Trend to the equilibrium for the Fokker-Planck system with a strong external magnetic field

TL;DR

This paper studies the Fokker-Planck equation under a strong magnetic field, proving global existence and exponential convergence to equilibrium in various functional spaces.

Contribution

It extends previous methods to establish convergence to equilibrium for the Fokker-Planck system with a magnetic field in broader spaces.

Findings

Global solutions near Maxwellian are constructed.

Exponential convergence to equilibrium is proven.

Results are extended to Lebesgue and Sobolev spaces.

Abstract

We consider the Fokker-Planck equation with a strong external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like the Lebesgue space and the Sobolev space with polynomial weight, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

Equations

$${\partial }_{t}F+v\cdot {\nabla }_{x}F=0,$$

$${\partial }_{t}F+v\cdot {\nabla }_{x}F+\mathcal{F}(t,x)\cdot {\nabla }_{v}F=Q(F),$$

$$\{\begin{array}{l}{\partial }_{t}F+v\cdot {\nabla }_{x}F-(v\wedge B_e)\cdot {\nabla }_{v}F={\nabla }_v\cdot ({\nabla }_v+v)F\\ F(0,x,v)=F_0(x,v),\end{array}$$

$$\mu (v):=\frac{1}{(2\pi {)}^{3/2}}{e}^{-v^2/2}.$$

$${\partial }_t\mu +v\cdot {\nabla }_x\mu =0,{\nabla }_v\cdot (-{\nabla }_v+v)\mu =0\text{ and }(v\wedge B_e)\cdot {\nabla }_v\mu =0.$$

$$\forall h\in L^2(dxd\mu ),\Vert h{\Vert }_{L^2(dxd\mu )}^2$$

$$\forall h\in H^1(dxd\mu ),\Vert h{\Vert }_{H^1(dxd\mu )}^2$$

$$\forall g,h\in L^2(dxd\mu ),⟨h,g⟩=\iint hgdxd\mu .$$

$$F=\mu +\mu f.$$

$$\{\begin{array}{l}{\partial }_{t}f+v\cdot {\nabla }_{x}f-(v\wedge B_e)\cdot {\nabla }_{v}f=-(-{\nabla }_v+v)\cdot {\nabla }_{v}f\\ f(0,x,v)=f_0(x,v)\end{array}$$

$$P_1:=X_0-L,$$

$$X_0=v\cdot {\nabla }_x-(v\wedge B_e)\cdot {\nabla }_v\text{ and }L=(-{\nabla }_v+v)\cdot {\nabla }_v$$

$$\{\begin{array}{l}{\partial }_{t}f+P_{1}f=0\\ f(0,x,v)=f_0(x,v)\end{array}$$

$$\forall t\ge 0,\Vert f(t){\Vert }_{L^2(dxd\mu )}\le c{e}^{-\kappa t}\Vert f_0{\Vert }_{L^2(dxd\mu )}.$$

$$\forall t\ge 0,\Vert f(t){\Vert }_{H^1(dxd\mu )}\le c{e}^{-\kappa t}\Vert f_0{\Vert }_{H^1(dxd\mu )}.$$

$$\Vert F{\Vert }_{L^p(m)}:=\Vert Fm{\Vert }_{L^p}={\left({\int }_{{\mathbb{R}}^3\times {\mathbb{T}}^3}{F}^p(x,v)m^p(v)dvdx\right)}^{\frac{1}{p}},$$

$$⟨H⟩=\iint Hdxdv,$$

$$\forall t\ge 0,\Vert F(t)-\mu ⟨F_0⟩{\Vert }_{L^p(m)}\le c_{k,p}{e}^{at}\Vert F_0-\mu ⟨F_0⟩{\Vert }_{L^p(m)}.$$

$${\tilde{W}}^{1,p}(m)=\{h\in L^p(m)\text{ such that }⟨v⟩h,{\nabla }_{v}h\text{ and }{\nabla }_{x}h\in L^p(m)\}.$$

$$\Vert h{\Vert }_{{\tilde{W}}^{1,p}(m)}={\left(\Vert h{\Vert }_{L^p(m⟨v⟩)}^p+\Vert {\nabla }_{v}h{\Vert }_{L^p(m)}^p+\Vert {\nabla }_{x}h{\Vert }_{L^p(m)}^p\right)}^{\frac{1}{p}}.$$

$$k>3(1-\frac{1}{p})+\frac{7}{2}+\max \left(\Vert B_e{\Vert }_{L^{\infty }({\mathbb{T}}^3)},\frac{1}{2}\Vert {\nabla }_x{B}_e{\Vert }_{L^{\infty }({\mathbb{T}}^3)}\right).$$

$$\forall t\ge 0,\Vert F(t)-\mu ⟨F_0⟩{\Vert }_{{\tilde{W}}^{1,p}(m)}\le C{e}^{at}\Vert F_0-\mu ⟨F_0⟩{\Vert }_{{\tilde{W}}^{1,p}(m)}$$

$$P_0=v\cdot {\nabla }_x-(v\wedge B_e)\cdot {\nabla }_v-{\nabla }_v\cdot ({\nabla }_v+v).$$

$$P_{1}u=({\mu }^{-1}{P}_0\mu )u\forall u\in D(P_1).$$

$$\begin{gathered} P_{1/2}:=v\cdot {\nabla }_x-(v\wedge B_e)\cdot {\nabla }_v+(-{\nabla }_v+\frac{v}{2})\cdot ({\nabla }_v+\frac{v}{2}) \\ =X_0+b^{\ast }b, \end{gathered}$$

$$\{\begin{array}{l}{\partial }_{t}u+P_{1/2}u=0\\ u(0,x,v)=u_0(x,v).\end{array}$$

$$⟨P_{1/2}u,u⟩$$

$$=\underset{=0}{\underbrace{\iint v\cdot {\nabla }_{x}u\times udxdv}}-\underset{=0}{\underbrace{\iint (v\wedge B_e)\cdot {\nabla }_{v}u\times udxdv}}+⟨b^{\ast }bu,u⟩$$

$$=\Vert bu{\Vert }^2$$

$$\ge 0,$$

$$T=P_{1/2}+{\lambda }_{0}Id$$

$$⟨u,(P_{1/2}+{\lambda }_{0}Id)h⟩=0,\forall h\in D(P_{1/2}).$$

$$(-{\Delta }_v+\frac{v^2}{4}+1-X_0)u=0,\text{ in }{\mathcal{D}}^{\prime }({\mathbb{R}}^3\times {\mathbb{T}}^3).$$

Full text

Zeinab Karaki

Université de Nantes

Laboratoire de Mathematiques Jean Leray

2, rue de la Houssinière

BP 92208 F-44322 Nantes Cedex 3, France

1. 1 Introduction and main results

2. 1.1 Introduction

3. 1.2 Fokker-Planck equation with a given external magnetic field

4. 1.2.1 Presentation of the equation

5. 1.2.2 The main results

6. 2 Study of the operator $P_{1}$

7. 3 Trend to the equilibrium

8. 3.1 Hypocoercivity in the space $L^{2}(dxd\mu)$

9. 3.2 Hypocoercivity in the space $H^{1}(dxd\mu)$

10. 4 Enlargement of the functional space

11. 4.1 Intermediate results

12. 4.2 Study of the magnetic-Fokker-Planck operator on the spaces $L^{p}(m)$ and $\tilde{W}^{1,p}(m)$:

13. 4.2.1 Proof of Theorem 1.5

14. 4.2.2 Proof of Theorem 1.7

15. A Property of the operator $(v\wedge B_{e})\cdot\nabla_{v}$

16. B Non positivity of a certain integral

1. Introduction and main results

1.1. Introduction

In this article, we are interested in inhomogeneous kinetic equations. These equations model the dynamics of a charged particle system described by a probability density $F(t,x,v)$ representing at time $t\geq 0$ the density of particles at position $x\in\mathbb{T}^{3}$ and at velocity $v\in\mathbb{R}^{3}$.

In the absence of force and collision, the particles move in a straight line at constant speed according to the principle of Newton, and $F$ is the solution of the Vlasov equation

[TABLE]

where $\nabla_{x}$ is the gradient operator with respect to the variable $x$, and the symbol ¡¡$\cdot$¿¿ represents the scalar product in the Euclidean space $\mathbb{R}^{3}$. When there are forces and shocks, this equation must be corrected. This leads to various kinetic equations, the most famous being those of Boltzmann, Landau and Fokker-Planck. The general model for the dynamics of the charged particles, assuming that they undergo shocks modulated by a collision kernel $Q$ and under the action of an external force $\mathcal{F}\in\mathbb{R}^{3}$, is written by the following kinetic equation:

[TABLE]

where $Q$, possibly non-linear, acts only in velocity and where $\mathcal{F}$ can even depend on $F$ via Poisson or Maxwell equations.

According to the $H$-theorem of Boltzmann in 1872, there exists a quantity $H(t)$ called entropy which varies monotonous over time, while the gas relaxes towards the thermodynamic equilibrium characterized by the Maxwellian: it is a solution time independent of equation $(1)$ having the same mass as the initial system. The effect of the collisions will bring the distribution $F(t)$ to the Maxwellian with time. A crucial question is then to know the rate of convergence and this question has been widely studied over the past 15 years, in particular with the so called hypocoercive strategy (see [21] or [10] for an introductive papers).

1.2. Fokker-Planck equation with a given external magnetic field

1.2.1. Presentation of the equation

We are interested in the Fokker-Planck inhomogeneous linear kinetic equation with a fixed external magnetic field $x\mapsto B_{e}(x)\in\mathbb{R}^{3}$ which depends only on the spatial variables $x\in\mathbb{T}^{3}\equiv[0,2\pi]^{3}$. The Cauchy problem is the following:

[TABLE]

Here $F$ is the distribution function of the particles, and represents the density of probability of presence of particles at time $t\geq 0$ at the position $x\in\mathbb{T}^{3}$ and with a speed $v\in\mathbb{R}^{3}$. (Where ¡¡$\wedge$¿¿ indicates the vector (cross) product.)

We define the Maxwellian

[TABLE]

It is the (only) time independent solution of the system (2), since

[TABLE]

Concerning (2), we are interested in the return to the global equilibrium $\mu$ and the convergence of $F$ to $\mu$ in norms $L^{2}(dxd\mu)$ and $H^{1}(dxd\mu)$ defined by

[TABLE]

where $d\mu:=\mu(v)dv$ and the (real) Hilbertian scalar product $\langle.,.\rangle$ on the space $L^{2}(dxd\mu)$ defined by

[TABLE]

To answer such questions, when $F$ is close to the equilibrium $\mu$, we define $f$ to be the standard perturbation of $F$ defined by

[TABLE]

We then rewrite equation (2) in the following form:

[TABLE]

We now introduce the main assumption on $B_{e}$.

Hypothesis 1.1**.**

The magnetic field $B_{e}$ is indefinitely derivable on $\mathbb{T}^{3}$.

1.2.2. The main results

First we will show that the problem (3) is well-posed in the $L^{2}(dxd\mu)$ space, in the sense of the associated semi-group (See [20]). We associate with the problem (3) the operator $P_{1}$ defined by

[TABLE]

The problem (3) is then written

[TABLE]

Theorem 1.2**.**

Under Hypothesis 1.1 and with $f_{0}\in C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3})$, the problem (6) admits a unique solution $f\in C^{1}([0,+\infty[,L^{2}(dxd\mu))\cap C([0,+\infty[,C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3}))$ .

We also show the exponential convergence towards equilibrium in the norm $L^{2}(dxd\mu)$.

Theorem 1.3**.**

Let $f_{0}\in L^{2}(dxd\mu)$ such that $\langle f_{0}\rangle=\displaystyle\iint f_{0}(t,x,v)\,dxd\mu=0$. If $B_{e}$ satisfies Hypothesis 1.1, then there exist $\kappa>0$ and $c>0$ (two explicit constants independent of $f_{0}$) such that

[TABLE]

Note that in the preceding statement the mean $\langle f_{0}\rangle$ is preserved over time.

We give a result about the return to the global equilibrium $\mu$ with an exponential decay rate in the space $H^{1}(dxd\mu)$.

Theorem 1.4**.**

There exist $c,\kappa>0$ such that $\forall f_{0}\in H^{1}(dxd\mu)$ with $\langle f_{0}\rangle=0$, the solution $f$ of the system (3) satisfies

[TABLE]

We are interested in extending the results about the exponential decay of the semi-group to much larger spaces, following the work of Gualdani-Mischler-Mouhot in [6]. The following result gives convergence in $L^{p}(m)$ norms of $F$ to $\mu$, where the space $L^{p}(m)$ for $p\in[1,2]$ is the weighted Lebesgue space associated with the norm

[TABLE]

for some given weight function $m=m(v)>0$ on $\mathbb{R}^{3}$. Since there is no ambiguity we again denote

[TABLE]

the mean with respect to the usual $L^{1}$ norm. The main result of this paper in this direction is the following.

Theorem 1.5**.**

Let $p\in[1,2]$, let $m=\langle v\rangle^{k}:=(1+|v|^{2})^{k/2}$, $k>3(1-\displaystyle\frac{1}{p})$ , and assume Hypothesis 1.1 . Then for all $0>a>3(1-\displaystyle\frac{1}{p})-k$ and for all $F_{0}\in L^{p}(m)$, there exists $c_{k,p}>0$ such that the solution $F$ of the problem (2) satisfies the decay estimate

[TABLE]

It is also possible to obtain the same type of results in weighted Sobolev space $\tilde{W}^{1,p}(m)$ which is defined by

[TABLE]

We equip the previous space with the following standard norm:

[TABLE]

Hypothesis 1.6**.**

Let $p\in[1,2]$, the polynomial weight $m(v)=\langle v\rangle^{k}$ is such that

[TABLE]

The second main result of this paper is the following.

Theorem 1.7**.**

Let $m$ be a weight that satisfies Hypothesis 1.6 with $p\in[1,2]$ and assume Hypothesis 1.1. If $F_{0}\in\tilde{W}^{1,p}(m)$, then there is a solution $F$ of the problem (2), such that $F(t)\in\tilde{W}^{1,p}(m)$ for all $t\geq 0$, and it satisfies the following decay estimate:

[TABLE]

with $0>a>\max(a_{m,1}^{i},a_{m,2}^{i},-\kappa),i\in\{1,2,3\}$, where $a_{m,1}^{i}$ and $a_{m,2}^{i}$ are functions defined afterwards in (45)-(47) and (54)-(56) and $\kappa$ is defined in Theorem 1.4.

We will end this part by a brief review of the literature related to the analysis of kinetic PDEs using hypocoercivity methods. In some studies [11, 19, 21, 22], the treated hypocoercivity method is very close to that of hypoellipticity following the method of Kohn, which deals simultaneously with regularity properties and trend to the equilibrium.

The hypocoercive results were developped for simple models in [5, 9, 17, 22], the methods used were close in spirit to the ones developed in Guo [7, 8] in functional spaces with exponential weights.

In recent years, the theory of factorization and enlargement of Banach spaces was introduced in [6] and [16]. This theory allows us to extend hypocoercivity results into much larger spaces with polynomial weights. We refer for example to [3] and [16], where the authors show, using a factorization argument, the return to equilibrium with an exponential decay rate for the Fokker-Planck equation with an external electrical potential, or [13] for the inhomogeneous Boltzmann equation without angular cutoff case.

We conclude this sectio with some comments on our result. For the proof of Theorem 1.3 , we follow the micro-macro method proposed in [10]. Note that for the proof of Theorem 1.3, the black box method proposed in [5] (see also [3]) could perhaps be employed, anyway the presence of the Magnetic field induces same difficulties. To prove Theorem 1.5 and 1.7, we apply the abstract theorem of enlargement from [6, 16] to our Fokker-Planck-Magnetic linear operator. We deduce the semi-group estimates of Theorem 1.3 on large spaces like $L^{p}(\langle v\rangle^{k})$ and $\tilde{W}^{1,p}(\langle v\rangle^{k})$ with $p\in[1,2]$.

We hope that this first work will help in future investigations of non-linear problems like the Vlasov-Poisson-Fokker-Planck or Vlasov-Maxwell-Fokker-Planck equations (see [7, 12] and [8, 23]).

**Plan of the paper: ** This article is organized as follows. In Section 2, we prove that the Fokker-Planck-magnetic operator $P_{1}$ is a generator of a strongly continuous semi-group over the space $L^{2}(dxd\mu)$. In section 3, we show hypocoercivity in the weighted spaces $L^{2}$ and $H^{1}$ with an exponential weight. Finally, section 4 is devoted to the proofs of Theorems 1.5 and 1.7 with factorization and enlargement of the functional space arguments.

2. Study of the operator $P_{1}$

In this part, we show that the problem (6) is well-posed in the space $L^{2}(dxd\mu)$ in the sense of semi-groups. By the Hille-Yosida Theorem, it is sufficient to show that $P_{1}$ is maximal accretive in the space $L^{2}(dxd\mu)$.

Notation 2.1**.**

We define $P_{0}$ by

[TABLE]

The perturbation of the Cauchy problem (2) reduces the study of the operator $P_{1}$ defined in (4) to the study of $P_{0}$, since $P_{1}$ is obtained via a conjugation of the operator $P_{0}$ by the function $\mu$, that is to say

[TABLE]

Similarly, we can define the operator $P_{\theta}$ as the conjugation of the operator $P_{1}$ by the function $\mu^{\theta}$ with $\theta\in]0,1]$. Note that any result on the operator $P_{\theta}$ is also true on the operator $P_{1}$ in the corresponding conjugated space.

We will work in this section on operator $P_{1/2}$ which is defined by

[TABLE]

here $b=(\nabla_{v}+\frac{v}{2})$ and $X_{0}$ is defined in (5). We now show that operator $P_{1/2}$ is maximal accretive in the space $L^{2}(dxdv)$ and note that this gives the same result for $P_{1}$ in the space $L^{2}(dxd\mu)$. We study the following problem:

[TABLE]

Proposition 2.2**.**

Suppose that $B_{e}$ satisfies the Hypothesis 1.1. Then the closure with respect to the norm $L^{2}(\mathbb{T}^{3}\times\mathbb{R}^{3})$ of the magnetic-Fokker-Planck operator $\overline{P}_{1/2}$ on the space $C^{\infty}_{0}(\mathbb{R}^{3}\times\mathbb{T}^{3})$ is maximally accretive.

Proof.

We adapt here the proof given in [19, page 44]. We apply the abstract criterion by taking $H=L^{2}(\mathbb{T}^{3}\times\mathbb{R}^{3})$ and the domain of $P_{1/2}$ defined by $D(P_{1/2})=C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3}).$ First, we show the accretivity of the operator $P_{1/2}$. When $u\in D(P_{1/2})$, we have to show that $\langle P_{1/2}u,u\rangle\geq 0$. Indeed,

[TABLE]

since operators $(v\wedge B_{e})\cdot\nabla_{v}$ and $v\cdot\nabla_{x}$ are skew-adjoint see Lemma A.1.

Let us now show that there exists $\lambda_{0}>0$ such that the operator

[TABLE]

has dense image in $H$. We take $\lambda_{0}=\frac{3}{2}+1$ (following [19]). Let $u\in H$ satisfy

[TABLE]

We have to show that $u=0$.

First, we observe that equality (12) implies that

[TABLE]

Under Hypothesis 1.1, and following Hormander [14, 15] or Helffer-Nier [19, Chapter 8], operator $-\Delta_{v}+\dfrac{v^{2}}{4}+1-X_{0}$ is hypoelliptic, so $u\in C^{\infty}(\mathbb{R}^{3}\times\mathbb{T}^{3})$.

Now we introduce the family of truncation functions $\xi_{k}$ indexed by $k\in\mathbb{N}^{*}$ and defined by

[TABLE]

where $\xi$ is a $C^{\infty}$ function satisfying $0\leq\xi\leq 1$, $\xi=1$ on $B(0,1)$, and $\operatorname{Supp}\xi\subset B(0,2).$

For all $u,w\in C^{\infty}(\mathbb{T}^{3}\times\mathbb{R}^{3})$, we have

[TABLE]

When $u$ satisfies (12) in particular, when $h=\xi_{k}^{2}w$ , we get for all $w\in C^{\infty}$

[TABLE]

In particular, we take the test function $w=u$, so

[TABLE]

By an integration by parts, we obtain

[TABLE]

Which gives the existence of a constant $c>0$ such that, for all $k\in\mathbb{N}^{2}$,

[TABLE]

This leads to

[TABLE]

Choosing $\eta\leq\frac{1}{8}$, we get

[TABLE]

Taking $k\longrightarrow+\infty$ in (13), leads to $u=0$. ∎

Proof of Theorem 1.2.

According to Remark 2.1, the operator $P_{1}$ has a closure $\overline{P_{1}}$ from $C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3})$. This gives Theorem 1.2, by a direct application of Hille-Yosida’s theorem (cf. [20] for more details for the semi-group theory) to the problem (3), with $D(P_{1})=C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3})$ and $H=L^{2}(dxd\mu)$. ∎

From now on, we write $P_{\theta}$ for the closure of the operator $P_{\theta}$ from the space $C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3})$ with respect the norm $L^{2}(\mathbb{T}^{3}\times\mathbb{R}^{3})$.

3. Trend to the equilibrium

3.1. Hypocoercivity in the space $L^{2}(dxd\mu)$

The purpose of this subsection is to show the exponential time decay of the $L^{2}(dxd\mu)$ entropy for $P_{1}$, based on macroscopic equations. First, we try to find the macroscopic equations associated with system (3). We write $f$ in the following form:

[TABLE]

where  $r(f)(x)=\displaystyle\int f(x,v)\,d\mu(v)$ and $m(f)(x)=\displaystyle\int vf(x,v)\,d\mu(v)$ will be use later.

Definition 3.1**.**

In the following, we define

[TABLE]

and introduce a class of Hilbert spaces

[TABLE]

where $\mathcal{S}^{{}^{\prime}}$ is the space of temperate distributions.

We recall that the operator $\Lambda_{x}^{2}$ is an elliptic, self-adjoint, invertible operator from $\mathbb{H}^{2}(dxd\mu)$ to $L^{2}(dxd\mu)$ and $\Lambda_{x}\geq Id$. (cf [11, section 6] for a proof of these properties).

Lemma 3.2**.**

Let $f$ be the solution of the system (3), with the decomposition given in (14). Then we have

[TABLE]

Where $Op_{1}$ denotes a bounded generic operator of $L^{2}$ to $\mathbb{H}^{-1}$.

Proof.

We suppose is $f$ is a Schwarz function. In order to show equation $(14)$, we integrate equation (3) with respect to the measure $d\mu$ . We get

[TABLE]

since, $L1=0$ , $L$ is a self-adjoint operator and

[TABLE]

Then, we obtain

[TABLE]

hence equality $(14).$

To show $(15)$, we multiply equation (3) by $v$ before performing an integration with respect to the measure $d\mu$, we obteinning

[TABLE]

where $\nabla_{x}\int(v\otimes v)f\,d\mu=\sum_{i=1}^{3}\sum_{j=1}^{3}\,\int\,v_{j}v_{i}\partial_{x_{i}}f\,d\mu$. Now, we will calculate term by term the left-hand side of the equality (17). We first observe that

[TABLE]

Furethermore,

[TABLE]

It remains to compute component by component $\displaystyle\int v\left((v\wedge B_{e}\cdot\nabla_{v}f)\right)d\mu$. We have for all $1\leq j\leq 3$,

[TABLE]

Therefore $\displaystyle\int v\left((v\wedge B_{e}\cdot\nabla_{v}f)\right)d\mu=-m\wedge B_{e}$, where $m$ is defined in (14).

By combining all the previous equalities in (17), we obtain

[TABLE]

∎

Remark 3.3**.**

Under Hypothesis 1.1, since $B_{e}\in L^{\infty}(\mathbb{T}^{3})$,

[TABLE]

so the macroscopic equation (14) takes the following form:

[TABLE]

Now we are ready to build a new entropy, defined for any $u\in L^{2}(dxd\mu)$ by

[TABLE]

Using the Cauchy-Schwarz inequality gives us directly that

Lemma 3.4**.**

If $\varepsilon\leq\frac{1}{2}$, then

[TABLE]

Now, we can prove the main result of hypocoercivity leading to the proof of Theorem 1.3.

Proposition 3.5**.**

There exists $\kappa>0$ such that, if $f_{0}\in L^{2}(dxd\mu)$ and $\langle f_{0}\rangle=0$, then the solution of system (3) satisfies

[TABLE]

Proof.

We write

[TABLE]

We will omit the dependence of $f$ with respect to $t$. For the first term, we notice that

[TABLE]

by the spectral property of the operator $L$. For the second term, using the macroscopic equations, we get

[TABLE]

Now, using $\|m\|\leq\|h\|$, the Cauchy-Schwarz inequality and the following estimate:

[TABLE]

we obtain

[TABLE]

Poincaré’s inequality on $L^{2}(dx)$ takes the form

[TABLE]

where $\langle\phi\rangle=\int\phi(x)\,dx$ and $c_{P}>0$ is the spectral gap of $-\Delta_{x}$ on the torus (see [10, Lemma 2.6] for the proof of the previous inequality). Using this, we obtain, by applying the previous estimate to $r$ ( since $\langle r\rangle=\langle f\rangle=\langle f_{0}\rangle=0$),

[TABLE]

gathering (19) and (20), we get

[TABLE]

Now we choose $\varepsilon$ such that $C\varepsilon\leq\frac{1}{2}$, we get

[TABLE]

Which gives the result with $\kappa=\frac{\varepsilon}{4}\frac{c_{P}}{c_{P}+1}>0$. ∎

We can deduce the proof of Theorem 1.3.

Proof of Theorem 1.3.

Starting from Lemma 3.4 and Proposition 3.5, we have, for $f$ the solution of the system (3),

[TABLE]

This completes the proof of Theorem 1.3. ∎

3.2. Hypocoercivity in the space $H^{1}(dxd\mu)$

We will establish some technical lemmas, which will help us to deduce the exponential time decay of the norm $H^{1}(dxd\mu)$, noting that we work in $3$ dimensions.

The following lemma gives the exact values of some commutators will be used later.

Lemma 3.6**.**

The following equalities

1. (1)

$[\partial_{v_{i}},v\cdot\nabla_{x}]=\partial_{x_{i}}\quad\forall i\in\{1,2,3\}.$** 2. (2)

$[\partial_{v_{i}},(-\partial_{v_{j}}+v_{j})]=\delta_{ij}\quad\forall i,j\in\{1,2,3\}.$** 3. (3)

$[\nabla_{v},(v\wedge B_{e})\cdot\nabla_{v}]=B_{e}\wedge\nabla_{v}$. 4. (4)

$[\nabla_{x},(v\wedge B_{e})\cdot\nabla_{v}]=(v\wedge\nabla_{x}B_{e})\cdot\nabla_{v}.$**

Proof.

Let $f\in C^{\infty}_{0}(\mathbb{T}^{3}\times\mathbb{R}^{3})$. The first two equalities are obvious. We directly go to the proof of $(3)$ in component. Writting $B_{e}=(B_{1},B_{2},B_{3})$,

[TABLE]

Similarly we can show that, for all $1\leq i\leq 3$,

[TABLE]

This proves the equality $(3)$. Now, we will show $(4)$,

[TABLE]

∎

Now, we are ready to build a new entropy that will allow us to show the exponential decay of the norm $H^{1}(dxd\mu)$. We define this modified entropy by

[TABLE]

where $C>D>E>1$ are constants fixed below. We first show that $\mathcal{E}(u)$ is equivalent to the norm $H^{1}(dxd\mu)$ of $u$.

Lemma 3.7**.**

If $E^{2}<D$, then $\forall u\in H^{1}(dxd\mu)$

[TABLE]

Proof.

Let $u\in H^{1}(dxd\mu)$. Using the Cauchy-Schwarz inequality, we get

[TABLE]

which implies,

[TABLE]

This implies (21) if $E^{2}<D$. ∎

Note that using the same approach as in Section $3$, we can show the existence of a solution of the problem (3), which will be denoted as $f$, in the space $H^{1}(dxd\mu)$ in the sense of an associated semi-group. Using the preceding results, we are able to study the decrease of the modified entropy $\mathcal{E}(f(t))$.

Proposition 3.8**.**

Suppose that $B_{e}$ satisfies the Hypothesis 1.1, then there exist $C,D,E$ and $\kappa>0$, such that for all $f_{0}\in H^{1}(dxd\mu)$ with $\langle f_{0}\rangle=0$, the solution $f$ of the system (3) satisfies

[TABLE]

Proof.

The time derivatives of the four terms defining $\mathcal{E}(f(t))$ will be calculated separately. For the first term we have

[TABLE]

The second term writes

[TABLE]

We used the fact that the operators $v\cdot\nabla_{x}$ and $(v\wedge B_{e})\cdot\nabla_{v}$ are skew-adjoint in $L^{2}(dxd\mu)$ by Lemma A.1. According to equalities $(1)$ and $(3)$ of Lemma 3.6, we then obtain

[TABLE]

The time derivative of the third term can be calculated as follows:

[TABLE]

We calculate each term of equality (22). For the first term, using equalities $(1),(2)$ and $(3)$ of Lemma 3.6, we obtain

[TABLE]

For the second term of equality (22), using equality $(4)$ of Lemma 3.6, we have

[TABLE]

Combining the proceding equalities of the two terms in (22), we get

[TABLE]

According to Lemma A.1, the operators $v\cdot\nabla_{x}$ and $(v\wedge B_{e})\cdot\nabla_{v}$ are skew-adjoint in $L^{2}(dxd\mu)$, we have

[TABLE]

Using equality (23)-(24), we obtain

[TABLE]

Eventually, the time derivative of the last term takes the following form

[TABLE]

By collecting all the ties, we get

[TABLE]

Now, we need the following technical lemma.

Lemma 3.9**.**

We have the following equalities in $L^{2}(dxd\mu)$:

1. i.

$\langle(v\wedge\nabla_{x}B_{e})\cdot\nabla_{v}f,\nabla_{v}f\rangle=-\langle\nabla_{v}\wedge(\nabla_{x}B_{e}\cdot\nabla_{v}f),\nabla_{v}f\rangle.$** 2. ii.

$\langle(v\wedge\nabla_{x}B_{e})\cdot\nabla_{v}f,\nabla_{x}f\rangle=-\langle\nabla_{v}\wedge(\nabla_{x}B_{e}\cdot\nabla_{x}f),\nabla_{v}f\rangle.$**

where $\nabla_{x}B_{e}$ is the Jacobian matrix of the function

[TABLE]

and

[TABLE]

Proof.

Using the fact that $v\,\mu(v)=\nabla_{v}(\mu(v))$ and integrations by part, we obtain the result by simple computations. ∎

Let’s go back to the proof of Proposition 3.8. Using Lemma 3.9, the time derivative of $\mathcal{E}(f(t))$ takes the following form:

[TABLE]

Now we estimate the scalar products in the previous equality in $L^{2}(dxd\mu)$. For all $\eta,\eta^{{}^{\prime}}$ and $\eta^{{}^{\prime\prime}}>0$, we have

[TABLE]

and using than $\|\nabla^{2}_{v}f\|^{2}\leq\|(-\nabla_{v}+v)\cdot\nabla_{v}f\|^{2}+\|\nabla_{v}f\|^{2}$, we obtain

[TABLE]

The last scalar product is bounded by

[TABLE]

Combining all the previous estimates, we have

[TABLE]

We notice that

[TABLE]

We choose $\eta$, $\eta^{{}^{\prime\prime}}$, $E$, $D$ and $C$ such that

1. (1)

$\eta\leq 1$ and $\eta^{{}^{\prime\prime}}\leq 1$ . 2. (2)

$E\geq 2$. 3. (3)

$D\geq\frac{1}{2}(E^{2}+\frac{\eta^{{}^{\prime}}}{2}).$ 4. (4)

$C\geq A$.

Under the previous conditions, we get

[TABLE]

Using the Poincaré inequality in space and velocity variables, we then obtain

[TABLE]

Which completes Proposition 3.5 with $\displaystyle\kappa=\frac{E}{8}\frac{c_{p}}{2C}>0$. ∎

Proof of Theorem 1.4..

Using Lemma 3.7 and Proposition 3.8, we get $\kappa>0$ and $1<E<D<C$ such that

[TABLE]

This completes the proof of Theorem 1.4. ∎

4. Enlargement of the functional space

4.1. Intermediate results

In this section, we extend the results of exponential time decay of the semi-group to enlarged spaces (which we will define later), following the recent work of Gualdani, Mischler, Mouhot in [6].

Notation: Let $E$ be a Banach space.

We denote by $\mathcal{C}(E)$ the space of unbounded, closed operators with dense domains in $E$. 2. -

We denote by $B(E)$ the space of bounded operators in $E$. 3. -

Let $a\in\mathbb{R}$. We define the complex half-plane

[TABLE] 4. -

Let $L\in\mathcal{C}(E)$. $\Sigma(L)$ denote the spectrum of the operator $L$ and $\sigma_{d}(L)$ its discrete spectrum. 5. -

Let $\xi\in\Sigma_{d}(L)$, for $r$ sufficiently small we define the spectral projection associated with $\xi$ by

[TABLE] 6. -

Let $a\in\mathbb{R}$ be such that $\Delta_{a}\cap\Sigma(L)=\{\xi_{1},\xi_{2},...,\xi_{k}\}\subset\Sigma_{d}(L)$. We define $\Pi_{L,a}$ as the operator

[TABLE]

We need the following definition on the convolution of semigroup (corresponding to composition at the level of the resolvent operators).

Definition 4.1** (Convolution of time dependent operators).**

Let $X_{1},X_{2}$ and $X_{3}$ be Banach spaces. For two given functions

[TABLE]

we define the convolution $\mathcal{S}_{2}*\mathcal{S}_{1}\in L^{1}(\mathbb{R}^{+};B(X_{1},X_{3}))$ by

[TABLE]

When $\mathcal{S}=\mathcal{S}_{1}=\mathcal{S}_{2}$ and $X_{1}=X_{2}=X_{3}$ , we define inductively $\mathcal{S}^{(*1)}=\mathcal{S}\text{ and }\mathcal{S}^{(*\ell)}=\mathcal{S}*\mathcal{S}^{(*(\ell-1))}\text{ for any }\ell\geq 2.$

We say that $L\in\mathcal{C}(E)$ is hypodissipative if it is dissipative for some norm equivalent to the canonical norm of $E$ and we say that $L$ is dissipative for the norm $\|\cdot\|_{E}$ on $E$ if

[TABLE]

We refer to the paper [6, Section 2.3] for an introduction to this subject. Now, we recall the crucial Theorem of enlargement of the functional space.

Theorem 4.2** (Theorem 2.13 in [6]).**

Let $E$ and $\mathcal{E}$ be two Banach spaces such that $E\subset\mathcal{E}$, $L\in\mathcal{C}(E)$ and $\mathcal{L}\in\mathcal{C}(\mathcal{E})$ such that $\mathcal{L}_{|_{E}}=L$. We suppose that there exist $\mathcal{A}$ and $\mathcal{B}\in\mathcal{C}(\mathcal{E})$ such that $\mathcal{L}=\mathcal{A}+\mathcal{B}$ (with corresponding restrictions $A,B$ on $E$). Suppose there exists $a\in\mathbb{R}$ and $n\in\mathbb{N}$ such that

• $(H_{1})$

Locating the spectrum of $L$:* *

[TABLE]

and $L-a$ is dissipative on $\mathrm{Im}(Id_{E}-\Pi_{L,0})$

• $(H_{2})$

Dissipativity of $\mathcal{B}$ and bounded character of $\mathcal{A}$:* $(\mathcal{B}-a)$ is hypodissipative on $\mathcal{E}$ and $\mathcal{A}\in B(\mathcal{E})$ and $A\in B(E)$.*

• $(H_{3})$

Regularization properties of $T_{n}(t)=\left(\mathcal{A}S_{\mathcal{B}}(t)\right)^{(*n)}:$**

[TABLE]

Then for all $a^{\prime}>a$, we have the following estimate:

[TABLE]

To finish this subsection, we give a lemma providing a practical criterion to prove hypothesis $(H_{3})$ in the previous theorem.

Lemma 4.3** (Lemma $2.4$ in [16]).**

Let $E$ and $\mathcal{E}$ be two Banach spaces with $E\subset\mathcal{E}$ dense with continuous embedding, and consider $L\in\mathcal{C}(E)$ and $\mathcal{L}\in\mathcal{C}(\mathcal{E})$ with $\mathcal{L}_{\mid E}=L$ and $a\in\mathbb{R}$. Let us assume that:

1. a)

$\mathcal{B}-a$* is hypodissipative on $\mathcal{E}$ and $B-a$ on $E$.* 2. b)

$\mathcal{A}\in B(\mathcal{E})$* and $A\in B(E)$.* 3. c)

There are constants $b\in\mathbb{R}$ and $\Theta\geq 0$ such that

[TABLE]

Then for all $a^{\prime}>a$, there exist some explicit constants $n\in\mathbb{N}$ and $C_{a^{\prime}}\geq 1$, such that

[TABLE]

4.2. Study of the magnetic-Fokker-Planck operator on the spaces $L^{p}(m)$ and $\tilde{W}^{1,p}(m)$:

This part consists in building the general framework of the problem.

Recall first the equation of Fokker-Planck (2) written in original variable:

[TABLE]

and where we recall that $P_{0}$ was introduced in Section 2 and with

[TABLE]

and $B_{e}$ is the external magnetic field satisfying Hypothesis 1.1. As mentioned in Section $2$, the Maxwellian $\mu$ is a solution of the system (2). We will need the following modified Poincaré inequality:

[TABLE]

where $\lambda_{p}>0$ which depends on the dimension (see [16, Lemma 3.6]). See also [17], [2] and [1].

Now we will define define the expanded functional space.

Definition 4.4**.**

Let $m=m(v)>0$ on $\mathbb{R}^{3}$ be a weight of class $C^{\infty}$ and recall that

• $\bullet$

The space $L^{p}(m)$ for $p\in[1,2]$, is the Lebesgue space with weight associated with the norm

[TABLE]

• $\bullet$

We define the technical function $\Psi_{m,p}$ by

[TABLE]

where $K(x,v)=v\wedge B_{e}(x)+v$.

We will show the decay of the semi-group associated with the problem (2) in the spaces $L^{p}(m)$ where $p\in[1,2]$, when $m$ verifies the following hypothesis:

1. $\mathrm{(W_{p})}$

The weight $m$ satisfies $L^{2}(\mu^{-\frac{1}{2}})\subset L^{p}(m)$ with continuous injection and

[TABLE]

Remark 4.5**.**

In the following, we note $m_{0}=\mu^{-1/2}$ the exponential weight. By direct computation, $L^{2}(\mu^{-1/2})\subset L^{q}(m_{0})$ for any $q\in[1,2]$ with continuous injection and there exists $b\in\mathbb{R}$ such that

[TABLE]

(See Lemma $3.7$ in [6] for a proof of the previous property). Under the previous hypothesis, by direct computation we obtain that the semi-group $\mathcal{S}_{L_{0}}$ is bounded from $L^{p}(m_{0})$ to $L^{p}(m_{0}).$

We work now in $L^{p}(m)$ with a polynomial weight $m$ satisfying Hypothesis $\mathrm{(W_{p})}$.

Lemma 4.6**.**

Let $m=\langle v\rangle^{k}:=(1+|v|^{2})^{k/2}$ and $p\in[1,2]$. Then hypothesis $\mathrm{(W_{p})}$ is true when $k$ satisfies the following estimate:

[TABLE]

Proof.

For the proof, see Lemma $3.7$ in [6]. ∎

4.2.1. Proof of Theorem 1.5

From now on, we write $L_{0}$ for the operator $-P_{0}$, the Fokker- Planck operator considered on the space $L^{2}(m_{0})$ defined in (25) (respectively $\mathcal{L}_{0}$ for $-P_{0}$ the Fokker-Planck operator considered on the space $\mathcal{E}=L^{p}(m)$, with $m=\langle v\rangle^{k}$, where $k>3(1-\displaystyle\frac{1}{p})$ and $p\in[1,2]$) . We will prove Theorem 1.5 by applying Theorem 4.2 to $\mathcal{L}_{0}$. To verify Hypotheses $(H_{2})$ and $(H_{3})$ of Theorem 4.2, we need two lemmas about the dissipativity and regularization properties of $\mathcal{L}_{0}$ following [6].

Definition 4.7**.**

We split operator $\mathcal{L}_{0}$ into two pieces: for $M$, $R>1$, we define the operator $\mathcal{B}$ by

[TABLE]

where $\chi_{R}(v)=\chi(v/R)$, and $0\leq\chi\in C_{0}^{\infty}(\mathbb{T}^{3}\times\mathbb{R}^{3})$ is such that $\chi(v)=1\,\text{ when }\,|v|\leq 1.$ We also denote by $A$ and $B$ the restriction of the operators $\mathcal{A}$ and $\mathcal{B}$ to the space $E$.

Lemma 4.8** (Dissipativity of $\mathcal{B}$).**

Under Assumption $\mathrm{(W_{p})}$, for all $0>a>a_{m,p}$, we can choose $R,M>1$ such that the operator $\mathcal{B}-a$ satisfies the dissipativity estimate for some $C>0$

[TABLE]

Proof.

The proof follows the one given in Lemma $3.8$ in [6]. Let $F$ be smooth, rapidly decaying and positive function $F$. Since of $\Psi_{m,p}$ is independent of the magnetic field, by integration by parts with respect to $v$ and using Remark A.2, we have

[TABLE]

Let now take $a>a_{m,p}$. As $m$ satisfies the hypothesis $(W_{p})$, there exist $M$ and $R$ two large constants such that

[TABLE]

and we obtain

[TABLE]

This completes the proof of Lemma 4.8. ∎

From now on, $a$, $M$ and $R$ are fixed. We note that $\mathcal{B}^{*}$ is the dual operator of $\mathcal{B}$ relative to the pivot space $L^{2}(\mathbb{T}^{3}\times\mathbb{R}^{3})$, which is defined as follows:

[TABLE]

Lemma 4.9** (Regularization properties).**

There exists $b\in\mathbb{R}$ and $C>0$ such that, for all $t\geq 0$,

[TABLE]

where $p^{\prime}$ and $q^{\prime}$ are the conjugates of $p$ and $q$ respectively

Proof.

We consider $F(t)$ the solution of the evolution equation

[TABLE]

We introduce the following entropy defined for all $t\in[0,T]$, with $T\ll 1$ and $r>1$ to be fixed later:

[TABLE]

with

[TABLE]

where $B>\alpha>D,\beta$, $E<\sqrt{\beta D}$ and $r$ is an integer that will be determined later. We will omit the dependence of $F$ on $t$. Using the methods and computations of the proof of Proposition 3.8 and adapting the techniques used in [10], we choose the constants $\alpha,D$ and $E>0$ large enough such that there exist a constant $C_{\mathcal{G}}>0$ (depending on $\|B_{e}\|_{L^{\infty}(\mathbb{T}^{3})}$ and $\|\nabla_{x}B_{e}\|_{L^{\infty}(\mathbb{T}^{3})}$) such that

[TABLE]

Here, $C_{\chi}>0$ is a uniform constant in $R>1$ but depends on $M$.

[TABLE]

We choose the constants $\beta$ and $T>0$ such that

[TABLE]

.

We deduce that

[TABLE]

Now, the Nash inequality [18] implies that there exists $C_{d}>0$ such that

[TABLE]

We need to have an estimate based on $\|\nabla_{x,v}F\|_{L^{2}(m_{0})}$. Firstly,

[TABLE]

On the other hand, we use the fact that $v\,m_{0}^{2}=\nabla_{v}(m_{0}^{2})$ to estimate $\|vF\|_{L^{2}(m_{0})}$. We get

[TABLE]

and integrating by parts in $v$ in the previous estimate, we obtain

[TABLE]

Applying Cauchy-Schwarz inequality, we get

[TABLE]

Therefore

[TABLE]

Using the previous estimate and inequality (31), we have

[TABLE]

Using the previous inequality and the fact that $\nabla_{x}(Fm_{0})=m_{0}\,\nabla_{x}F$ (since $m_{0}$ does not depend on $x$), there exists $C^{{}^{\prime}}_{d}>0$ such that the estimate (30) becomes

[TABLE]

Using Young’s inequality with $p=(d+1)\text{ and }q=(d+1)/d$ , we get, for all $\varepsilon>0$,

[TABLE]

Using the previous estimate, we choose $\varepsilon>0$ small enough that there is a $C^{{}^{\prime\prime}}>0$

[TABLE]

According to Remark 4.5 there exists $b\in\mathbb{R}$ such that $\forall p\in[1,2]$

[TABLE]

Finally, using the previous estimate when $p=1$ and choosing $r=3d+1$, we deduce that there exists $B^{{}^{\prime\prime}}>0$ such

[TABLE]

Thanks to Gronwall’s Lemma, there exists $B^{\prime\prime\prime}>0$ such that

[TABLE]

Then,

[TABLE]

As a consequence, using the continuity of $S_{\mathcal{B}}(t)$ on $L^{p}(m_{0})$ with $p=2$,

[TABLE]

and eventually for all $t\in(0,+\infty)$

[TABLE]

Let us now consider $p$ and $q$ satisfying $1\leq p\leq q\leq 2$. $S_{\mathcal{B}}(t)$ is continuous from $L^{p}(m_{0})$ into $L^{q}(m_{0})$ using the Riesz-Thorin Interpolation Theorem. Moreover, if we denote by $C_{p,q}(t)$ the norm of $S_{\mathcal{B}}(t):L^{p}(m_{0})\to L^{q}(m_{0})$, we get the following estimate:

[TABLE]

This shows the first estimate.

Now we will show the second estimate. According to the first estimate, we have

[TABLE]

which means

[TABLE]

where $h=m_{0}F_{0}$. Then by duality, we get

[TABLE]

where $p^{{}^{\prime}}$ and $q^{{}^{\prime}}$ are the conjugates of $p$ and $q$ respectively. Which gives the result by reusing the definition of weighted dual spaces

[TABLE]

This completes the proof. ∎

Corollary 4.10**.**

Let $m$ be a weight that satisfies Hypothesis 1.6, then there exists $\Theta\geq 0$ such that for all $F_{0}\in L^{p}(m)$ with $p\in[1,2]$, we have the following estimate

[TABLE]

Proof.

We first prove the second inequality. Let $F_{0}\in L^{p}(m)$ with $m$ a polynomial weight satisfying Hypothesis 1.6. For all $1\leq p\leq 2$ and for all $t\in]0,1]$ and $\ v\in\mathbb{R}^{3}$, using Lemma 4.9 with $q=2$, we get

[TABLE]

where $\Theta=(3d+1)(1/p-1/2)>0$.

To show the first estimate, we proceed step by step.

**Step 1: ** First, we will show the following estimate:

[TABLE]

Indeed, using the continuous and dense injection $L^{p^{\prime}}(m_{0})\subset L^{p^{\prime}}(m)$, we obtain

[TABLE]

then using Lemma 4.9 with $q^{\prime}=2$, we obtain

[TABLE]

where $\Theta=(3d+1)(1/p-1/2)$.

Step 2: Of the inequality (34), it follows that for $g=\mathcal{A}F_{0}$, we get

[TABLE]

which means, denoting $h=mF_{0}$

[TABLE]

by a duality argument and noting that $\mathcal{A}^{*}=\mathcal{A}$, we get

[TABLE]

Finally, according to our definition of weighted dual spaces and replacing $h$ by $mF_{0}$, we obtain

[TABLE]

To obtain the result, we notice that

[TABLE]

and we combine the previous estimate with the estimate (35), which completes the proof of the first estimate. ∎

Now we prove Theorem 1.5.

Proof of Theorem 1.5.

For $p\in[1,2]$. We consider $\mathcal{E}=L^{p}(m)$, $E=L^{2}(m_{0})$, and denote $\mathcal{L}_{0}$ and $L_{0}$ the Fokker-planck operator considered respectively on $\mathcal{E}$ and $E$ (defined in (25)). We split the operator as $\mathcal{L}_{0}=\mathcal{A}+\mathcal{B}$ as in (28). Let us proceed step by step:

$\bullet$** Step 1: Verification of condition $(H_{1})$ of Theorem 4.2**

Theorem 1.3 shows us the existence of the semi-group $S_{L_{0}}(t)$, associated with the Fokker-Planck operator defined in (25) on the space $L^{2}(m_{0})$ and the constants $\kappa$ and $c>0$, for which, for all $F_{0}\in L^{2}(m)$ such that $\langle F_{0}\rangle=0$,

[TABLE]

Which implies the dissipativity of the operator $L_{0}-a$ on $E$, for all $0>a>-\kappa$.

$\bullet$** Step 2: Verification of condition $(H_{2})$ of Theorem 4.2.

**According to Lemma 4.8, the operator $\mathcal{B}-a$ is dissipative on $\mathcal{E}$, for all $0>a>a_{m,p}$, and by definition of the operator $\mathcal{A}$ and $A$, we have $\mathcal{A}\in B(\mathcal{E})$ and $A\in B(E)$.

$\bullet$** Step 3: Verification of condition $(H_{3})$ of Theorem 4.2.**

According to Corollary 4.10, the operators $\mathcal{A}S_{\mathcal{B}}$ and $S_{\mathcal{B}}\mathcal{A}$ satisfy the property $c)$ of Lemma 4.3. By applying Lemma 4.3,

[TABLE]

Then for all $a^{\prime}>a$, there exist constructible constants $n\in\mathbb{N}$ and $C_{a^{\prime}}\geq 1$ , such that

[TABLE]

$\bullet$** Step 4: End of the Proof

**All the hypotheses of Theorem 4.2 are satisfied. We deduce that $\mathcal{L}_{0}-a$ is a dissipative operator on $\mathcal{E}$ for all $a>\max(a_{m,p},-\kappa)$, with the semi-group $S_{\mathcal{L}_{0}}(t)$ satisfying estimate (7).

∎

4.2.2. Proof of Theorem 1.7

This part is dedicated to the proof of the exponential time decay estimates of the semi-group associated with the Cauchy problem (2) with an external magnetic field $B_{e}$, with an initial datum in $\tilde{W}^{1,p}(m)$ defined in (8).

For the proof of Theorem 1.7, we consider the space $\mathcal{E}=\tilde{W}^{1,p}(m)$ and $E=H^{1}(m_{0})$.

Definition 4.11**.**

We split operator $\mathcal{L}_{0}$ into two pieces and define for all $R,M>0$

[TABLE]

where $M>0$, $\chi_{R}(v)=\chi(v/R)$ $R>1,$ and $\chi\in C^{\infty}_{0}(\mathbb{R}^{3})\text{ such that }\chi(v)=1$ $|v|\leq 1.$ We also denote $A$ and $B$ the restriction of operators $\mathcal{A}$ and $\mathcal{B}$ on the space $E$ respectively.

Lemma 4.12** (Dissipativity of $\mathcal{B}$).**

Under Assumptions 1.1 and 1.6, there exists $M$ and $R>0$ such that for all $0>a>\max(a_{m,1}^{i},a_{m,2}^{i})$ (defined in (45)-(47) and (54)-(56)) such that operator $\mathcal{B}-a$ is dissipative in $\tilde{W}^{1,p}(m)$ where $p\in[1,2]$. In other words, the semi-group $\mathcal{S}_{\mathcal{B}}$ satisfies the following estimate:

[TABLE]

Proof.

Let $F_{0}\in\tilde{W}^{1,p}(m)$. We consider $F$ the solution of the evolution equation

[TABLE]

Recall that the norm on the space $\tilde{W}^{1,p}(m)$ is given by

[TABLE]

where $\tilde{m}=m\langle v\rangle$. Differentiating the previous equality with respect to $t$, we get

[TABLE]

We now estimate each term of the equality (39).

For the first term in (39), we apply Lemma 4.8 and get

[TABLE]

Secondly, we differentiate the equation (38) with respect to $v$, and then we use the equalities of Lemma 3.6. We get the following equation (recall $d=3$):

[TABLE]

This gives

[TABLE]

Then, proceeding exactly as in the proof of Lemma 4.8 and applying Young’s inequality, we obtain for all $\eta_{1}>0$

[TABLE]

Finally, we estimate the last term of the equality (39). We treat two cases, and then we use an interpolation argument to complete the proof.

$\bullet$** Case 1: $p=1$.

**We differentiate the equation (38) with respect to $x_{i}$ for all $i=1,2,3$, then we use the equalities of Lemma 3.6. We will have the following equation:

[TABLE]

Using the previous equation, we obtain

[TABLE]

Using the computations made in Lemma 4.8 for $p=1$, using Lemma B.1 in the appendix B, and performing an integration by parts with respect to $v$, we get

[TABLE]

where, we used the fact that $(v\wedge\partial_{x_{i}}B_{e})\cdot\nabla_{v}m=0$. Then, defining the norm

[TABLE]

and using the previous definition, we have

[TABLE]

Collecting all the estimates, we obtain

[TABLE]

We define then (for $M$ and $R$ to be fixed below).

[TABLE]

(Recall that $\limsup\limits_{|v|\to+\infty}\Psi_{m,1}=-k$). We denote then

[TABLE]

We now assume that $k$ satisfies

[TABLE]

Hypothesis (48) implies that $a_{m,1}^{i}<0$, for all $i=1,2,3$. Consequently, for $\eta_{1}$ sufficiently small, we may then find $M$ and $R>0$ large enough so that, for all $0>a>\max(a_{m,1}^{1},a_{m,1}^{2},a_{m,1}^{3})$, we have

[TABLE]

Hence the operator $\mathcal{B}-a$ is dissipative on $\tilde{W}^{1,1}(m)$ .

$\bullet$** Case 2: $p=2$.

**Again, we differentiate the equation (38) with respect to $x$, and we use the equalities of Lemma 3.6 to obtain the following equation:

[TABLE]

Using the calculations made in Lemma 4.8 and the previous equation, we obtain

[TABLE]

Then, by integration by parts with respect to $v$, we get

[TABLE]

According to the Cauchy-Schwarz inequality, for every $\varepsilon>0$, there is a $C_{\varepsilon}>0$ such that

[TABLE]

We choose $\varepsilon=\dfrac{1}{4}$, and we finally get

[TABLE]

Collecting all the estimates, we thus obtain

[TABLE]

Again, we define then, for $M$ and $R$ to be fixed in the next paragraph

[TABLE]

Again, we denote

[TABLE]

Assuming $k$ satisfies

[TABLE]

we obtain that $a_{m,2}^{i}<0$ for all $i=1,2,3$. Consequently, we may find $M$ and $R>0$ large enough so that for all $0>a>\max(a_{m,2}^{1},a_{m,2}^{2},a_{m,2}^{3})$

[TABLE]

Hence the operator $\mathcal{B}-a$ is dissipative on $\tilde{W}^{1,2}(m)$ for such $M$ and $R$.

For the general case $1\leq p\leq 2$: The cases $1$ and $2$ show us that the operator $S_{\mathcal{B}}(t)$ is continuous on $\tilde{W}^{1,1}(m)$ (on $\tilde{W}^{1,2}(m)$) with the operator $\mathcal{B}$ is given by

[TABLE]

where $M$ and $R>0$ agree with the conditions given in case $1$ and case $2$. Applying the Riesz-Thorin interpolation Theorem and using Hypothesis 1.6, we obtain that the operator $S_{\mathcal{B}}(t)$ is continuous on $\tilde{W}^{1,p}(m)$ for all $1\leq p\leq$ 2, with the following dissipative estimate:

[TABLE]

This completes the proof. ∎

From now on, $M$ and $R$ are fixed.

Lemma 4.13** (Property of regularization).**

There exist $b$ and $C>0$ such that, for all $p,q$ with $1\leq p\leq q\leq 2,$ we have

[TABLE]

[TABLE]

Here $2\leq q^{{}^{\prime}}\leq p^{{}^{\prime}}\leq+\infty$ are the conjugates of $p$ and $q$ respectively.

Proof.

Let $F$ be the solution of the evolution equation

[TABLE]

In to the proof of Lemma 4.9, the following relative entropy has been introduced

[TABLE]

with $\mathcal{G}$ defined by

[TABLE]

We have shown, for constants $\alpha,D,E$ and $\beta>0$ well chosen, that there exist $C>0$ and $r=3d+1$ such that

[TABLE]

Using the previous estimate and the definition of $\mathcal{H}$ and $\mathcal{G}$, we get

[TABLE]

Therefore,

[TABLE]

Finally, to complete the proof, we use the Riesz-Thorin Interpolation Theorem in the real case on the operator $S_{\mathcal{B}}(t)$ . We obtain the continuity of $S_{\mathcal{B}}(t)$ from $\tilde{W}^{1,p}(m_{0})$ to $\tilde{W}^{1,q}(m_{0})$, with $S_{\mathcal{B}}$ satisfying the estimate (58).

The estimate (59) follows from (58) by duality. ∎

Corollary 4.14**.**

Let $m$ be a weight that satisfies Hypothesis 1.6. Then there exists $\Theta\geq 0$ such that for all $F_{0}\in\tilde{W}^{1,p}(m)$ with $p\in[1,2]$

[TABLE]

Proof.

The proof is similar to that of Corollary 4.10. ∎

Proof of Theorem 1.7.

The estimate (10) is an immediate consequence of Theorem 4.2 together with Theorem 1.4, Lemma 4.12, Lemma 4.13 and Corollary 4.14. ∎

Appendix A Property of the operator $(v\wedge B_{e})\cdot\nabla_{v}$

In the following Lemma we show that operator $(v\wedge B_{e})\cdot\nabla_{v}$ and $v\cdot\nabla_{x}$ are formally skew-adjoint operators in the space $L^{2}(dxd\mu)$.

Lemma A.1**.**

Let $B_{e}$ be the external magnetic field, then, with adjoints in the space $L^{2}(dxd\mu)$,

[TABLE]

and

[TABLE]

Proof.

Let $f$ and $g$ $\in C^{\infty}_{0}(\mathbb{R}^{3}\times\mathbb{T}^{3})$. We have

[TABLE]

Using the fact

[TABLE]

we obtain

[TABLE]

since $(v\wedge B_{e})\cdot v=0$. By integration by parts, we have then

[TABLE]

For the second equality, we obtain

[TABLE]

by integration by parts with respect to $x$. Since $\mu$ is independent of $x$, we have then

[TABLE]

This completes the proof. ∎

Remark A.2**.**

We can generalize the results of the preceding Lemma for all function $m$ which are radial in $v$ and independant of $x$. We obtain that $v\cdot\nabla_{x}$ and ($v\wedge B_{e})\cdot\nabla_{v}$ are formally skew-adjoint operators in the space $L^{2}(m)$.

Appendix B Non positivity of a certain integral

The following well-know lemma is used in the proof of the dissipativity of the operator $\mathcal{B}-a$ in the spaces $L^{p}(m)$ and $\tilde{W}^{1,p}(m)$ in Section 4. This lemma is a special case of the general study done in the article [4].

Lemma B.1**.**

Let $g$ be a smooth function and let $p\geq 1$. Then the following integral is well-posed and satisfy the following estimate

[TABLE]

Proof.

Formal integration by parts with respect to $v$ justifies the property for all $p>1$. For $p=1$, we regularize and use convexity of the function $\Psi:s\to|s|$. ∎

Acknowledgments

I would like to thank my advisors Frédéric Hérau and Joseph Viola for their invaluable help and support during the maturation of this paper. I thank also the Centre Henri Lebesgue ANR-11-LABX-0020-01 and the Faculty of Sciences (Section I) of Lebanese University for its support .