mathrm{Id}-\Pi)F\|^{2}\,,$
(27)

and also that

$|\langle\mathsf{A}F,F\rangle|\leq\frac{1}{2}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\leq\frac{1}{4}\,\|F\|^{2}\,.$ (28)

As a consequence of this last identity, we obtain

$\left|\mathsf{H}_{1}[F]-\tfrac{1}{2}\,\|F\|^{2}\right|=\delta\,\big{|}\langle\mathsf{A}F,F\rangle\big{|}\leq\frac{\delta}{4}\,\|F\|^{2}\,,$

which, under the condition $\delta<2$, is a proof of (7) with the improved
constant

$c_{\pm}=\frac{2\pm\,\delta}{4}\,.$ (29)

Now let us come back to the proof of (6). Collecting (25), (26), and (27) with
the definition of $\mathsf{D}[F]$, we find that

$\mathsf{D}[F]\geq(\lambda_{m}-\,\delta)\,X^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,Y^{2}-\,\delta\,C_{M}\,X\,Y$

with $X:=\|(\mathrm{Id}-\Pi)F\|$ and $Y:=\|\Pi F\|$. Using (28), we observe
that

$\mathsf{H}_{1}[F]\leq\frac{1}{2}\left(X^{2}+Y^{2}\right)+\frac{\delta}{2}\,X\,Y\,.$

Hence the largest value of $\lambda$ for which

$\mathsf{D}[F]\geq\lambda\,\mathsf{H}_{1}[F]$

can be estimated by the largest value of $\lambda$ for which

$\mathcal{Q}(X,Y):=(\lambda_{m}-\,\delta)\,X^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,Y^{2}-\,\delta\,C_{M}\,X\,Y-\frac{\lambda}{2}\left(X^{2}+Y^{2}\right)-\frac{\lambda}{2}\,\delta\,X\,Y\\\
=\left(\lambda_{m}-\,\delta-\frac{\lambda}{2}\right)X^{2}-\,\delta\left(C_{M}+\frac{\lambda}{2}\right)X\,Y+\left(\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}-\frac{\lambda}{2}\right)Y^{2}$

is a nonnegative quadratic form. It is characterized by the discriminant
condition $h_{\star}(\delta,\lambda)\leq 0$, and the condition
$\lambda_{m}-\,\delta-\lambda/2>0$ which determines $\mathcal{T}_{m}$ with the
two other conditions: $\delta>0$ and $\lambda>0$. From (6), we deduce the
decay of $\mathsf{H}_{1}[F(t,\cdot)]$ and the decay of $\|F(t)\|^{2}$ by (7)
using (29). $\square$

###### Remark 1

The estimate (8) of (BDMMS, , Proposition 4) is easily recovered as follows.
Using

$\mathsf{D}[F]\geq(\lambda_{m}-\delta)\,X^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,Y^{2}-\delta\,C_{M}\,X\,Y\\\
\geq(\lambda_{m}-\delta)\,X^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,Y^{2}-\frac{\delta}{2}\left(C_{M}^{2}\,X^{2}+Y^{2}\right)$

and

$\mathsf{H}_{1}[F]\leq\frac{2+\delta}{4}\left(X^{2}+Y^{2}\right)\,,$

with $\delta$ defined as in (8), we obtain

$\mathsf{D}[F]\geq\frac{\lambda_{m}}{4}\,X^{2}+\frac{\delta\,\lambda_{M}}{2\,(1+\lambda_{M})}\,Y^{2}\\\
\geq\frac{1}{4}\,\min\left\\{\lambda_{m},\frac{2\,\delta\,\lambda_{M}}{1+\lambda_{M}}\right\\}\|F\|^{2}\geq\frac{2\,\delta\,\lambda_{M}}{3\,(1+\lambda_{M})}\,\mathsf{H}[F]\,.$

Hence we have that $\frac{1}{4}\,\|F\|^{2}\geq\frac{1}{3}\,\mathsf{H}[F]$
because $4/(2+\delta)\geq 8/5>4/3$ if $\delta<1/2$. This estimate is non-
optimal and it is improved in the proof of Proposition 2.

###### Remark 2

In the discussion of the positivity of $\mathcal{Q}$, we can observe that
$(X,Y)$ is restricted to the upper right quadrant corresponding to $X>0$ and
$Y>0$. The discriminant condition $h_{\star}(\delta,\lambda)\leq 0$ and the
condition $\lambda_{m}-\,\delta-\lambda/2>0$ guarantee that
$\mathcal{Q}(X,Y)\geq 0$ _for any_ $X$, $Y\in{\mathbb{R}}$, which is of course
a sufficient condition. It is also necessary because the coefficient of
$Y^{2}$ is positive (otherwise one can find some $X>0$ and $Y>0$ such that
$\mathcal{Q}(X,Y)<0$) and then by solving a second degree equation, one could
again find a region in the upper right quadrant such that $\mathcal{Q}$ takes
negative values.

Hence we produce a necessary and sufficient condition for $\mathcal{Q}$ to be
a nonnegative quadratic form. This does not mean that the condition of
Proposition 2 is necessary because we have made various estimates, which are
not generically optimal, in order to reduce the problem to the discussion of
the sign of $\mathcal{Q}$. In special cases, we can indeed improve upon
Proposition 2. We will discuss such improvements in the next section.

###  II.1.3 Mode-by-mode hypocoercivity

#### Fourier representation and mode-by-mode estimates

Let us consider the Fourier transform in $x$, take the Fourier variable
$\xi\in{\mathbb{R}}^{d}$ as a parameter, and study, _for a given $\xi$_,
Equation (23). For a given $\xi\in{\mathbb{R}}^{d}$, let us implement the
strategy of Theorem I.1.1 and Proposition 2 applied to $(t,v)\mapsto
F(t,\xi,v)$, with the choices (24). The operator $\mathsf{A}$ is defined by

$(\mathsf{A}F)(v)=-\frac{\,i\,\xi}{1+|\xi|^{2}}\cdot\int_{{\mathbb{R}}^{d}}w\,F(w)\,dw\,\mathcal{M}(v)\,.$

Taking advantage of the explicit form of $\mathsf{A}$, we can reapply the
method of § II.1.2 with explicit numerical values, and actually improve upon
the previous results. Let us give some details, which will be useful for
benchmarks and numerical computations. Again we aim at relating the Lyapunov
functional

$\mathsf{H}_{1}[F]:=\frac{1}{2}\,\|F\|^{2}+\delta\,\mathrm{Re}\langle\mathsf{A}F,F\rangle$

defined as in (3) with $\mathsf{D}[F]$ defined by (4), _i.e._ ,

$\mathsf{D}[F]:=-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi
F,F\rangle\\\
-\,\delta\,\mathrm{Re}\langle\mathsf{TA}F,F\rangle+\delta\,\mathrm{Re}\langle\mathsf{AT}(\mathrm{Id}-\Pi)F,F\rangle-\delta\,\mathrm{Re}\langle\mathsf{AL}F,F\rangle\,.$

In other words, we want to estimate the optimal constant $\lambda(\xi)$ in the
_entropy – entropy production inequality_

$\mathsf{D}[F]\geq\lambda(\xi)\,\mathsf{H}_{1}[F]$ (30)

corresponding to the best possible choice of $\delta$, for a given
$\xi\in{\mathbb{R}}^{d}$.

If $\mathsf{L}=\mathsf{L}_{1}$, $\lambda_{m}=1$ is given by the Gaussian
Poincaré inequality. If $\mathsf{L}=\mathsf{L}_{2}$, it is straightforward to
check that $\lambda_{m}=1$. In both cases, it follows from the definition of
$\mathsf{T}$ that $\lambda_{M}=|\xi|^{2}$. With $X:=\|(\mathrm{Id}-\Pi)F\|$
and $Y:=\|\Pi F\|$, using (25) we have

$-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi F,F\rangle\geq
X^{2}+\frac{\delta\,|\xi|^{2}}{1+|\xi|^{2}}\,Y^{2}\,.$ (31)

By a Cauchy-Schwarz estimate, we know that

$\left|\xi\cdot\int_{{\mathbb{R}}^{d}}w\,F(w)\,dw\right|=|\xi|\,\left|\int_{{\mathbb{R}}^{d}}\frac{\xi}{|\xi|}\cdot
w\,\sqrt{\mathcal{M}}\;\frac{(\mathrm{Id}-\Pi)F}{\sqrt{\mathcal{M}}}\,dw\right|\leq|\xi|\,\|(\mathrm{Id}-\Pi)F\|$

and therefore obtain that

$\|\mathsf{A}F\|\leq\frac{|\xi|}{1+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|\quad\mbox{and}\quad\|\mathsf{AL}F\|\leq\frac{|\xi|}{1+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|\,,$
(32)

where the second estimate is a consequence of $\mathsf{AL}F=-\,\mathsf{A}F$
when $\mathsf{L}=\mathsf{L}_{1}$ or $\mathsf{L}=\mathsf{L}_{2}$. Notice that
the estimate of $\|\mathsf{A}F\|$ is sharper than the one used in the
introduction.

Using (32), we have that

$|\mathrm{Re}\langle\mathsf{A}F,F\rangle|\leq\frac{|\xi|}{1+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\leq\frac{1}{2}\,\frac{|\xi|}{1+|\xi|^{2}}\,\|F\|^{2}$
(33)

and obtain an improved version of (7) given by

$\frac{1}{2}\left(1-\frac{\delta\,|\xi|}{1+|\xi|^{2}}\right)\|F\|^{2}\leq\mathsf{H}_{1}[F]\leq\frac{1}{2}\left(1+\frac{\delta\,|\xi|}{1+|\xi|^{2}}\right)\|F\|^{2}\,.$
(34)

We also deduce from (32) that

$\mathsf{H}_{1}[F]\leq\frac{1}{2}\left(X^{2}+Y^{2}\right)+\frac{\delta\,|\xi|}{1+|\xi|^{2}}\,X\,Y$
(35)

and, using $\mathsf{AL}F=-\,\mathsf{A}F$ and (33),

$|\mathrm{Re}\langle\mathsf{AL}F,F\rangle|\leq\frac{|\xi|}{1+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\,.$ (36)

As for estimating $\|\mathsf{A}F\|$, by a Cauchy-Schwarz estimate we obtain

$\|\mathsf{TA}F\|\leq\frac{|\xi|^{2}}{1+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|\,,$

so that

$\delta\,|\mathrm{Re}\langle\mathsf{TA}F,F\rangle|\leq\frac{\delta\,|\xi|^{2}}{1+|\xi|^{2}}\,X^{2}\,.$
(37)

As in BDMMS , we can also estimate

$\displaystyle\|\mathsf{AT}(\mathrm{Id}-\Pi)F\|$
$\displaystyle=\tfrac{\left|\int_{{\mathbb{R}}^{d}}\left(v^{\prime}\cdot\xi\right)^{2}\,(\mathrm{Id}-\Pi)F(v^{\prime})\,dv^{\prime}\right|}{1+|\xi|^{2}}$
$\displaystyle\leq\tfrac{\left(\int_{{\mathbb{R}}^{d}}\left(v^{\prime}\cdot\xi\right)^{4}\mathcal{M}(v^{\prime})\,dv^{\prime}\right)^{1/2}}{1+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|=\frac{\sqrt{3}\,|\xi|^{2}}{1+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|\,.$

This inequality and (32) establish that (H4) holds with
$C_{M}=\frac{|\xi|\,(1+\sqrt{3}\,|\xi|)}{1+|\xi|^{2}}$. Let us finally notice
that

$\delta\,|\mathrm{Re}\langle\mathsf{AT}(\mathrm{Id}-\Pi)F,F\rangle|+\delta\,|\mathrm{Re}\langle\mathsf{AL}F,F\rangle|\leq\delta\,\frac{|\xi|\left(1+\sqrt{3}\,|\xi|\right)}{1+|\xi|^{2}}\,X\,Y\,.$
(38)

#### Improved estimates with some plots

In this section, our purpose is to provide constructive estimates of the rate
$\lambda$ in Theorem I.1.1 and get improved estimates using various
refinements in the mode-by-mode approach. Let us start with the one given in
(8).

With $s:=|\xi|$, we read from § II.1.3 that

$\lambda_{m}=1\,,\quad\lambda_{M}=s^{2}\quad\mbox{and}\quad
C_{M}=\frac{s\left(1+\sqrt{3}\,s\right)}{1+s^{2}}\,.$ (39)

In that case, the estimate (8) becomes $\lambda\geq\lambda_{0}(s)$ for
$\delta=\delta_{0}(s)$ with

$\lambda_{0}(s):=\frac{1}{3}\,\frac{s^{2}}{\left(1+\sqrt{3}\,s\right)^{2}}\quad\mbox{and}\quad\delta_{0}(s):=\frac{1}{2}\,\frac{1+s^{2}}{\left(1+\sqrt{3}\,s\right)^{2}}\,.$

With (39) in hand, we can also apply the result of Proposition 2. In order to
take into account the dependence on $s$, the function $h_{\star}$ has to be
replaced by a function $h_{1}$ defined by

$h_{1}(\delta,\lambda,s):=\delta^{2}\left(\frac{s\left(1+\sqrt{3}\,s\right)}{1+s^{2}}+\frac{\lambda}{2}\right)^{2}-4\left(1-\,\delta-\frac{\lambda}{2}\right)\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)\,,$

so that the whole game is now reduced, for a given value of $s>0$, to study
the conditions on $(\delta,\lambda)\in\mathcal{T}_{m}$ such that
$h_{1}(\delta,\lambda,s)\leq 0$. In particular, we are interested in computing
the largest value $\lambda_{1}(s)$ of $\lambda$ for which there exists
$\delta>0$ for which $h_{1}(\delta,\lambda,s)\leq 0$ with
$(\delta,\lambda)\in\mathcal{T}_{m}$, and denote it by $\delta_{1}(s)$. The
triangle $\mathcal{T}_{m}$ is shown in Fig. 1 and the curves
$s\mapsto\lambda_{1}(s)$ and $s\mapsto\delta_{1}(s)$ in Figs. 2 and 3.
Solutions are numerically contained in $\mathcal{T}_{m}$ in the sense that
$s\mapsto\big{(}\delta_{1}(s),\lambda_{1}(s)\big{)}\in\mathcal{T}_{m}$ for any
$s>0$.

$\delta$$\lambda$$s=5$$h_{1}(\delta_{1},\lambda_{1})<0$$(\delta_{1}(s),\lambda_{1}(s))$
Figure 1:  With $\lambda_{m}$, $\lambda_{M}$ and $C_{M}$ given by (39), the
admissible range $\mathcal{T}_{m}$ of the parameters $(\delta,\lambda)$ is
shown in grey for $s=5$. The darker area is the region in which
$h_{1}(\delta,\lambda,s)$ takes negative values, and
$(\delta_{1}(s),\lambda_{1}(s))$ are the coordinates of the maximum point of
the curve which separates the two regions in the triangle $\mathcal{T}_{m}$.

As already noted, some estimates in § II.1.3 (namely (32), (34), (35), (36)
and (37)) are slightly more accurate then the estimates of the proof of
Proposition 2. By collecting (31), (35), (37) and (38), we obtain

$\mathsf{D}[F]-\lambda\,\mathsf{H}_{1}[F]\\\
\geq\left(1-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)X^{2}-\frac{\delta\,s}{1+s^{2}}\left(1+\sqrt{3}\,s+\lambda\right)X\,Y+\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)Y^{2}$

is nonnegative for any $X$ and $Y$ under the discriminant condition which
amounts to the nonpositivity of

$h_{2}(\delta,\lambda,s):=\delta^{2}\,s^{2}\left(\frac{1+\sqrt{3}\,s+\lambda}{1+s^{2}}\right)^{2}-4\left(1-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)\,,$

in the triangle

$\mathcal{T}_{m}(s):=\Big{\\{}(\delta,\lambda)\in\left(0,\lambda_{m}\,\tfrac{1+s^{2}}{s^{2}}\right)\times(0,2\,\lambda_{m})\,:\,\lambda<2\left(\lambda_{m}-\tfrac{\delta\,s^{2}}{1+s^{2}}\right)\Big{\\}}$

with $\lambda_{m}=1$. Exactly the same discussion as for
$s\mapsto\lambda_{1}(s)$ and $s\mapsto\delta_{1}(s)$ determines the curves
$s\mapsto\lambda_{2}(s)$ and $s\mapsto\delta_{2}(s)$ shown in Figs. 2 and 3.
Solutions satisfy
$s\mapsto\big{(}\delta_{2}(s),\lambda_{2}(s)\big{)}\in\mathcal{T}_{m}(s)$ for
any $s>0$.

$s$$\lambda_{0}(s)$$\lambda_{1}(s)$$\lambda_{2}(s)$ Figure 2:  With
$\lambda_{m}$, $\lambda_{M}$ and $C_{M}$ given by (39), curves
$s\mapsto\lambda_{i}(s)$ with $i=0$, $1$ and $2$ are shown. The improvement of
$\lambda_{2}$ upon $\lambda_{0}$ is of the order of a factor $5$.
$s$$\delta_{2}(s)$$\delta_{1}(s)$$\delta_{0}(s)$ Figure 3:  With
$\lambda_{m}$, $\lambda_{M}$ and $C_{M}$ given by (39), curves
$s\mapsto\delta_{i}(s)$ with $i=0$, $1$ and $2$ are shown. The dotted curve
$s\mapsto\delta_{0}(s)$ shows the estimate (8) of (BDMMS, , Proposition 4). It
can be checked numerically that the numerical curves
$s\mapsto\big{(}\delta_{i}(s),\lambda_{i}(s)\big{)}$ with $i=1$, $2$ satisfy
the constraints, _i.e._ , stay in their respective triangles for all $s>0$, as
shown in Fig. 1.

###  II.1.4 Further observations

In this section, we collect various observations, which are of practical
interest, and rely all on the same computations as the ones of Sections II.1.2
and II.1.3.

#### Explicit estimates

The explicit computation of $\delta_{2}$ and $\lambda_{2}$ is delicate as it
involves finding the roots of high degree polynomials, but it is possible to
obtain a very good approximation as follows. After estimating $\lambda\,X\,Y$
by $\lambda\left(X^{2}+Y^{2}\right)/2$, we obtain that

$\mathsf{D}[F]-\lambda\,\mathsf{H}_{1}[F]\\\
\geq\left(1-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)X^{2}-\frac{\delta\,s}{1+s^{2}}\left(1+\sqrt{3}\,s+\lambda\right)X\,Y+\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\right)Y^{2}\\\
\geq\left(1-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\left(1+\frac{\delta\,s}{1+s^{2}}\right)\right)X^{2}-\frac{\delta\,s}{1+s^{2}}\left(1+\sqrt{3}\,s\right)X\,Y\\\
+\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\left(1+\frac{\delta\,s}{1+s^{2}}\right)\right)Y^{2}=:\widetilde{\mathcal{Q}}(X,Y)$

is nonnegative for any $X$ and $Y$, under the discriminant condition which
amounts to the nonpositivity of

$\tilde{h}_{2}(\delta,\lambda,s):=\delta^{2}\,s^{2}\left(\frac{1+\sqrt{3}\,s}{1+s^{2}}\right)^{2}\\\
-4\left(1-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\left(1+\frac{\delta\,s}{1+s^{2}}\right)\right)\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda}{2}\left(1+\frac{\delta\,s}{1+s^{2}}\right)\right)\,.$
(40)

By doing a computation as in § II.1.3, we can find an explicit result, which
goes as follows.

###### Proposition 3

Assume (39). The largest value of $\lambda>0$ for which there is some
$\delta>0$ such that the quadratic form $\widetilde{\mathcal{Q}}$ is
nonnegative is

$\textstyle\tilde{\lambda}_{2}(s):=\frac{7\,s^{2}-\sqrt{21\,s^{4}+4\,(3+5\,\sqrt{3})\,s^{3}+\,(22+8\,\sqrt{3})\,s^{2}+4\,(1+\sqrt{3})\,s+1}+2\,(1+\sqrt{3})\,s+1}{7\,s^{2}+2\,(2+\sqrt{3})\,s+2}$

with corresponding $\delta$ given by

$\textstyle\tilde{\delta}_{2}(s):=\frac{s^{2}+1}{s}\,\frac{\tilde{\lambda}_{2}(s)^{2}-\tilde{\lambda}_{2}(s)+2\,s}{7\,s^{2}+2\,\sqrt{3}\,s+1-\tilde{\lambda}_{2}(s)^{2}}\,.$

The proof is tedious but elementary and we shall skip it. By construction, we
know that

$\lambda_{2}(s)\geq\tilde{\lambda}_{2}(s)\quad\forall\,s>0$

and the approximation of $\lambda_{2}(s)$ by $\tilde{\lambda}_{2}(s)$ is
numerically quite good (with a relative error of the order of about 10 %),
with exact asymptotics in the limits as $s\to 0_{+}$, in the sense that
$\lambda_{2}(s)/s^{2}\sim\tilde{\lambda}_{2}(s)/s^{2}$, and $s\to+\infty$. See
Figs. 4 and 5. The approximation of $\delta_{2}(s)$ by $\tilde{\delta}_{2}(s)$
is also very good.

Figure 4:  Plot of $s\mapsto\lambda_{2}(s)$ and of of
$s\mapsto\tilde{\lambda}_{2}(s)$, represented, respectively, by the plain and
by the dotted curves.
$s$$s\mapsto\big{(}\lambda_{2}(s)-\tilde{\lambda}_{2}(s)\big{)}\,\big{(}1+s^{-2}\big{)}$
Figure 5:  The curves $s\mapsto\lambda_{2}(s)$ and
$s\mapsto\tilde{\lambda}_{2}(s)$ have the same asymptotic behaviour as $s\to
0_{+}$ and as $s\to+\infty$.

Let us summarize some properties which, as a special case, are of interest for
Sections II.1.4 and II.2.2.

###### Lemma 2

With the notation of Proposition 3, the function $\tilde{\lambda}_{2}$ is
monotone increasing, the function $\tilde{\delta}_{2}$ is monotone increasing
for $s>0$ large enough, and

$\lim_{s\to
0_{+}}\frac{\tilde{\lambda}_{2}(s)}{s^{2}}=2\,,\quad\lim_{s\to+\infty}\tilde{\lambda}_{2}(s)=1-\sqrt{3/7}\approx
0.345346\quad\mbox{and}\quad\lim_{s\to+\infty}\tilde{\delta}_{2}(s)=2/7\,.$

The proof of this result is purely computational and will be omitted here.

#### Mode-by-mode diffusion limit

We consider the diffusion limit which corresponds to the parabolic scaling
applied to the abstract equation (1), that is, the limit as $\varepsilon\to
0_{+}$ of

$\varepsilon\,\frac{dF}{dt}+\mathsf{T}F=\frac{1}{\varepsilon}\,\mathsf{L}F\,.$

We will not go to the details and should simply mention that this amounts to
replace $\lambda$ by $\lambda\,\varepsilon$ when we look for a rate $\lambda$
which is asymptotically independent of $\varepsilon$. We also have to replace
(H4) by the assumption

$\|\mathsf{AT}(\mathrm{Id}-\Pi)F\|+\frac{1}{\varepsilon}\,\|\mathsf{AL}F\|\leq
C_{M}^{\varepsilon}\,\|(\mathrm{Id}-\Pi)F\|$ (H4ε)

in order to clarify the dependence on $\varepsilon$. Since $\mathsf{A}$,
$\mathsf{T}$, $\Pi$ and $\mathsf{L}$ do not depend on $\varepsilon$, this
simply means that we can write
$C_{M}^{\varepsilon}=C_{M}^{(1)}+\frac{1}{\varepsilon}\,C_{M}^{(2)}$ where
$C_{M}^{(1)}$ and $C_{M}^{(2)}$ are the bounds corresponding to

$\|\mathsf{AT}(\mathrm{Id}-\Pi)F\|\leq
C_{M}^{(1)}\,\|(\mathrm{Id}-\Pi)F\|\quad\mbox{and}\quad\|\mathsf{AL}F\|\leq
C_{M}^{(2)}\,\|(\mathrm{Id}-\Pi)F\|\,.$

With these considerations taken into account, proving an entropy - entropy
production inequality is equivalent to proving the nonnegativity of

$\mathsf{D}[F]-\lambda\,\varepsilon\,\mathsf{H}_{1}[F]\\\
\geq\left(\frac{1}{\varepsilon}-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda\,\varepsilon}{2}\right)X^{2}-\frac{\delta\,s}{1+s^{2}}\left(\frac{1}{\varepsilon}+\sqrt{3}\,s+\lambda\,\varepsilon\right)X\,Y+\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda\,\varepsilon}{2}\right)Y^{2}$

for any $X$ and $Y$, and the discriminant condition amounts to the
nonpositivity of

$\frac{\delta^{2}\,s^{2}}{\varepsilon^{2}}\left(\frac{1+\sqrt{3}\,\varepsilon\,s+\lambda\,\varepsilon^{2}}{1+s^{2}}\right)^{2}-4\left(\frac{1}{\varepsilon}-\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda\,\varepsilon}{2}\right)\left(\frac{\delta\,s^{2}}{1+s^{2}}-\frac{\lambda\,\varepsilon}{2}\right)\,.$

In the limit as $\varepsilon\to 0_{+}$, we find that the optimal choice for
$\lambda$ is given by $(\lambda_{\varepsilon}(s),\delta_{\varepsilon}(s))$
with

$\lim_{\varepsilon\to
0_{+}}\lambda_{\varepsilon}(s)=2\,s^{2}\quad\mbox{and}\quad\delta_{\varepsilon}(s)=2\,(1+s^{2})\,\varepsilon\,\big{(}1+o(1))\big{)}\,.$

Notice that $\lambda(s)=2\,s^{2}$ corresponds to the expected value of the
spectrum associated with the heat equation obtained in the diffusion limit.
This also corresponds to the limiting behaviour as $s\to 0_{+}$ of
$\tilde{\lambda}_{2}$ obtained in Lemma 2.

#### Towards an optimized mode-by-mode hypocoercivity approach ?

In our method, the essential property of the operator
$\mathsf{A}:=\Big{(}\mathrm{Id}+(\mathsf{T}\Pi)^{*}\mathsf{T}\Pi\Big{)}^{-1}(\mathsf{T}\Pi)^{*}$
is the equivalence of $\langle\mathsf{AT}\Pi F,F\rangle$ with $\|\Pi F\|^{2}$
given by the estimate

$\frac{\lambda_{M}}{1+\lambda_{M}}\,\|\Pi F\|^{2}\leq\langle\mathsf{AT}\Pi
F,F\rangle\leq\|\Pi F\|^{2}\,.$

These inequalities arise from the _macroscopic coercivity_ condition (H2) and,
using the spectral theorem, from the elementary estimate $z/(1+z)\leq 1$ for
any $z\geq 0$. On the one hand the Lyapunov functional
$\mathsf{H}_{1}[F]:=\tfrac{1}{2}\,\|F\|^{2}+\delta\,\mathrm{Re}\langle\mathsf{A}F,F\rangle$
is equivalent to $\|F\|^{2}$ for $\delta>0$ small enough because $\mathsf{A}$
is a bounded operator. On the other hand,
$\mathsf{D}[F]=-\,\frac{d}{dt}\mathsf{H}_{1}[F]$ can be compared directly with
$\|F\|^{2}$ because, up to terms that can be controlled, as in the proof of
Proposition 2, in the limit as $\delta\to 0_{+}$, $\mathsf{D}[F]$ is bounded
from below by

$-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi
F,F\rangle\geq\lambda_{m}\,\|(\mathrm{Id}-\Pi)F\|^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,\|\Pi
F\|^{2}$

by Assumptions (H1) and (H2). Notice that this estimate holds for any
$\delta>0$. The choice of $z/(1+z)$ picks a specific scale and one may wonder
if $z/(\varepsilon+z)$ would not be a better choice for some value of
$\varepsilon>0$ to be determined. By “better”, we simply have in mind to get a
larger decay rate as $t\to+\infty$, without trying to optimize on the constant
$C$ in (5). It turns out that _the answer is negative_ , as $\varepsilon$ can
be scaled out. Let us give some details.

Let us replace $\mathsf{A}$ by

$\mathsf{A}_{\varepsilon}:=\Big{(}\varepsilon^{2}\,\mathrm{Id}+(\mathsf{T}\Pi)^{*}\mathsf{T}\Pi\Big{)}^{-1}(\mathsf{T}\Pi)^{*}$

for some $\varepsilon>0$ that can be adjusted, without changing the general
strategy, and consider the Lyapunov functional

$\mathsf{H}_{1,\varepsilon}[F]:=\tfrac{1}{2}\,\|F\|^{2}+\delta\,\mathrm{Re}\langle\mathsf{A}_{\varepsilon}F,F\rangle$

for some $\delta>0$, so that

$\begin{array}[]{rl}\displaystyle\mathsf{D}_{\varepsilon}[F]:=&\displaystyle-\,\frac{d}{dt}\mathsf{H}_{1,\varepsilon}[F]\\\\[6.0pt]
=&-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{A_{\varepsilon}T}\Pi
F,F\rangle\\\\[4.0pt]
&-\,\delta\,\mathrm{Re}\langle\mathsf{TA_{\varepsilon}}F,F\rangle+\delta\,\mathrm{Re}\langle\mathsf{A_{\varepsilon}T}(\mathrm{Id}-\Pi)F,F\rangle-\delta\,\mathrm{Re}\langle\mathsf{A_{\varepsilon}L}F,F\rangle\,.\end{array}$
(41)

For a given $\xi\in{\mathbb{R}}^{d}$ considered as a parameter, if $f$ solves
(21) and if $F=\hat{f}$, then we are back to the framework of § II.1.3. In
this framework, the operator $\mathsf{A}_{\varepsilon}$ is given by

$(\mathsf{A}_{\varepsilon}F)(v)=-\frac{\,i\,\xi}{\varepsilon^{2}+|\xi|^{2}}\cdot\int_{{\mathbb{R}}^{d}}w\,F(w)\,dw\,\mathcal{M}(v)\,.$

We have to adapt the computations of § II.1.3 to $\varepsilon\neq 1$.

As a first remark, we notice that we do not need any estimate of
$\|\mathsf{A}_{\varepsilon}F\|$: all quantities in (41) involving
$\mathsf{A}_{\varepsilon}$ are directly computed except of
$\mathrm{Re}\langle\mathsf{TA_{\varepsilon}}F,F\rangle$. Estimating
$\mathrm{Re}\langle\mathsf{TA_{\varepsilon}}F,F\rangle$ provides a bound which
is independent of $\varepsilon$ for the following reason. When we solve
$G=\mathsf{A}_{\varepsilon}F$, _i.e._ , if
$(\mathsf{T}\Pi)^{*}F=\varepsilon^{2}\,G+(\mathsf{T}\Pi)^{*}\,\mathsf{T}\Pi\,G$,
then

$\langle\mathsf{TA_{\varepsilon}}F,F\rangle=\langle
G,(\mathsf{T}\Pi)^{*}\,F\rangle=\varepsilon^{2}\,\|G\|^{2}+\|\mathsf{T}\Pi
G\|^{2}=\varepsilon^{2}\,\|\mathsf{A}_{\varepsilon}F\|^{2}+\|\mathsf{TA}_{\varepsilon}F\|^{2}\,.$

By the Cauchy-Schwarz inequality, we know that

$\langle
G,(\mathsf{T}\Pi)^{*}F\rangle=\langle\mathsf{TA}_{\varepsilon}F,(\mathrm{Id}-\Pi)F\rangle\leq\|\mathsf{TA}_{\varepsilon}F\|\,\|(\mathrm{Id}-\Pi)F\|\,,$

which proves that $\|\mathsf{TA}_{\varepsilon}F\|\leq\|(\mathrm{Id}-\Pi)F\|$
and, as a consequence,

$|\mathrm{Re}\langle\mathsf{TA_{\varepsilon}}F,F\rangle|\leq\|(\mathrm{Id}-\Pi)F\|^{2}\,.$

It is clear that the right-hand side is independent of $\varepsilon>0$. A
better estimate is obtained by computing as in (37). By doing so, we obtain

$|\mathrm{Re}\langle\mathsf{TA_{\varepsilon}}F,F\rangle|\leq\frac{|\xi|^{2}}{\varepsilon^{2}+|\xi|^{2}}\,\|(\mathrm{Id}-\Pi)F\|^{2}\,.$

As in § II.1.3, we have $\lambda_{m}=1$, $\lambda_{M}=|\xi|^{2}$ and the same
computations show that

$\displaystyle|\mathrm{Re}\langle\mathsf{A}_{\varepsilon}F,F\rangle|\leq\frac{|\xi|}{\varepsilon^{2}+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\,,$
$\displaystyle|\mathrm{Re}\langle\mathsf{A_{\varepsilon}L}F,F\rangle|\leq\frac{|\xi|}{\varepsilon^{2}+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\,,$
$\displaystyle|\mathrm{Re}\langle\mathsf{A_{\varepsilon}T}(\mathrm{Id}-\Pi)F,F\rangle|\leq\frac{\sqrt{3}\,|\xi|^{2}}{\varepsilon^{2}+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\,.$

This establishes that (H4) holds with

$C_{M}=\frac{|\xi|\left(1+\sqrt{3}\,|\xi|\right)}{\varepsilon^{2}+|\xi|^{2}}\,,$
$\left|\mathsf{H}_{1,\varepsilon}[F]-\frac{1}{2}\,\|F\|^{2}\right|\leq\delta\,\left|\mathrm{Re}\langle\mathsf{A}_{\varepsilon}F,F\rangle\right|\leq\frac{\delta\,|\xi|}{\varepsilon^{2}+|\xi|^{2}}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\,,$

and as a consequence it yields an improved version of (7) which reads

$\frac{1}{2}\left(1-\frac{\delta\,|\xi|}{\varepsilon^{2}+|\xi|^{2}}\right)\|F\|^{2}\leq\mathsf{H}_{1,\varepsilon}[F]\leq\frac{1}{2}\left(1+\frac{\delta\,|\xi|}{\varepsilon^{2}+|\xi|^{2}}\right)\|F\|^{2}\,.$
(42)

Notice that the lower bound holds with a positive left-hand side for any $\xi$
only under the additional condition that

$\delta<2\,\varepsilon\,.$

Anyway, if we allow $\delta$ to depend on $\xi$, the whole method still
applies, including for proving the hypocoercive estimate on $\|F\|^{2}$, if
the condition

$\varepsilon^{2}-\delta\,|\xi|+|\xi|^{2}\geq 0$

is satisfied for every $\xi$.

With $X:=\|(\mathrm{Id}-\Pi)F\|$, $Y:=\|\Pi F\|$, and $s=|\xi|$, we look for
the largest value of $\lambda$ for which the right-hand side in

$\mathsf{D}_{\varepsilon}[F]-\lambda\,\mathsf{H}_{1,\varepsilon}[F]\\\
\geq\left(1-\frac{\delta\,s^{2}}{\varepsilon^{2}+s^{2}}-\frac{\lambda}{2}\right)X^{2}-\frac{\delta\,s}{\varepsilon^{2}+s^{2}}\left(1+\sqrt{3}\,s+\lambda\right)X\,Y+\left(\frac{\delta\,s^{2}}{\varepsilon^{2}+s^{2}}-\frac{\lambda}{2}\right)Y^{2}$
(43)

is nonnegative for any $X$ and $Y$. Recall that $s$ is fixed and $\delta$ is a
parameter to be adjusted. If we change the parameter $\delta$ into
$\delta_{*}$ such that

$\frac{\delta\,s}{\varepsilon^{2}+s^{2}}=\frac{\delta_{*}\,s}{1+s^{2}}\,,$
(44)

then the nonnegativity problem of the r.h.s. in (43) is reduced to the same
problem with $\varepsilon=1$, provided that no additional constraint is added.
Let us define

$h_{3}(\delta,\lambda,\varepsilon,s):=\delta^{2}\,s^{2}\left(\frac{1+\sqrt{3}\,s+\lambda}{\varepsilon^{2}+s^{2}}\right)^{2}-4\left(1-\frac{\delta\,s^{2}}{\varepsilon^{2}+s^{2}}-\frac{\lambda}{2}\right)\left(\frac{\delta\,s^{2}}{\varepsilon^{2}+s^{2}}-\frac{\lambda}{2}\right)$

with $(\delta,\lambda)$ in the triangle

$\mathcal{T}_{m}^{\varepsilon}(s):=\Big{\\{}(\delta,\lambda)\in\left(0,\big{(}1+\varepsilon^{2}\,s^{-2}\big{)}\,\lambda_{m}\right)\times(0,2\,\lambda_{m})\,:\,\lambda<2\left(\lambda_{m}-\tfrac{\delta\,s^{2}}{\varepsilon^{2}+s^{2}}\right)\Big{\\}}$

and $\lambda_{m}=1$. Exactly the same method as in § II.1.3 determines the
curves $s\mapsto\lambda_{3}(s)$ and $s\mapsto\delta_{3}(s,\varepsilon)$, but
we have $\lambda_{3}(s)=\lambda_{2}(s)$ for any $s>0$ while $\delta_{2}(s)$
and $\delta_{3}(s,\varepsilon)$ can be deduced from each other using (44).
Solutions have to satisfy the constraint
$s\mapsto\big{(}\delta_{3}(s,\varepsilon),\lambda_{3}(s)\big{)}\in\mathcal{T}_{m}^{\varepsilon}(s)$
for any $s>0$. It is straightforward to check that
$(\delta,\lambda)\in\mathcal{T}_{m}^{\varepsilon}(s)$ if and only if
$(\delta_{*},\lambda)\in\mathcal{T}_{m}^{1}(s)=\mathcal{T}_{m}(s)$, where
$\delta_{*}$ is determined by (44). Altogether, our observations can be
reformulated as follows.

###### Lemma 3

Assume (39). Then for any $s>0$, we have

$\max\Big{\\{}\lambda>0\,:\,(\delta,\lambda)\in\mathcal{T}_{m}^{\varepsilon}(s)\,,\;h_{3}(\delta,\lambda,\varepsilon,s)\leq
0\Big{\\}}$

is independent of $\varepsilon>0$.

To conclude this subsection, we note that, while the Lyapunov functionals
$\mathsf{H}_{1,\varepsilon}$ are clearly different for different values of
$\varepsilon>0$, mode-by-mode, _i.e._ , for a given value of $s=|\xi|$, they
all yield the same exponential decay rate $\lambda=\lambda_{2}(s)$, when
choosing the best parameter $\delta=\delta_{3}(s,\varepsilon)$. Similarly, no
improvement on the constant $C$ as in (5) is achieved by adjusting
$\varepsilon>0$ when $\varepsilon$ is taken into account in (42), for proving
the equivalence of $\mathsf{H}_{1,\varepsilon}[F]$ and $\|F\|^{2}$. This
reflects a deep scaling invariance of the method.

##  II.2 Convergence rates and decay rates

In this section, we come back to the study of (21) and consider two
situations. A periodic solution on a _small torus_ has a behaviour driven by
high frequencies corresponding to $|\xi|$ large, while the decay rate of a
solution on the whole Euclidean space is asymptotically determined by the low
frequency regime with $\xi\to 0$. In the latter case, we use estimates as in
Nash type inequalities and relate the time decay with the behaviour of
$\lambda(\xi)$ in a neighbourhood of $\xi=0$.

###  II.2.1 Exponential convergence rate on a small torus

In this section, let us assume that $\mathcal{X}=[0,L)^{d}$ (with periodic
boundary conditions) and consider the limit as $L\to 0_{+}$. With the notation
of § II.1.1 and the Fourier transform (22), the periodicity implies that
$\xi\in(2\,\pi/L)\,\mathbb{Z}^{d}$ and in particular, for any fixed
$j\in{\mathbb{Z}}^{d}$ and $\xi=2\,\pi\,j/L$, we have $|\xi|\to+\infty$ as
$L\to 0_{+}$, unless $j=0$. Let us denote by $\lambda_{L}(\xi)$ the optimal
constant in (30) when $\xi$ is limited to
$(2\,\pi/L)\,\mathbb{Z}^{d}\setminus\\{0\\}$. We recall that

$\lambda_{\star}:=\liminf_{L\to
0_{+}}\inf_{\xi\in(2\,\pi/L)\,\mathbb{Z}^{d}\setminus\\{0\\}}\lambda_{L}(\xi)\geq
1-\sqrt{3/7}$

according to Lemma 2. As a consequence, we have the following result.

###### Proposition 4

For any $\varepsilon>0$, small, there exists some $L_{\varepsilon}>0$ such
that, if $\mathcal{X}=[0,L)^{d}$ for an arbitrary $L\leq L_{\varepsilon}$, if
$f$ solves (21) with
$f_{0}\in\mathrm{L}^{2}(\mathcal{X}\times{\mathbb{R}}^{d},dx\,d\gamma)$ and
$\mathsf{L}=\mathsf{L}_{1}$ or $\mathsf{L}=\mathsf{L}_{2}$, then we have

$\|f(t,\cdot,\cdot)-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(dx\,d\gamma)}^{2}\leq(1+\varepsilon)\,\|f_{0}-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(dx\,d\gamma)}^{2}\,e^{-\min\\{2,\lambda_{\star}-\varepsilon\\!\\}\,t}\quad\forall\,t\geq
0\,,$

with
$\bar{f}:=\frac{1}{L^{d}}\iint_{\mathcal{X}\times{\mathbb{R}}^{d}}f_{0}(x,v)\,dx\,dv$.

###### Proof

Let us notice that $g(t,v):=\hat{f}(t,0,v)=\int_{\mathcal{X}}f(t,x,v)\,dx$
solves

$\partial_{t}g=\mathsf{L}g\,.$

As a consequence either of the definition of $\mathsf{L}=\mathsf{L}_{2}$, or
of the Gaussian Poincaré inequality

$\|g-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(d\gamma)}^{2}\leq\|\nabla
g\|_{\mathrm{L}^{2}(d\gamma)}^{2}$

if $\mathsf{L}=\mathsf{L}_{1}$, we know that

$\|g(t,\cdot)-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(d\gamma)}^{2}\leq\|g(0,\cdot)-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(d\gamma)}^{2}\,e^{-2t}\quad\forall\,t\geq
0\,.$

By the Plancherel formula, we have

$\|f(t,\cdot,\cdot)-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(dx\,d\gamma)}^{2}=\|g(t,\cdot)-\bar{f}\,\mathcal{M}\|_{\mathrm{L}^{2}(d\gamma)}^{2}+(2\,\pi)^{-d}\kern-18.0pt\sum_{\xi\in(2\,\pi/L)\,\mathbb{Z}^{d}\setminus\\{0\\}}\kern-18.0pt\|\hat{f}(t,\xi,\cdot)\|_{\mathrm{L}^{2}(d\gamma)}^{2}\,.$

The conclusion follows from
$C(s)=\left(1+s^{2}+\tilde{\delta}_{2}(s)\,s\right)/\left(1+s^{2}-\tilde{\delta}_{2}(s)\,s\right)$,

$\|\hat{f}(t,\xi,\cdot)\|_{\mathrm{L}^{2}(d\gamma)}^{2}\leq
C(|\xi|)\,\|\hat{f}_{0}(\xi,\cdot)\|_{\mathrm{L}^{2}(d\gamma)}^{2}\,e^{-\lambda_{L}(\xi)\,t}$

for any
$(t,\xi)\in{\mathbb{R}}^{+}\times(2\,\pi/L)\,\mathbb{Z}^{d}\setminus\\{0\\}$,
and the estimates of Lemma 2. $\square$

###  II.2.2 Algebraic decay rate in the whole Euclidean space

As a refinement of BDMMS , we investigate the decay estimates for the solution
to (21) on $\mathcal{X}={\mathbb{R}}^{d}$. Here we rely on _Nash type
estimates_.

To start with, let us consider a model problem. Assume that
$s\mapsto\lambda(s)$ is a positive non-decreasing bounded function on
$(0,+\infty)$ and, for any $s>0$, let

$h_{\lambda}(M,R,s):=\lambda(R)\left(\omega_{d}\,R^{d}\,M^{2}-s\right)\,,\quad\lambda^{*}(M,s):=-\min_{R>0}h_{\lambda}(M,R,s)\,,$

where $M$ is a positive parameter and $\omega_{d}=|\mathbb{S}^{d-1}|/d$. Since
$h_{\lambda}(M,R,s)\sim-\lambda(R)\,s$ as $R\to 0_{+}$ and
$h_{\lambda}(M,R,s)\geq c\,R^{d}$ for some $c>0$ as $R\to+\infty$, there is
indeed some $R>0$ such that $\lambda^{*}(M,s)=-\,h_{\lambda}(M,R,s)$ and
$\lambda^{*}(M,s)$ is positive for any $(M,s)\in(0,+\infty)^{2}$. We also
define the monotone decreasing function

$\psi_{\\!\lambda,M}(s):=-\int_{1}^{s}\frac{dz}{\lambda^{*}(M,z)}\quad\forall\,s\geq
0\,.$

Our first result is a decay rate on ${\mathbb{R}}^{d}$ for a solution of
$\partial_{t}u=\mathcal{L}u$ where the operator $\mathcal{L}$ acts on the
Fourier space as the multiplication of $\xi\mapsto\hat{u}(\xi)$ with some
scalar function $-\lambda(\xi)/2$, for any $\xi\in{\mathbb{R}}^{d}$. With just
a spectral inequality, we obtain the following estimate.

###### Lemma 4

Assume that $s\mapsto\lambda(s)$ is a positive non-decreasing bounded function
on $(0,+\infty)$ such that, with the above notation, $\lim_{s\to
0_{+}}\psi_{\\!\lambda,\mu}(s)=+\infty$ for all $\mu>0$. If $u\in
C({\mathbb{R}}^{+},\mathrm{L}^{1}\cap\mathrm{L}^{2}(dx))$ is such that
$M:=\|u(t,\cdot)\|_{\mathrm{L}^{1}(dx)}$ does not depend on $t$ and

$\frac{d}{dt}|\hat{u}(t,\xi)|^{2}\leq-\,\lambda(|\xi|)\,|\hat{u}(t,\xi)|^{2}\quad\forall\,(t,\xi)\in{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}\,,$

then

$\|u(t,\cdot)\|_{\mathrm{L}^{2}(dx)}^{2}\leq\psi_{\\!\lambda,M}^{-1}\left(t+\psi_{\\!\lambda,M}\left(\|u(0,\cdot)\|_{\mathrm{L}^{2}(dx)}^{2}\right)\right)\quad\forall\,t\in{\mathbb{R}}^{+}\,.$

Here $\hat{u}$ denotes the Fourier transform of $u$ in $x$.

###### Proof

The inspiration for the proof comes from (Nash58, , page 935). Let

$\displaystyle y(t):=\int_{{\mathbb{R}}^{d}}|\hat{u}(t,\xi)|^{2}\,d\xi$
$\displaystyle\leq\int_{|\xi|\leq
R}\kern-3.0pt\|\hat{u}(t,.)\|_{\mathrm{L}^{\infty}(d\xi)}^{2}\,d\xi+\frac{1}{\lambda(R)}\int_{{\mathbb{R}}^{d}}\lambda(|\xi|)\,|\hat{u}(t,\xi)|^{2}d\xi$
$\displaystyle\leq\omega_{d}\,R^{d}\,\|u(t,.)\|_{\mathrm{L}^{1}(dx)}^{2}-\frac{1}{\lambda(R)}\,\frac{d}{dt}\int_{{\mathbb{R}}^{d}}|\hat{u}(t,\xi)|^{2}\,d\xi\,.$

Hence,

$y^{\prime}\leq h_{\lambda}(M,R,y)\,,$

for any $R>0$. Taking the minimum of the r.h.s. over $R>0$, we obtain

$y^{\prime}\leq-\,\lambda^{*}(M,y)\,,$

and the conclusion follows after elementary computations. $\square$

###### Example 1

Let us consider a case that we have already encountered in Sections II.1.4 and
II.2.2. If $\lambda(s)=2\,s^{2}$ for $s\in(0,1)$, we find that

$\lambda^{*}(M,s)=2\,d\left(\frac{2}{\omega_{d}\,M^{2}}\right)^{\frac{2}{d}}\left(\frac{s}{d+2}\right)^{1+\frac{2}{d}}\quad\forall\,s\in(0,1)\,.$

With $c_{d}:=\frac{1}{2}\,\big{(}(d+2)/2\big{)}^{1+2/d}\,\omega_{d}^{2/d}$, we
find

$\psi_{\\!\lambda,M}(s)=c_{d}\,M^{\frac{4}{d}}\left(s^{-\frac{2}{d}}-1\right)$

and deduce from Lemma 4 that

$\|u(t,\cdot)\|_{\mathrm{L}^{2}(dx)}^{2}\leq\left(\|u(0,\cdot)\|_{\mathrm{L}^{2}(dx)}^{-\frac{4}{d}}+\frac{t}{c_{d}}\,\|u(0,\cdot)\|_{\mathrm{L}^{1}(dx)}^{-\frac{4}{d}}\right)^{-\frac{d}{2}}\quad\forall\,t\in{\mathbb{R}}^{+}\,.$

This estimate is similar to the estimate that one would deduce from Nash’s
inequality in the case of the heat equation on ${\mathbb{R}}^{d}$. Notice that
differentiating this estimate at $t=0$ gives a proof of Nash’s inequality,
with the same constant as in Nash’s proof in Nash58  (see Bouin_2020  for a
discussion of the optimal constant).

The assumption on $u$ in Lemma 4 is a coercivity estimate for the operator
$\mathcal{L}$ with Fourier representation $-\lambda(|\xi|)/2$, which allows us
to use a Bihari-LaSalle estimate, _i.e._ , a nonlinear version of Grönwall’s
lemma. For the main application in this paper, we have to rely on a
hypocoercivity estimate, which is slightly more complicated.

###### Lemma 5

Assume that $s\mapsto\lambda(s)$ is a positive non-decreasing bounded function
on $(0,+\infty)$ such that, with the above notation, $\lim_{s\to
0_{+}}\psi_{\\!\lambda,\mu}(s)=+\infty$ for all $\mu>0$. Let $u\in
C({\mathbb{R}}^{+},\mathrm{L}^{1}\cap\mathrm{L}^{2}(dx))$ be such that, for
some bounded continuous function $s\mapsto C(s)$ such that $C(s)\geq 1$ for
any $s>0$,

$|\hat{u}(t,\xi)|^{2}\leq
C(|\xi|)\,|\hat{u}(0,\xi)|^{2}\,e^{-\,\lambda(|\xi|)\,t}\quad\forall\,(t,\xi)\in{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}$

and $\|u(t,\cdot)\|_{\mathrm{L}^{1}(dx)}\leq M$ for some $M$ which does not
depend on $t$. Then, for any $t\geq 0$, we have

$\|u(t,\cdot)\|_{\mathrm{L}^{2}(dx)}^{2}\leq\Psi_{M,Q}(t)\,,$ (45)

where $Q:=\|u(0,\cdot)\|_{\mathrm{L}^{2}(dx)}$ and

$\Psi_{M,Q}(t):=\inf_{R>0}\left(\int_{0}^{R}C(s)\,e^{-\,\lambda(s)\,t}\,s^{d-1}\,ds\,\omega_{d}\,d\,M^{2}+\sup_{s\geq
R}C(s)\,e^{-\,\lambda(R)\,t}\,Q^{2}\right).$ (46)

###### Proof

For any $R>0$, we have

$\int_{|\xi|\leq R}|\hat{u}(t,\xi)|^{2}\,d\xi\leq\int_{|\xi|\leq
R}C(|\xi|)\,e^{-\,\lambda(|\xi|)\,t}\,d\xi\,\|\hat{u}(0,\cdot)\|_{\mathrm{L}^{\infty}({\mathbb{R}}^{d},d\xi)}^{2}$

with
$\|\hat{u}(0,\cdot)\|_{\mathrm{L}^{\infty}({\mathbb{R}}^{d},d\xi)}\leq\|u(0,\cdot)\|_{\mathrm{L}^{1}({\mathbb{R}}^{d},dx)}$
on the one hand, and

$\int_{|\xi|>R}|\hat{u}(t,\xi)|^{2}\,d\xi\leq\sup_{s>R}C(s)\,e^{-\,\lambda(R)\,t}\|\hat{u}(0,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},d\xi)}^{2}$

on the other hand. The result follows by optimizing on $R>0$. $\square$

The result of Lemma 5 is not as explicit as the result of Lemma 4, but it is
useful to investigate, for instance, the limit as $t\to+\infty$: if
$\lim_{s\to 0_{+}}C(s)=C(0)>0$ and $\lambda(s)=2\,s^{2}$ for any $s\in(0,1)$,
then one can prove that

$\|u(t,\cdot)\|_{\mathrm{L}^{2}(dx)}^{2}\leq
O\left(t^{-d/2}\right)\quad\mbox{as}\quad t\to+\infty\,.$

In the spirit of BDMMS , let us draw some consequences for the solution of
(21).

###### Theorem II.2.1

If $f$ solves (21) for some nonnegative initial datum
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},dx\,d\gamma)\cap\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d},d\gamma;\mathrm{L}^{1}({\mathbb{R}}^{d},dx)\big{)}$
and $\mathsf{L}=\mathsf{L}_{1}$ or $\mathsf{L}=\mathsf{L}_{2}$, then we have
the estimate

$\|f(t,\cdot,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},dx\,d\gamma)}^{2}\leq(2\,\pi)^{-d}\,\Psi_{M,Q}(t)$

with
$M=\|f_{0}\|_{\mathrm{L}^{2}\left({\mathbb{R}}^{d},d\gamma;\mathrm{L}^{1}({\mathbb{R}}^{d},dx)\right)}$,
$Q=\|f_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},dx\,d\gamma)}$,
and $\Psi_{M,Q}(t)$ defined by (46) using $C(s)=(2+\delta(s))/(2-\delta(s))$
and $\lambda(s)$, for any pair $(\delta,\lambda)$ of continuous functions on
$(0,+\infty)$ taking values in $(0,2)\times(0,+\infty)$, with
$s\mapsto\lambda(s)$ monotone non-decreasing, such that the _entropy – entropy
production inequality_ (30) and the equivalence (34) hold.

Here we abusively write $\lambda(\xi)=\lambda(s)$ and $\delta(\xi)=\lambda(s)$
with $s=|\xi|$.

###### Proof

We estimate
$\|f(t,\cdot,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},dx\,d\gamma)}^{2}$
using (22) and Plancherel’s theorem

$\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|f(t,x,v)|^{2}\,dx\,d\gamma=\frac{1}{(2\,\pi)^{d}}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|\hat{f}(t,\xi,v)|^{2}\,d\xi\,d\gamma\,.$

Applying the results of Theorem I.1.1 with

$C(\xi)=\frac{1+|\xi|^{2}+\delta(\xi)\,|\xi|}{1+|\xi|^{2}-\delta(\xi)\,|\xi|}\,,$

we learn that

$\int_{{\mathbb{R}}^{d}}|\hat{f}(t,\xi,v)|^{2}\,d\gamma\leq
C(\xi)\int_{{\mathbb{R}}^{d}}|\hat{f}_{0}(\xi,v)|^{2}\,d\gamma\,e^{-\lambda(\xi)\,t}\,.$

We can apply the same strategy as for Lemma 5, with

$\displaystyle\iint_{B_{R}\times{\mathbb{R}}^{d}}C(\xi)\,|\hat{f}_{0}(\xi,v)|^{2}\,e^{-\lambda(\xi)\,t}\,d\xi\,d\gamma\leq\int_{|\xi|\leq
R}C(\xi)\,e^{-\,\lambda(\xi)\,t}\,d\xi\,M^{2}\,,$
$\displaystyle\iint_{B_{R}^{c}\times{\mathbb{R}}^{d}}C(\xi)\,|\hat{f}_{0}(\xi,v)|^{2}\,e^{-\lambda(\xi)\,t}\,d\xi\,d\gamma\leq\sup_{\xi\in
B_{R}^{c}}C(\xi)\,e^{-\lambda(R)\,t}\,Q^{2}\,,$

using
$\sup_{\xi\in{\mathbb{R}}^{d}}|\hat{f}_{0}(\xi,v)|\leq\int_{{\mathbb{R}}^{d}}f_{0}(x,v)\,dx$
for the first inequality, and the monotonicity of $\lambda$. $\square$

In practice, any good estimate, for instance the estimate based on the
functions $(\tilde{\delta}_{2},\tilde{\lambda}_{2})$ of Proposition 3,
provides us with explicit and constructive decay rates of the solution to (21)
on ${\mathbb{R}}^{d}$. As a concluding remark, it has to be made clear that
the method is not limited to the operators $\mathsf{L}_{1}$ and
$\mathsf{L}_{2}$.

Part III. The Goldstein–Taylor model

##  III.1 General setting and Fourier decomposition

Consider the _two velocities Goldstein–Taylor (GT) model_ (_cf._ (DMS-2part, ,
§ 1.4)) with constant relaxation coefficient $\sigma>0$, position variable
$x\in\mathcal{X}\subseteq{\mathbb{R}}$, and $t>0$:

$\displaystyle\begin{aligned}
\partial_{t}f_{+}(t,x)+\partial_{x}f_{+}(t,x)&=\frac{\sigma}{2}\,\big{(}f_{-}(t,x)-f_{+}(t,x)\big{)}\,,\\\
\partial_{t}f_{-}(t,x)-\partial_{x}f_{-}(t,x)&=-\frac{\sigma}{2}\,\big{(}f_{-}(t,x)-f_{+}(t,x)\big{)}\,,\\\
f_{\pm}(x,0)&=f_{\pm,0}(x)\,,\end{aligned}$ (47)

where $f_{\pm}(t,x)$ are the density functions of finding a particle with a
velocity $\pm 1$ in a position $x$ at time $t>0$ and
$f_{\pm,0}\in\mathrm{L}^{1}_{+}(\mathcal{X})$ is the initial configuration.
This model is the prime example of discrete velocity BGK equations, as
described in § I.2, Example 2, with $b=(1/2,1/2)^{T}$ and
$V=\operatorname{diag}(1,-1)$. We consider two situations for $\mathcal{X}$,
the one-dimensional torus and the real line, _i.e._ ,
$\mathcal{X}\in\\{\mathbb{T},{\mathbb{R}}\\}$.

Rewriting (47) in the macroscopic variables of (mass and flux densities)

$u(t,x):=f_{+}(t,x)+f_{-}(t,x)\geq 0\,,\quad v(t,x):=f_{+}(t,x)-f_{-}(t,x)\,,$

leads to the transformed equations

$\displaystyle\begin{aligned} \partial_{t}u(t,x)&=-\,\partial_{x}v(t,x)\,,\\\
\partial_{t}v(t,x)&=-\,\partial_{x}u(t,x)-\sigma\,v(u,x)\,,\end{aligned}$ (48)

for $x\in\mathcal{X}$, $t\geq 0$. Integrating these equations along
$\mathcal{X}$ directly shows that the total mass is conserved for all times,
_i.e._ $\int_{\mathcal{X}}u(t,x)\,dx\equiv\int_{\mathcal{X}}u(x,0)\,dx$, and
that the total flux is decaying exponentially, _i.e._
$\int_{\mathcal{X}}v(t,x)\,dx=e^{-\sigma\,t}\int_{\mathcal{X}}v(x,0)\,dx$ for
$t\geq 0$.

A Fourier transformation in the space variable $x\in\mathcal{X}$ leads to ODEs
of form (10), given explicitly as

$\partial_{t}\hat{y}(t,\xi)=-C(\xi,\sigma)\,\hat{y}(t,\xi)$ (49)

with

$\hat{y}(t,\xi):=\begin{pmatrix}\hat{u}(t,\xi)\\\
\hat{v}(t,\xi)\end{pmatrix}\quad\mbox{and}\quad
C(\xi,\sigma):=\begin{pmatrix}0&i\,\xi\\\ i\,\xi&\sigma\end{pmatrix}\,,$

for the Fourier modes $\xi\in{\mathbb{Z}}$ in the case of
$\mathcal{X}=\mathbb{T}$, and $\xi\in{\mathbb{R}}$ for
$\mathcal{X}={\mathbb{R}}$.

The matrix $C(\xi,\sigma)$ from (49) has the eigenvalues

$\lambda_{\pm}(\xi,\sigma):=\frac{\sigma}{2}\pm\sqrt{\frac{\sigma^{2}}{4}-\xi^{2}}$

and hence its _modal spectral gap_ is given by

$\mu(\xi,\sigma):=\mathrm{Re}\left({\frac{\sigma}{2}-\sqrt{\frac{\sigma^{2}}{4}-\xi^{2}}}\right)\,,\quad\xi\neq
0\,.$ (50)

For $\mathcal{X}={\mathbb{R}}$, the modal spectral gap takes all values in the
intervall $(0,\sigma/2]$ with $\lim_{\xi\to 0}\mu(\xi,\sigma)=0$. To obtain
decay estimates with the sharp decay rate of solutions $y(t,\xi)$ to (48) it
is therefore important to achieve precise estimates of the decay behavior as
$\xi\to 0$. For $\mathcal{X}=\mathbb{T}$, the spectral gap for solutions to
(48) corresponds to the _uniform-in- ${\mathbb{Z}}$ spectral gap_, _i.e._

$\overline{\mu}(\sigma):=\min_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\mu(\xi,\sigma)\,.$

The set of modal spectral gaps which coincide with the uniform spectral gap is
denoted by $\Xi(\sigma)$ and depends on the values of $\sigma>0$:

  * •

For $\sigma\in(0,2]$ it follows that

$\overline{\mu}(\sigma)=\frac{\sigma}{2}\,,\quad\Xi(\sigma)={\mathbb{Z}}\setminus\\{0\\}\,.$

  * •

For $\sigma>2$ the lowest modes determine the uniform-in-${\mathbb{Z}}$
spectral gap,

$\overline{\mu}(\sigma)=\mu(\pm
1,\sigma)=\frac{\sigma}{2}-\sqrt{\frac{\sigma^{2}}{4}-1}\,,\quad\Xi=\\{-1,1\\}\,.$
(51)

Now, we consider the two hypocoercivity methods from § I.1 and § I.2 for
solutions $\hat{y}(t,\xi)$ of (49) for fixed but arbitrary modes $\xi$.

Approach of § I.2: For equations of form (10), we consider the modal Lyapunov
functionals $\|\hat{y}(t,\xi)\|^{2}_{P(\xi,\sigma)}$ with deformation matrices
$P(\xi,\sigma)$. These functionals satisfy the explicit estimates of form
(16), which go as follows:

  * •

For fixed $|\xi|\neq\sigma/2$, $|\xi|>0$ the matrix $C(\xi,\sigma)$ is not
defective and it follows from Lemma 1 that

$\|\hat{y}(t,\xi)\|^{2}_{P(\xi,\sigma)}\leq
e^{-2\,\mu(\xi,\sigma)\,t}\,\|\hat{y}(\xi,0)\|^{2}_{P(\xi,\sigma)}\,,$ (52)

with $P(\xi,\sigma)=P^{(1)}(\xi,\sigma)$ for $|\xi|>\sigma/2$ and
$P(\xi,\sigma)=P^{(2)}(\xi,\sigma)$ for $|\xi|<\sigma/2$, where

$P^{(1)}(\xi,\sigma):=\begin{pmatrix}1&-\frac{i\,\sigma}{2\,\xi}\\\
\frac{i\,\sigma}{2\,\xi}&1\end{pmatrix}\,,\quad
P^{(2)}(\xi,\sigma):=\begin{pmatrix}1&-\frac{2\,i\,\xi}{\sigma}\\\
\frac{2\,i\,\xi}{\sigma}&1\end{pmatrix}\,.$ (53)

  * •

For $|\xi|=\sigma/2$ the matrix $C(\xi,\sigma)$ is defective. Then, due to
(Arnold2014, , Lemma 4.3), for any $\varepsilon>0$ there exists an
$\varepsilon$-dependent matrix that yields the purely exponential decay
$\mu(\sigma/2)-\varepsilon$. For later purposes it will be sufficient to
investigate the case $\sigma=2$ with $\xi=1$, see § III.2.2. Hence, we will
not state the general form here.

Approach of § I.1: With notation from Theorem I.1.1, the Goldstein–Taylor
equation in Fourier modes (49) can be written as

$\partial_{t}\hat{y}(t,\xi)=\big{(}\mathsf{L}(\sigma)-\mathsf{T}(\xi)\big{)}\,\hat{y}(t,\xi)\,.$

The Hermitian collision matrix and the anti-Hermitian transport matrix are,
respectively, given as

$\mathsf{L}(\sigma):=\begin{pmatrix}0&0\\\
0&-\,\sigma\end{pmatrix}\,,\quad\mathsf{T}(\xi):=\begin{pmatrix}0&i\,\xi\\\
i\,\xi&0\end{pmatrix}\,.$

The projection on the space of local-in-$x$ equilibria (satisfying
$\mathsf{L}(\sigma)\Pi=0$) is given by the matrix

$\Pi:=\begin{pmatrix}1~{}&0\\\ 0~{}&0\end{pmatrix}\,.$

We introduce the operator $\mathsf{A}(\xi)$ as in (2) for each mode $\xi$:

$\mathsf{A}(\xi):=\Big{(}\mathrm{Id}+\big{(}\mathsf{T}(\xi)\,\Pi\big{)}^{*}\,\mathsf{T}(\xi)\,\Pi\Big{)}^{-1}\,\big{(}\mathsf{T}(\xi)\,\Pi\big{)}^{*}=\begin{pmatrix}0&-\frac{i\,\xi}{1+\xi^{2}}\\\
0&0\end{pmatrix}\,.$

The modal Lyapunov functional (3) is given as

$\displaystyle\mathsf{H}_{1}(\xi,\delta)[\hat{y}(\xi)]$
$\displaystyle:=\frac{1}{2}\,\|\hat{y}(\xi)\|^{2}+\delta\,\mathrm{Re}\big{(}\hat{y}(\xi)^{*}\,\mathsf{A}(\xi)\,\hat{y}(\xi)\big{)}$
$\displaystyle=\frac{1}{2}\,\|\hat{y}(\xi)\|^{2}+\delta\hat{y}(\xi)^{*}\mathsf{A}_{H}(\xi)\,\hat{y}(\xi)$
$\displaystyle=\frac{1}{2}\,\hat{y}(\xi)^{*}\begin{pmatrix}1&-\frac{i\,\xi\,\delta}{1+\xi^{2}}\\\
\frac{i\,\xi\,\delta}{1+\xi^{2}}&1\end{pmatrix}\,\hat{y}(\xi)\,,$ (54)

where we denote the Hermitian part of the matrix $\mathsf{A}$ by
$\mathsf{A}_{H}:=\frac{1}{2}(\mathsf{A}+\mathsf{A}^{*})$.

##  III.2 Comparison of the two hypocoercivity methods for
$\mathcal{X}=\mathbb{T}$

In the next step we shall assemble, for both hypocoercivity methods, the modal
Lyapunov functionals to form a global one. When appropriately optimizing both
of these functionals, we shall see that they actually coincide and achieve
optimal decay estimates in the class of all quadratic forms.

###  III.2.1 The optimal global Lyapunov functional

We start by applying the strategies outlined in § I.2 to assemble a global
$\mathrm{L}^{2}(\mathbb{T})$ functional. For simplicity, let us first assume
that the matrix $C(\xi,\sigma)$, $\xi\in{\mathbb{Z}}$ of (49) is
diagonalizable for all modes, _i.e._ $\sigma\not\in 2\,{\mathbb{Z}}$. A brief
discussion of the defective cases is deferred to the end of this section.

We first consider Strategy 1 of § I.2 that leads to the functional

$\mathsf{H}_{2}[y]:=\sum_{|\xi|>\sigma/2}\,\|\hat{y}(\xi)\|^{2}_{P^{(1)}(\xi)}+\sum_{|\xi|<\sigma/2}\,\|\hat{y}(\xi)\|^{2}_{P^{(2)}(\xi)}\,,\quad
y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}\,,$ (55)

according to definition (17). Assuming that the system has total mass $0$,
_i.e._
$\int_{\mathbb{T}}u(x,0)\,dx=0$, we obtain that solutions $y(t)$ of (48), (49)
satisfy the estimate:

$\|y(t)\|^{2}\leq\overline{c}_{P}\,e^{-2\,\overline{\mu}(\sigma)\,t}\,\|y(0)\|^{2}\,,$
(56)

where

$\bar{c}_{P}:=\max\left\\{\sup_{|\xi|>\sigma/2}\left[\mbox{cond}\left(P^{(1)}(\xi)\right)\right],\sup_{|\xi|<\sigma/2}\left[\mbox{cond}\left(P^{(2)}(\xi)\right)\right]\right\\}\,.$
(57)

To improve upon the multiplicative constant $\overline{c}_{P}$ in (56), we
continue with Strategy 2 of § I.2.

  * •

For the case $\sigma<2$ the functional $\mathsf{H}_{2}$ and (57) directly
yield the optimal multiplicative constant. With notation from § I.2 this
follows from $\Xi={\mathbb{Z}}\setminus\\{0\\}$ and
$\overline{c}_{P}=c_{\Xi}=\mbox{cond}\big{(}P^{(1)}(\pm
1)\big{)}=(2+\sigma)/(2-\sigma)$. In this case the two eigenvalues of the ODE
system matrix $C(\xi)$ are distinct and form a complex conjugate pair. Hence,
the multiplicative constant $\overline{c}_{P}$ is the optimal constant within
the family of form (56), as has been shown in (AAS19, , Theorem 3.7).

  * •

For the case $\sigma>2$, $\sigma\not\in 2\,{\mathbb{Z}}$, the lowest modes
have the slowest decay: $\Xi=\\{-1,1\\}$ with
$c_{\Xi}=\mbox{cond}\big{(}P^{(2)}(\pm 1)\big{)}=(\sigma+2)/(\sigma-2)$. The
multiplicative constant $c_{\Xi}$ is not the smallest possible multiplicative
constant in (56). However, according to (AAS19, , Theorem 4.1) it is the best
possible multiplicative constant achievable by Lyapunov functionals that are
quadratic forms. As $\sigma>2$ it follows that $\bar{c}_{P}>c_{\Xi}$, and
hence we replace the functionals $\|\cdot\|^{2}_{\kern 2.0ptP^{(1)}(\xi)}$ and
$\|\cdot\|^{2}_{\kern 2.0ptP^{(2)}(\xi)}$ for the faster decaying modes
$\xi\not\in\Xi$, $\xi\neq 0$. Let us define

$\overline{P}^{(1)}(\xi):=\begin{pmatrix}1&-\frac{2\,i}{\sigma\xi}\\\
\frac{2\,i}{\sigma\xi}&1\end{pmatrix},\quad\xi\not\in\Xi,\xi\neq 0,$

and notice that $\overline{P}^{(1)}(\xi)$ satisfies the matrix inequality

$C^{*}(\xi)\,\overline{P}^{(1)}(\xi)+\overline{P}^{(1)}(\xi)\,C(\xi)\geq
2\,\overline{\mu}\,\overline{P}^{(1)}(\xi)\,,\quad\xi\not\in\Xi\,,\;\xi\neq
0\,,$ (58)

where $\overline{\mu}$ is the explicitly given uniform-in-${\mathbb{Z}}$
spectral gap (51). Furthermore, as
$\mbox{cond}\big{(}\overline{P}^{(1)}(\xi)\big{)}\leq\mbox{cond}\big{(}\overline{P}^{(1)}(\pm
1)\big{)}=(\sigma+2)/(\sigma-2)$, it satisfies the estimate
$\mbox{cond}\big{(}\overline{P}^{(1)}(\xi)\big{)}\leq c_{\Xi}(\sigma)$. Thus,
the choice $\|\cdot\|_{\kern 2.0pt\overline{P}^{(1)}(\xi)}$ for
$\xi\not\in\Xi(\sigma)$ and $\xi\neq 0$ leads (via (19)) to the global
functional for $y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}$, given as

$\widetilde{\mathsf{H}}_{2}[y]:=\sum_{\xi\in\Xi}\|\hat{y}(\xi)\|_{\kern
2.0ptP^{(2)}(\xi)}^{2}+\sum_{\xi\not\in\Xi,\xi\neq 0}\|\hat{y}(\xi)\|_{\kern
2.0pt\overline{P}^{(1)}(\xi)}^{2}=\sum_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\|\hat{y}(\xi)\|_{\kern
2.0pt\overline{P}^{(1)}(\xi)}^{2}\,,$

where the equality follows as $P^{(2)}(\pm 1)=\overline{P}^{(1)}(\pm 1)$.
$\widetilde{\mathsf{H}}_{2}$ yields decay with sharp rate
$2\,\overline{\mu}(\sigma)$ given by (51) and, within the family of quadratic
forms, the optimal multiplicative constant $c_{\Xi}(\sigma)$ in (56).

In summary, for arbitrary $\sigma>0$, $\sigma\not\in 2\,{\mathbb{Z}}$,
Strategy 2 of § I.2 yields the global Lyapunov functional

$\widetilde{\mathsf{H}}_{2}(\sigma)[y]:=\sum_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\|\hat{y}(\xi)\|_{\kern
2.0pt\overline{P}(\xi,\theta(\sigma))}^{2}\,,\quad
y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}\,,$ (59)

where

$\overline{P}(\xi,\theta):=\begin{pmatrix}1&-\frac{i\,\theta}{2\,\xi}\\\
\frac{i\,\theta}{2\,\xi}&1\end{pmatrix}\,,\quad\theta\left(\sigma\right):=\begin{cases}\sigma\,,&0<\sigma<2\,,\\\
\frac{4}{\sigma}\,,&\sigma>2\,.\end{cases}$ (60)

Next, we turn to the method of § I.1 and derive another global
$\mathrm{L}^{2}(\mathbb{T})$ functional that is based on the modal functionals
(54):

$\mathsf{H}_{1}(\delta)[y]:=\sum_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\mathsf{H}_{1}(\xi,\delta)[\hat{y}(\xi)]\,,\quad
y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}\,.$

In (DMS-2part, , § 1.4) the parameter $\delta\in(0,2)$ was chosen independent
of $\xi$. But optimizing the resulting decay rate of $\mathsf{H}_{1}(\delta)$
w.r.t. the parameter $\delta\in(0,2)$ yields non-sharp decay rates (as derived
in (DMS-2part, , § 1.4) for $\lambda_{m}=\sigma=1$). Hence, we shall optimize
here each modal functional $\mathsf{H}_{1}(\xi,\delta(\xi))$ w.r.t. the
parameter $\delta$.

For $y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}$ and arbitrary $\sigma>0$,
$\sigma\not\in 2\,{\mathbb{Z}}$, the resulting functional is given as

$\widetilde{\mathsf{H}}_{1}(\sigma)[y]:=2\sum_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\mathsf{H}_{1}\left(\xi,\overline{\delta}(\xi,\sigma)\right)[\hat{y}(\xi)]\,,$
(61)

with the optimal parameter
$\overline{\delta}(\xi,\sigma):=\frac{\theta(\sigma)\,(1+\xi^{2})}{2\,\xi^{2}}\in(0,2)$
and $\theta(\sigma)$ defined in (60). The following theorem relates
$\widetilde{\mathsf{H}}_{1}$ to the previously defined functional
$\widetilde{\mathsf{H}}_{2}$, given respectively by (61) and (59).

###### Theorem III.2.1

For $y\in(\mathrm{L}^{2}(\mathbb{T}))^{2}$ and arbitrary $\sigma>0$,
$\sigma\not\in 2\,{\mathbb{Z}}$ it follows that

$\widetilde{\mathsf{H}}_{1}(\sigma)[y]=\widetilde{\mathsf{H}}_{2}(\sigma)[y]\,.$

###### Proof

Thanks to previous considerations, the proof is now straightfoward. For each
mode $\xi\in{\mathbb{Z}}\setminus\\{0\\}$, the identity

$2\,\mathsf{H}_{1}\left(\xi,\overline{\delta}(\xi,\sigma)\right)[\hat{y}]=\|\hat{y}\|_{\kern
2.0pt\overline{P}(\xi,\theta(\sigma))}^{2}\,,\quad\hat{y}\in{\mathbb{C}}^{2}$
(62)

follows by setting $\delta=\overline{\delta}(\xi,\theta)$ in (54).

###  III.2.2 The defective cases

For $\sigma\neq 2$ the defective modes $|\xi|=\sigma/2$ do not exhibit the
slowest decay of all modes, _i.e._ $\xi\not\in\Xi$ as defined in § I.2. The
functional $\|\cdot\|^{2}_{\kern 2.0pt\overline{P}^{(1)}(\xi)}$ yields the
sufficient decay rate $2\,\bar{\mu}(\sigma)$, along with multiplicative
constants that are small enough, _i.e._ ,
$\mbox{cond}\big{(}\overline{P}^{(1)}(\xi)\big{)}\leq c_{\Xi}(\sigma)$. It
follows that Strategy 2 of § I.2 again yields the functional
$\widetilde{\mathsf{H}}_{2}$ as defined in (59).

The case $\sigma=2$ is the only case where the defective modes correspond to
the slowest modal decay, _i.e._ $\xi=\pm 1\in\Xi$. Then, for arbitrarily small
$\varepsilon>0$ the modified norm $\|\cdot\|^{2}_{\kern
2.0pt\overline{P}(\xi,\theta_{\varepsilon})}$, defined in (60), with

$\theta_{\varepsilon}:=2\,\frac{2-\varepsilon^{2}}{2+\varepsilon^{2}}\,,$ (63)

yields the exponential decay rate
$2\,\big{(}\overline{\mu}(2)-\varepsilon\big{)}$ for all modes $|\xi|\neq 0$.
Due to the lack of an eigenvector basis in the defective case, constructing
the matrix $\overline{P}(\xi,\theta_{\varepsilon})$ results in a decay
estimate of form (56) with multiplicative constant
$c_{\varepsilon}=\sqrt{2}/\varepsilon$. The blow-up $\lim_{\varepsilon\to
0_{+}}c_{\varepsilon}=+\infty$ reflects the fact that the true decay behaviour
of solutions in this defective setting is not purely exponential with rate
$2\,\overline{\mu}(2)$, but rather exponential times a polynomial in time $t$.
An approach based on more involved time-dependent Lyapunov functionals yields
estimates with the sharp defective decay behaviour. As the time-dependent
construction is besides our focus, we simply refer to AJW  for further
details.

###  III.2.3 Decay results for the case $\mathcal{X}=\mathbb{T}$

In this subsection we start by refining the general strategy of § II.1.3 to
extract the sharp decay rate for the GT model from the functional
$\widetilde{\mathsf{H}}_{1}$. Subsequently, we conclude the torus case by
expressing the global Lyapunov functional in the spatial variable.

In Theorem III.2.1 above, we establish that both functional constructions (as
described in § I) coincide for the GT equation if one chooses the appropriate
parameter $\delta(\xi)$ for $\widetilde{\mathsf{H}}_{1}$. Now, we compare both
approaches of § I to extract explicit decay rates from the functional.

The Strategy 2 of § I.2 for $\widetilde{\mathsf{H}}_{2}$ is based on modal
matrix inequalities, (58), which prove the sharp global decay rate
$2\,\overline{\mu}$ as already discussed in § III.2.1.

The general method of § I.1 for $\widetilde{\mathsf{H}}_{1}$ (and its
improvements of § II.1) is to estimate the entropy – entropy production
inequality $\mathrm{D}[y]-\lambda\,\widetilde{\mathsf{H}}_{1}[y]\geq 0$ in
terms of $\|(\mathrm{Id}-\Pi)\,y\|$ and $\|\Pi y\|$ that are then optimized
for $\lambda$. As assumed in § II.1, we restrict our discussion to
$\lambda_{m}=1$ which requires the relaxation rate $\sigma=1$. Applying
Proposition 3 to the modal equation (49) for $\xi=\pm 1$ yields the decay
estimates

$\mathsf{H}_{1}\left(\pm 1,\tilde{\delta}_{2}(1)\right)[\hat{y}(\pm 1,t)]\leq
e^{-\tilde{\lambda}(1)\,t}\,\mathsf{H}_{1}\left(\pm
1,\tilde{\delta}_{2}(1)\right)[\hat{y}(\pm 1,0)]$

with non-optimal modal decay rate $\tilde{\lambda}(1)\approx 0.165$ and
parameter $\tilde{\delta}_{2}(1)\approx 0.325$. Higher modes, $|\xi|>1$, yield
higher decay rates (_cf._ Lemma 2), but as

$\inf_{\xi\in{\mathbb{Z}}\setminus\\{0\\}}\tilde{\lambda}(|\xi|)=\tilde{\lambda}(1)<2\,\overline{\mu}(1)=1\,,$

the optimal global rate cannot be recovered. One cause for not reaching the
sharp rate is that Proposition 3 approximates the entropy - entropy production
inequality condition to obtain readable formulas (via the discriminant
$\tilde{h}_{2}$ as defined in (40)). But even omitting approximations when
optimizing $\delta$ does not yield sharp decay rates $\lambda(|\xi|)=1$ for
our example.

This is not surprising as § II.1.4 provides explicit estimates with a general
hypocoercive setting in mind. In order to obtain sharp decay rates for the GT
model, we sacrifice this generality and refine the strategy for the simple
structure at hand. The reduction of the continuous velocity space
$v\in{\mathbb{R}}$ (as defined in § II.1.1) to two discrete velocities in the
GT setting allows the following modifications: With the notation of § III.1 it
holds that

$\mathsf{A}(\xi)\,\mathsf{T}(\xi)(\mathrm{Id}-\Pi)\hat{y}=0\,,\quad\hat{y}\in{\mathbb{C}}^{2}\,.$

Thus, the constant $C_{M}$ as defined in (39) improves to
$C_{M}=\frac{|\xi|}{1+|\xi|^{2}}$. Additionally, as

$\mathsf{A}(\xi)\,(\mathsf{L}+\lambda\,\mathrm{Id})=\begin{pmatrix}0&\frac{i\,\xi\,(1-\lambda)}{1+\xi^{2}}\\\
0&0\end{pmatrix}\,,$

it follows that

$\Big{|}\mathrm{Re}\big{\langle}\mathsf{A}(\xi)\,(\mathsf{L}+\lambda\,\mathrm{Id})\hat{y}(\xi)\,,\hat{y}(\xi)\big{\rangle}\Big{|}\leq\frac{|\xi|}{1+|\xi|^{2}}\,(1-\lambda)\,X\,Y\,,$

for $0\leq\lambda\leq 1$, where $X:=\|(\mathrm{Id}-\Pi)\hat{y}\|=\|v\|$ and
$Y:=\|\Pi\hat{y}\|=\|u\|.$ Then, as a refinement of (II.1.3) for the GT
equation with $\mathsf{H}_{1}(\xi,\delta)$ from (54) it follows that

$\mathsf{D}(\xi,\delta)[\hat{y}]-\lambda\,\mathsf{H}_{1}(\xi,\delta)[\hat{y}]\\\
\geq\left(1-\frac{\delta\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right)\,X^{2}-\delta\,\mathrm{Re}\langle\mathsf{A}\,(\mathsf{L}-\lambda\,\mathrm{Id})\,F,F\rangle+\left(\frac{\lambda\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right)\,Y^{2}\\\
\geq\left(1-\frac{\delta\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right)\,X^{2}-\frac{\delta\,|\xi|\,(1-\lambda)}{1+\xi^{2}}\,X\,Y+\left(\frac{\lambda\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right)\,Y^{2}\,.$

The _refined discriminant condition_ is then given by the non-positivity of

$h_{\textrm{GT}}(\delta,\lambda):=\frac{\delta^{2}\,\xi^{2}}{(1+\xi^{2})^{2}}\,(1-\lambda)^{2}-4\left(1-\frac{\delta\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right)\left(\frac{\delta\,\xi^{2}}{1+\xi^{2}}-\frac{\lambda}{2}\right).$

It can be verified directly that
$\overline{\delta}(\xi):=\frac{1+\xi^{2}}{2\,\xi^{2}}$ for $\xi\neq 0$ yields
$h_{\textrm{GT}}\big{(}\overline{\delta}(\xi),1\big{)}=0$. Hence we recover
the sharp exponential decay rate $\lambda(|\xi|)=2\,\overline{\mu}(1)=1$ for
the modal equations (49) for all $\xi\in{\mathbb{Z}}\setminus\\{0\\}$ with
$\sigma=1$.

With this we have shown that refining the method of § II.1.3 for the GT model
(with $\sigma=1$) allows us to recover the sharp global decay rate from the
global functional $\widetilde{\mathsf{H}}_{1}$, as defined in (61).

In § III.2 above we show that both hypocoercive methods from § I lead to the
same global Lyapunov functional for arbitrary $\sigma>0$. We conclude this
subsection by leaving the modal formulation behind and expressing this global
functional in the spatial variable.

In AESW  the authors define an explicit spatial Lyapunov functional that
yields the sharp, purely exponential decay rates and best possible
multiplicative constant (reachable via quadratic forms) for each $\sigma>0$:

###### Definition 1

Let $u$, $v\in\mathrm{L}^{2}\left(\mathbb{T}\right)$ be real-valued and let
$\theta\in(0,2)$ be given. We then define the functional
$\mathsf{E}_{\theta}[u,v]$ as

$\mathsf{E}_{\theta}[u,v]:=\lVert
u\rVert^{2}_{\mathrm{L}^{2}(\mathbb{T})}+\lVert
v\rVert^{2}_{\mathrm{L}^{2}(\mathbb{T})}-\frac{\theta}{2\pi}\int_{0}^{2\pi}v\,\partial_{x}^{-1}u\,dx\,.$

Here, the anti-derivative of $u$ is defined as

$\partial_{x}^{-1}u(x):=\int_{0}^{x}u\,dy-\left(\int_{0}^{x}u(y)\,dy\right)_{\mathrm{avg}}\,,$
(64)

where
$u_{\operatorname{{\mathrm{avg}}}}:=\frac{1}{2\pi}\int_{0}^{2\pi}u\,dx=\hat{u}(0).$

###### Theorem III.2.2

For $u$, $v\in\mathrm{L}^{2}(\mathbb{T})$ and arbitrary $\sigma>0$,
$\sigma\not\in 2\,{\mathbb{Z}}$ it follows that

$\widetilde{\mathsf{H}}_{1}(\sigma)\left[u-u_{\operatorname{{\mathrm{avg}}}},v\right]=\mathsf{E}_{\theta(\sigma)}\left[u-u_{\operatorname{{\mathrm{avg}}}},v\right]\,,$
(65)

with $\widetilde{\mathsf{H}}_{1}$ defined in (61).

###### Proof

As in (AESW, , § 4.3), we use Parseval’s identity and the fact that
$(i\,k)^{-1}$ is the (discrete) Fourier symbol of $\partial_{x}^{-1}$ as
defined in (64). For the total entropy of arbitrary
$y:=(u,v)^{T}\in(\mathrm{L}^{2}(\mathbb{T}))^{2}$, with
$u_{\operatorname{{\mathrm{avg}}}}=0$, we deduce from (62) that

$\displaystyle\widetilde{\mathsf{H}}_{1}(\sigma)[y]+\|\hat{v}(0)\|^{2}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\|\hat{y}(\xi)\|_{\kern
2.0pt\overline{P}(\xi,\theta(\sigma))}^{2}+\|\hat{v}(0)\|^{2}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\Big{(}|u|^{2}+|v|^{2}-\theta(\sigma)\,v\,\partial_{x}^{-1}u\Big{)}\,dx$
$\displaystyle=\mathsf{E}_{\theta(\sigma)}[u,v]\,.$

In the following result we recall from (AESW, , Theorem 2.2.a) the optimal
exponential decay for $y(t)$ to the steady state
$y_{\infty}=(u_{\operatorname{{\mathrm{avg}}}},0)^{T}$, both in the functional
$\mathsf{E}_{\theta}$ and in the Euclidean norm. Mild solution refers to the
terminology of semigroup theory Pa92 .

###### Theorem III.2.3

Let $(u,v)\in C([0,\infty);(\mathrm{L}^{2}\left(\mathbb{T}\right))^{2})$ be a
mild real valued solution to (48) with initial datum $u_{0}$,
$v_{0}\in\mathrm{L}^{2}\left(\mathbb{T}\right)$ and define
$u_{\mathrm{avg}}:=\frac{1}{2\pi}\int_{0}^{2\pi}u_{0}(x)\,dx$.
$\bullet$ If $\sigma\not=2$ then

$\mathsf{E}_{\theta(\sigma)}[u(t)-u_{\mathrm{avg}},v(t)]\leq\mathsf{E}_{\theta(\sigma)}[u_{0}-u_{\mathrm{avg}},v_{0}]\,e^{-2\,\mu\left(\sigma\right)\,t}\quad\forall\,t\geq
0\,,$

where

$\theta\left(\sigma\right):=\begin{cases}\sigma\,,&0<\sigma<2\\\
\frac{4}{\sigma}\,,&\sigma>2\end{cases}\,,\quad\mu\left(\sigma\right):=\begin{cases}\frac{\sigma}{2}\,,&0<\sigma<2\\\
\frac{\sigma}{2}-\sqrt{\frac{\sigma^{2}}{4}-1}\,,&\sigma>2\end{cases}\,.$

Consequently we obtain the decay estimate

$\begin{gathered}\left\|f(t)-\binom{f_{\infty}}{f_{\infty}}\right\|_{\mathrm{L}^{2}(\mathbb{T})}\leq\mathscr{C}_{\sigma}\left\|f_{0}-\binom{f_{\infty}}{f_{\infty}}\right\|_{\mathrm{L}^{2}(\mathbb{T})}\,e^{-\mu\left(\sigma\right)\,t}\quad\forall\,t\geq
0\,,\end{gathered}$

where the decay rate $\mu(\sigma)$ is sharp and

$\mathscr{C}_{\sigma}:=\sqrt{\frac{2+\sigma}{|2-\sigma|}}\,,\quad
f(t):=\binom{f_{+}(t)}{f_{-}(t)}\,,\quad
f_{\infty}=\frac{1}{2}\,u_{\mathrm{avg}}\,,\quad
f_{0}:=\binom{f_{+,0}}{f_{-,0}}\,.$

$\bullet$ If $\sigma=2$ then for any $0<\varepsilon<1$

$\mathsf{E}_{\theta_{\varepsilon}}[u(t)-u_{\mathrm{avg}},v(t)]\leq\mathsf{E}_{\theta_{\varepsilon}}[u_{0}-u_{\mathrm{avg}},v_{0}]\,e^{-2\,\left(1-\varepsilon\right)\,t}\quad\forall\,t\geq
0$

with $\theta_{\varepsilon}$ defined as in (63) and we have that

$\begin{gathered}\left\|f(t)-\binom{f_{\infty}}{f_{\infty}}\right\|_{\mathrm{L}^{2}(\mathbb{T})}\leq\frac{\sqrt{2}}{\varepsilon}\,\left\|f_{0}-\binom{f_{\infty}}{f_{\infty}}\right\|_{\mathrm{L}^{2}(\mathbb{T})}\,e^{-\left(1-\varepsilon\right)\,t}\quad\forall\,t\geq
0\,.\end{gathered}$

##  III.3 Decay results for the case $\mathcal{X}={\mathbb{R}}$

We consider the GT model with position $x$ on the real line and prove two
global decay estimates with sharp algebraic rate. Our first goal is to obtain
modal decay estimates of general form (45) with modal constants $C(|\xi|)$ as
small as possible. As we discuss below, a straightforward application of Lemma
5 with Strategy 1 of § I.2 is not possible due to the appearance of a
defective eigenvalue in the modal equation (49). To avoid this difficulty we
shall use a non-sharp decay estimate as input to apply Lemma 5. Our second
goal is to construct a simple spatial functional that closely approximates our
first result. To achieve this, we construct modal Lyapunov functionals that
yield slightly less precise estimates but have the advantage of representing a
more convenient pseudo-differential operator.

To simplify notation, we assume that $\sigma=1$ in the present section. This
is no restriction, as the general case $\sigma>0$ in (47) can always be
reduced to the normalized one thanks to the rescaling $\tilde{t}=\sigma\,t$,
$\tilde{x}=\sigma\,x$.

A natural approach to obtain a decay estimate for $\mathcal{X}={\mathbb{R}}$
is an application of Lemma 5 to the decay estimates (52) with the matrices
$P^{(1)}$ and $P^{(2)}$ of (53) for $\sigma=1$. The extension of Strategy 1 of
§ I.2 to $\xi\in{\mathbb{R}}$ leads to (56). But as
$\operatorname{cond}\big{(}P^{(1)}(\xi)\big{)}\to\infty$ and
$\operatorname{cond}\big{(}P^{(2)}(\xi)\big{)}\to\infty$ for $|\xi|\to 1/2$,
it follows that the multiplicative constants in (56) become unbounded. This is
due to the defective limit of the modal equation (49) at $|\xi|=1/2$. The
modal Lyapunov functionals with sharp rate depend on the eigenspace structure,
which has a discontinuity at $|\xi|=1/2$ and the decay rates are not purely
exponential there. Hence, we cannot directly use Lemma 5 with sharp rates.

Therefore, the natural and in fact sharper approach is to start with the exact
modal decay function (instead of an exponential approximation): for $2\times
2$ ODE systems, this decay function was given in (AAS19, , Proposition 4.2):

$\|\hat{y}(t,\xi)\|_{2}^{2}\leq
h_{+}(t,\xi)\,\|\hat{y}(0,\xi)\|_{2}^{2}\quad\forall\,t\geq 0\,,$ (66)

where $h_{+}(t,\xi)$, the squared propagator norm associated with (49), is
explicitly given in AAS19 . Since this function is continuous at the defective
point $\xi=1/2$ for all $t\geq 0$ (see Fig. 6), one could easily extend Lemma
5 to this setting. But, since $h_{+}(t,\xi)$ is a quite involved function, the
minimization w.r.t. $R$ (as in (46)) could only be carried out numerically. In
order to come up with an explicit decay estimate, we shall therefore rather
approximate the modal (exponential) decay estimates that are used as a
starting point for Lemma 5.

$s$$h_{+}(t,s)$ Figure 6: The mapping $|\xi|=s\mapsto h_{+}(t,s)$ shows the
continuous modal dependency of the squared propagator norm of $C(\xi,1)$ for
fixed times $t$. Note that the kinks are no numerical artefact.

We now approximate the decay estimate for large frequencies $|\xi|$, but keep
the sharp estimates for $|\xi|$ small.

###### Lemma 6

Assume that $R\in(0,1/2)$ and let $\hat{y}(t,\xi)$ be a solution to (49).
Then,

$\|\hat{y}(t,\xi)\|^{2}\leq
c(\xi)\,e^{-\lambda(\xi)\,t}\,\|\hat{y}(0,\xi)\|^{2}\,,\quad\forall\,\xi\in{\mathbb{R}}\setminus\\{0\\}\,,\quad\forall\,t\geq
0\,,$ (67)

with $c(\xi)=\begin{cases}\frac{1+2\,|\xi|}{1-2\,|\xi|}\,,&|\xi|<R\,,\\\
\frac{|\xi|+2\,R^{2}}{|\xi|-2\,R^{2}}\,,&|\xi|\geq
R\,,\end{cases}\quad\lambda(\xi)=\begin{cases}2\,\mu(\xi)\,,&|\xi|<R\,,\\\
2\,\mu(R)\,,&|\xi|\geq R\,.\end{cases}$

###### Proof

For every $|\xi|<R$, the modal functional $\|\cdot\|^{2}_{P^{(2)}(\xi)}$, as
defined in (53), yields the sharp modal decay

$2\,\mu(\xi):=2\,\mu(\xi,1)=1-\sqrt{1-4\,\xi^{2}}\,,$

given by (50). The condition number of $P^{(2)}(\xi)$ is given as

$c(\xi):=\operatorname{cond}\left(P^{(2)}(\xi)\right)=\frac{1+2\,|\xi|}{1-2\,|\xi|}\leq\frac{1+2\,R}{1-2\,R}\,,\quad|\xi|<R\,.$
(68)

For $|\xi|\geq R$ we use the rescaled version of $\|\cdot\|^{2}_{\kern
2.0pt\overline{P}^{(1)}(\xi)}$ (from § • ‣ III.2.1, but now for
$\mathcal{X}={\mathbb{R}}$), given as $\|\cdot\|^{2}_{\kern
2.0pt\overline{P}(\xi)}$, with the matrix

$\overline{P}(\xi):=\begin{pmatrix}1&-\frac{2\,i\,R^{2}}{\xi}\\\
\frac{2\,i\,R^{2}}{\xi}&1\end{pmatrix}\,.$ (69)

As this matrix satisfies the inequality

$C^{*}(\xi)\overline{P}(\xi)+\overline{P}(\xi)C(\xi)\geq
2\mu(R)\overline{P}(\xi),\quad|\xi|\geq R,$ (70)

the functional $\|\cdot\|^{2}_{\overline{P}(\xi)}$ yields an exponential decay
$2\,\mu(R)$ for all modes $|\xi|\geq R$. The condition number of
$\overline{P}(\xi)$ is given as:

$c(\xi):=\operatorname{cond}\left(\overline{P}(\xi)\right)=\frac{|\xi|+2\,R^{2}}{|\xi|-2\,R^{2}}\leq\operatorname{cond}\left(\overline{P}(R)\right)=\frac{1+2\,R}{1-2\,R}\,,\quad|\xi|\geq
R\,,$ (71)

from which the desired result follows.

We can now apply Lemma 5 and Lemma 6 to obtain following global decay
estimate.

###### Proposition 5

Let $y:=(u,v)^{T}$ be a solution of the Goldstein–Taylor equation (48) on
${\mathbb{R}}$ with $\sigma=1$ and initial datum $y_{0}:=(u_{0},v_{0})^{T}$,
such that $u_{0}$,
$v_{0}\in\mathrm{L}^{1}({\mathbb{R}})\cap\mathrm{L}^{2}({\mathbb{R}})$. Let
the modal spectral gap, defined in (50), be denoted as $\mu(\xi):=\mu(\xi,1)$.
Then, for any $t\geq 0$ it follows that

$\|y(t)\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}\leq\inf_{0<R<\frac{1}{2}}\frac{1+2\,R}{1-2\,R}\left(2\min\left\\{\frac{B(t,R)}{\sqrt{t}},R\right\\}\,\|y_{0}\|_{\mathrm{L}^{1}({\mathbb{R}})}^{2}+e^{-\,2\,\mu(R)\,t}\,\|y_{0}\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}\right)$

with
$B(t,R):=\sqrt{t}\int_{0}^{R}e^{-2\,\mu(s)\,t}ds\in\big{[}0,\sqrt{\pi/8}\,\big{)}$.

###### Proof

Applying Lemma 5 to the modal decay estimates (67) and taking into account the
estimates (68) and (71) leads to the decay result where, for $B(t,R)$, we use
the estimate
$\mu(|\xi|)/\xi^{2}=\big{(}1/2-\sqrt{1/4-\xi^{2}}\,\big{)}/\xi^{2}\geq 1$ for
$0\leq|\xi|\leq 1/2$. The bound on $B$ follows from
$B(t,R)\leq\sqrt{t}\int_{0}^{\infty}e^{-2\,\xi^{2}\,t}\,d\xi$. $\square$

###### Remark 3

The decay result of Proposition 5 is neither explicit in the optimization with
respect to $R$ (for fixed $t$), nor optimal, as this would require an approach
starting from (66). It is however the best possible estimate of form (45)
achievable with quadratic forms for each mode. This follows, as for one, the
modal functionals for $|\xi|\leq R$, $\xi\neq 0$ are optimal for quadratic
forms (_cf._ the discussion on $P^{(2)}(\xi)$ in § III.2.1). Additionally, the
modal functionals for $|\xi|\geq R$ are sufficient (in light of Lemma 5) as
they yield the sufficient decay $2\,\overline{\mu}(R)$ and the sufficient
multiplicative constants $\sup_{|\xi|\geq
R}c(\xi)=\sup_{|\xi|<R}c(\xi)=(1+2\,R)/(1-2\,R)$. In analogy to § I.2 the
decay stated in Proposition 5 results from the global functional

$\widehat{\mathsf{H}}_{2}(R)[y]:=\int_{(-R,R)}\|\hat{y}(\xi)\|^{2}_{P^{(2)}(\xi)}\,d\xi+\int_{|\xi|\geq
R}\|\hat{y}(\xi)\|^{2}_{\kern 2.0pt\overline{P}(\xi)}\,d\xi\,.$

As our final result, we shall consider an alternative modal functional for the
GT equation on ${\mathbb{R}}$ that translates into a convenient representation
in the spatial variable. The trade-off is a less accurate global decay
estimate.

The result of Proposition 5 was based on the modal Lyapunov functional
$\|\cdot\|^{2}_{\kern 2.0pt\overline{P}(\xi)}$ for large modes and
$\|\cdot\|^{2}_{P^{(2)}(\xi)}$ for small modes. Now, we replace both
functionals by the single norm $\|\cdot\|_{\tilde{P}(\xi)}^{2}$ with the
positive definite Hermitian matrix

$\tilde{P}(\xi):=\begin{pmatrix}1&-\frac{2\,i\,\xi}{1+4\,\xi^{2}}\\\
\frac{2\,i\,\xi}{1+4\,\xi^{2}}&1\end{pmatrix}\,,\quad\xi\neq 0\,,$ (72)

which asymptotically approximates the matrices from (53) which yield sharp
modal decay. For the off-diagonal matrix elements we have

$\displaystyle\tilde{P}_{12}(\xi)-P^{(1)}_{12}(\xi)$
$\displaystyle=o\left(P^{(1)}_{12}(\xi)\right)\quad\text{ as
}\quad|\xi|\to+\infty\,,$ $\displaystyle\tilde{P}_{12}(\xi)-P^{(2)}_{12}(\xi)$
$\displaystyle=o\left(P^{(2)}_{12}(\xi)\right)\quad\text{ as }\quad|\xi|\to
0\,.$

It satisfies the matrix inequality (12) with $P=\tilde{P}(\xi)$ and the
spectral gap $\mu$ replaced by

$\tilde{\mu}(\xi)=\frac{1}{2}\left(1-\frac{1}{\sqrt{1+4\,\xi^{2}\,(1+4\,\xi^{2})}}\right)\,.$

The rate $2\,\tilde{\mu}(\xi)$ is an approximation to the sharp decay rate
$2\,\mu(\xi)$ of fifth order for modes $\xi$ close to 0, see Fig. 7.

$s=|\xi|$$\mu(s)$$\tilde{\mu}(s)$ Figure 7:  Exponential decay rate
$\tilde{\mu}$ in comparison to the sharp exponential rate $\mu$, shown as
functions of the spatial frequency $s=|\xi|$.

The condition number of $\tilde{P}(\xi)$ is given by

$\tilde{c}(\xi):=\operatorname{cond}\left(\tilde{P}(\xi)\right):=\frac{1+2\,|\xi|+4\,\xi^{2}}{1-2\,|\xi|+4\,\xi^{2}}\,,$
(73)

and hence we arrive at the modal decay estimates for $\xi\neq 0$:

$\|\hat{y}(t,\xi)\|^{2}\leq\tilde{c}(\xi)\,e^{-2\,\tilde{\mu}(\xi)\,t}\,\|\hat{y}(0,\xi)\|^{2}\,,\quad
t\geq 0\,.$ (74)

We define the global Lyapunov functional

$\mathsf{H}_{3}[y]:=\int_{\mathbb{R}}\|\hat{y}(\xi)\|^{2}_{\tilde{P}(\xi)}\,d\xi\,.$

As we shall see now, this can be rewritten in $x$-space (without resorting to
the $\xi$-modes) in terms of a fairly simple pseudo-differential operator,
similar to the functional $E_{\theta}[y]$ from Definition 1. Moreover, it is
easily related to the functional $\mathsf{H}_{1}[y]$ from § I.1: on the symbol
level it holds that functional
$\mathsf{H}_{3}(\xi)=2\,\mathsf{H}_{1}(2\,\xi,\delta=1)$, see (72), (54).

###### Proposition 6

  1. a)

For $u$, $v\in\mathrm{L}^{2}({\mathbb{R}})$, the functional $\mathsf{H}_{3}$
can be expressed as

$\mathsf{H}_{3}[u,v]=\|u\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}+\|v\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}-4\int_{\mathbb{R}}u(x)\,\partial_{x}\,\big{(}1-4\,\partial_{x}^{2}\big{)}^{-1}\,v(x)\,dx\,.$

  2. b)

Let $y:=(u,v)^{T}\in
C\big{(}[0,\infty);(\mathrm{L}^{2}\left({\mathbb{R}}\right))^{2}\big{)}$ be a
mild real valued solution to (48) with $\sigma=1$ and initial datum
$u_{0},\,v_{0}\in\mathrm{L}^{1}({\mathbb{R}})\cap\mathrm{L}^{2}({\mathbb{R}})$.
Then, the functional $\mathsf{H}_{3}$ yields the decay estimate

$\displaystyle\|y(t,x)\|_{\mathrm{L}^{2}({\mathbb{R}})}^{2}$
$\displaystyle\leq\inf_{0<R\leq\tfrac{\sqrt{5}-1}{4}}\left(\tfrac{1+2\,R+4\,R^{2}}{1-2\,R+4\,R^{2}}\min\left\\{2\,R,\sqrt{\tfrac{\pi}{2\,t}}\right\\}\|y_{0}\|_{\mathrm{L}^{1}({\mathbb{R}})}^{2}+3\,e^{-2\,\tilde{\mu}(R)\,t}\,\|y_{0}\|_{\mathrm{L}^{2}({\mathbb{R}})}^{2}\right)\,.$

###### Proof

With Plancherel’s identity it follows that

$\displaystyle\mathsf{H}_{3}[y]$
$\displaystyle=\int_{\mathbb{R}}\|\hat{y}(\xi)\|^{2}_{\tilde{P}(\xi)}\,d\xi=\|u\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}+\|v\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}+2\,\mathrm{Re}\left(\int_{{\mathbb{R}}}2\,i\,\xi\,\hat{u}(\xi)\,\frac{\overline{\hat{v}(\xi)}}{1+4\,\xi^{2}}\,d\xi\right)$
$\displaystyle=\|u\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}+\|v\|^{2}_{\mathrm{L}^{2}({\mathbb{R}})}-4\int_{\mathbb{R}}u(x)\,\partial_{x}\big{(}1-4\,\partial_{x}^{2}\big{)}^{-1}v(x)\,dx\,.$

To prove the decay estimate, we apply Lemma 5 to (74). The multiplicative
constant $\tilde{c}(\xi)$ from (73) is monotonously increasing for
$\xi\in[0,1/2]$ to its global maximum
$\operatorname{cond}\left(\tilde{P}(1/2)\right)=3$. For the integral in (46)
with $\tilde{c}(\xi)$, we estimate

$\displaystyle\int_{|\xi|\leq
R}\tilde{c}(\xi)\,e^{-2\,\tilde{\mu}(\xi)\,t}\,d\xi\leq\tilde{c}(R)\int_{|\xi|\leq
R}e^{-\xi^{2}\alpha(\xi)\,t}\,d\xi$

with

$\alpha(\xi):=\frac{1}{\xi^{2}}\left(1-\frac{1}{\sqrt{1+4\,\xi^{2}\,(1+4\,\xi^{2})}}\right)\,.$

One easily sees that $\alpha$ has a local minimum at $\xi=0$ with
$\alpha(0)=\alpha(\xi_{1})=1$, $\xi_{1}=\left(\sqrt{5}-1\right)/4\approx 0.3$,
_i.e._ for $0<R<\big{(}\sqrt{5}-1\big{)}/4$ it holds that $\alpha(\xi)\geq 1$
for all $|\xi|<R$. Thus, for $0<R<\big{(}\sqrt{5}-1\big{)}/4$ and $t>0$, the
desired result follows.

###### Acknowledgements.

This work has been partially supported by the Project EFI (ANR-17-CE40-0030)
of the French National Research Agency (ANR) and the Amadeus project
_Hypocoercivity_ no. 39453PH. J.D. and C.S. thank E. Bouin, S. Mischler and C.
Mouhot for stimulating discussions that took place during the preparation of
BDMMS : some questions raised at this occasion are the origin for this
contribution. A.A., C.S., and T.W. were partially supported by the FWF
(Austrian Science Fund) funded SFB F65. © 2020 by the authors. This paper may
be reproduced, in its entirety, for non-commercial purposes.

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