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

We consider the Fokker-Planck equation with an 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 certain Lebesgue spaces with polynomial weights and modified weighted Sobolev spaces, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

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 ≥ 0 the density of particles at position x ∈ T 3 and at velocity v ∈ R 3 . In the absence of force and collisions, 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 where ∇ x is the gradient operator with respect to the variable x, and the symbol "·" represents the scalar product in the Euclidean space R 3 . When there are forces and collisions, 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 charged particles, assuming that they undergo collision modulated by a collision kernel Q and under the action of an external force F ∈ R 3 , is written by the following kinetic equation: where Q, possibly non-linear, acts only in velocity and where 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 monotonically in time, while the gas relaxes towards the thermodynamic equilibrium characterized by the Maxwellian: it is a solution of equation (1), independent in time and 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 [28] or [14] for an introductive papers).

Presentation of the equation.
We are interested in the Fokker-Planck inhomogeneous linear kinetic equation with a fixed external magnetic field x → B e (x) ∈ R 3 which depends only on the spatial variables x ∈ T 3 ≡ [0, 2π] 3 . The Cauchy problem is the following: Here F is the distribution function of the particles, and represents the probability density of finding a particle with velocity v ∈ R 3 and position x ∈ T 3 at time t ≥ 0.

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 311
For the Fokker-Planck system with external magnetic field (2), the entropy functional in relation with the so called H-Theorem for a smooth sufficiently decaying (in space and velocity) positive probability density h. Then, we directly check that H(µ, µ) = 0. According to the non-negativity of the function s → s ln s − s + 1, H(h, µ) is non-negative since When F (t, .) is a solution of problem (2) [7,14]. This is a version of the H-Theorem about decay of entropy. Let us now check that the equilibrium µ is in fact unique. Let us suppose that F is a positive probability density, smooth, rapidly-decaying and a time independent solution of (2), F (t, .) = F (.). Then, we get So that and there exist ϕ such that F = µ(v) ϕ(x), and putting this into (4) yields since (v ∧ B e ) · ∇ v µ = 0. (Note that this fact will be of use through this article). By positivity and the fact that F is a probability density, we get F = µ. Concerning (2), we are interested in the return to the global equilibrium µ, in the sense that exponential decay to equilibrium for perturbations around the Maxwellian is considered, and the convergence of F to µ in norms in the largest possible spaces. Now, we fix some notation for the weighted Lebesgue spaces.

ZEINAB KARAKI
To answer such questions, when F is close to the equilibrium µ, we define f to be the standard perturbation of µ We then rewrite equation (2) in the following form: 1.2.2. The main results. First we will show that the problem (4) is well-posed in the L 2 (µ 1/2 ) space, in the sense of the associated semi-group (See [27]). We associate with the problem (4) the operator P 1 defined by The problem (4) is then written

Remark 1.3.
We note that in a work in progress, we show that operator P 1 is maximal accretive with non-regular magnetic field and more precisely we suppose just that B e ∈ L ∞ (T 3 ), which matches Hypothesis 1.8. For the time being, we rely on Theorem 1.2 which forces us to assume that B e is smooth.
We also show the exponential convergence towards equilibrium in the norm L 2 (µ 1/2 ).
, then there exist κ > 0 and c > 0 (two explicit constants independent of f 0 ) such that Note that in the preceding statement the mean f (t) µ is preserved over time.
We give a result about the return to the global equilibrium µ with an exponential decay rate in the space H 1 (µ 1/2 ). Theorem 1.5. We suppose that B e ∈ C ∞ (T 3 ). There exist c, κ > 0 such that for all f 0 ∈ H 1 (µ 1/2 ) with f 0 µ = 0, the solution f of the system (4) satisfies Herda and Rodrigues in [19] have showed an estimate of type H 1 for the solution of the Vlasov-Fokker-Planck equation with a constant magnetic field by the hypocoercivity method. Their goal was not to obtain a large-time behaviour result of solutions but to have a uniform estimates for a parameter limit (parameter of diffusion) (see also [3,18] for the study of the Vlasov-Poisson-Fokker-Planck with constant magnetic field). Kinetic equations with magnetic field have also been studied by Zhang and Yin in [31] who have showed existence theorems for the initial value problem of the cometary flow equation with an external electromagnetic force (see also [6] for the study of the cometary flow equation with a self-consistent electromagnetic field). Remark 1.6. We note that Theorems 1.4 and 1.5 give convergence to the equilibrium µ at exponential rate for the solution F on spaces with an exponential weight with the notation given in (3), so 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 [10]. The following result gives convergence in L p (m) norms of F to µ with m = v k . So that there is no ambiguity, we denote the mean with respect to the usual L 1 norm by The main result of this paper in this direction is the following.
there exists c k,p > 0 such that the solution F of the problem (2) satisfies the decay estimate It is also possible to obtain the same type of results in the weighted Sobolev spaceW 1,p (m) which is defined bỹ We equip the previous space with the following standard norm: The second main result of this paper is the following.  1 and a i m,2 are constants defined afterwards in (47)-(49) and (56)-(58) and κ is defined in Theorem 1.5.
Note that we need to consider a non-classical weighted Sobolev space (more weight is needed on the function than on its derivatives in x and v), which is necessary to close the dissipativity estimates inW 1,p (m). Note that this trick is not necessary in exponentially weighted spaces. (See also Remark 4.17.) 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 [15,26,28,29], 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 in L 2 specifically were developed in [8,13] for linear collisional kinetic models. The hypocoercivity L 2 and H 1 methods were developed in [14,28] for linear collisional kinetic opertors with one-dimensional kernels. See also [24] for an introductive papers for the hypocoercivity methods for the general collisional kinetic models. The methods used were close in spirit to the ones developed by Guo in [11,12] in functional spaces with exponential weights.
In recent years, the theory of factorization and enlargement of Banach spaces was introduced in [10] and [22]. We note that Mouhot in [23] initiated this type of strategy which has then been developed in an abstract setting by Gualdani, Mischler and Mouhot in [10]. This theory allows us to extend hypocoercivity results into much larger spaces with polynomial weights. We refer for example to [4] and [22], 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 [17] for the inhomogeneous Boltzmann equation without angular cutoff case.
The main objective of this article is to obtain exponential decay results for solutions of the Fokker-Planck equation with magnetic field in the largest possible space. To achieve this goal, we first develop both L 2 and H 1 hypocoercivity methods. Then by applying the enlargement method we extend the results to much larger spaces like L 1 with polynomial weights. We emphasize that, on the way to proving the previously mentioned results, we also prove quantitative regularity estimates.
We conclude this section with some comments on our result. For the proof of Theorem 1.4, we follow the micro-macro method proposed in [14]. Note that for the proof of Theorem 1.4, the black box method proposed in [8] (see also [4]) could perhaps be employed, anyway the presence of the Magnetic field induces some difficulties. To prove Theorem 1.7 and 1.9, we apply the abstract theorem of enlargement from [10,22] to our Fokker-Planck-Magnetic linear operator. We deduce the semi-group estimates of Theorem 1.4 on large spaces like L p ( v k ) and 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 [11,16] and [12,30]).

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 (µ 1/2 ). 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 1.7 and 1.9 with factorization and enlargement of the functional space arguments.
2. Study of the operator P 1 . In this part, we show that the problem (7) is well-posed in the space L 2 (µ 1/2 ) 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 (µ 1/2 ). Notation 2.1. We define P 0 by The linearization around the equilibrium µ of the Cauchy problem (2) reduces the study of the operator P 1 defined in (5) to the study of P 0 , since P 1 is obtained via a conjugation of the operator P 0 by the function µ, that is to say Similarly, we can define the operator P θ as the conjugation of the operator P 1 by the function µ θ with θ ∈]0, 1]. Note that any result on the operator P θ 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 and X 0 is defined in (6). We now show that operator P 1/2 is maximal accretive in the space L 2 (T 3 × R 3 ) and note that this gives the same result for P 1 in the space L 2 (µ 1/2 ). We study the following problem: Proposition 2.2. Suppose that B e ∈ L ∞ (T 3 ). Then the closure with respect to the norm L 2 (T 3 × R 3 ) of the magnetic-Fokker-Planck operator P 1/2 on the space is maximally accretive. Proof. We adapt here the proof given in [26, page 44]. We apply the abstract criterion by taking H = L 2 (T 3 × R 3 ) and the domain of P 1/2 defined by D( . First, we show the accretivity of the operator P 1/2 . When u ∈ D(P 1/2 ), we have to show that P 1/2 u, u ≥ 0 where we note by . , . the scalar product in the space L 2 (T 3 × R 3 ) and . the associated norm. Indeed, Let us now show that there exists λ 0 > 0 such that the operator We take λ 0 = 3 2 + 1 (following [26]). Let u ∈ H satisfy u, We have to show that u = 0. First, we observe that equality (13) implies that Under Hypothesis that B e ∈ L ∞ (T 3 ), and following Hormander [20,21] or Helffer- . Now we introduce the family of truncation functions ξ k indexed by k ∈ N * and defined by . We note that in [26], a cut-off in x and v was necessary to develop the argument whereas here it suffices to perform a cut- When u satisfies (13) in particular, when h = ξ 2 k w , we get for all w ∈ C ∞ In particular, we take the test function w = u, so By an integration by parts, we obtain Which gives the existence of a constant c > 0 such that, for all k ∈ N * ,

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 317
This leads to, for η > 0 to be chosen later, Choosing η ≤ 1 4 , we get Taking k −→ +∞ in (14), leads to u = 0.
Proof of Theorem 1.2. According to Remark 2.1, the operator P 1 has a closure P 1 from C ∞ 0 (T 3 ×R 3 ). This gives Theorem 1.2, by a direct application of Hille-Yosida's theorem (cf. [27] for more details for the semi-group theory) to the problem (4), with . From now on, we write P θ for the closure of the operator P θ from the space 3. Trend to the equilibrium.

Hypocoercivity in the space
The purpose of this subsection is to show the exponential time decay of the L 2 (µ 1/2 ) entropy for P 1 , based on macroscopic equations. First, we try to find the macroscopic equations associated with system (4). We write f in the following form: where Definition 3.1. In the following, we define

and introduce a class of Hilbert spaces
We recall that the operator Λ 2 x is an elliptic, self-adjoint, invertible operator from [15, section 6] for a proof of these properties). (4), with the decomposition given in (15). Then we have

Lemma 3.2. Let f be the solution of the system
Where Op 1 denotes a bounded generic operator from Proof. We suppose is f is a Schwarz function. In order to show equation (16), we integrate equation (4) with respect to the measure dµ := µ(v) dv . We get where we noted by . , . the scalar product in the space L 2 (µ 1/2 ) and . the associated norm. Then, by using the equality v f dµ = v h dµ with dµ = µ dv we obtain hence equality (16).
To show (17), we multiply equation (4) by v before performing an integration with respect to the measure dµ, we obtain where . Now, we will calculate term by term the left-hand side of the equality (18). We use that because ∀j = i, performing an integration by parts we get By using the equality (19), then we have

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 319
Therefore where m is defined in (15).
By combining all the previous equalities in (18), we obtain so the macroscopic equation (15) takes the following form Now we are ready to build a new entropy, defined for any u ∈ L 2 (µ 1/2 ) by Using the Cauchy-Schwarz inequality gives us directly that Now, we can prove the main result of hypocoercivity leading to the proof of Theorem 1.4.

Proposition 3.5.
There exists κ > 0 such that, if f 0 ∈ L 2 (µ 1/2 ) and f 0 = 0, then the solution of system (4) satisfies x r, m . We will omit the dependence of f with respect to t. For the first term, we notice that d dt by Poincaré's inequality and the spectral gap property of the operator L (see [9, Lemma 2.1] for more details on this subject). For the second term, using the macroscopic equations, we get x ∇ x m . Now, using m ≤ h , the Cauchy-Schwarz inequality and the following estimate:

ZEINAB KARAKI
Poincaré's inequality on L 2 (dx) takes the form where φ = φ(x) dx and c P > 0 is the spectral gap of −∆ x on the torus (see [14,Lemma 2.6] for the proof of the previous inequality). Using this, we obtain, by applying the previous estimate to r ( since gathering (21) and (22), we get We can deduce the proof of Theorem 1.4.
Proof of Theorem 1.4. Starting from Lemma 3.4 and Proposition 3.5, we have, for f the solution of the system (4), This completes the proof of Theorem 1.4.
3.2. Hypocoercivity in the space H 1 (µ 1/2 ). We will establish some technical lemmas, which will help us to deduce the exponential time decay of the norm H 1 (µ 1/2 ), noting that we work in three dimensions.
The following lemma gives the exact values of some commutators will be used later.
Lemma 3.6. The following equalities hold: . The first two equalities are obvious. We go directly to the proof of 3.
Similarly we can show that, for all 1 ≤ i ≤ 3,

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 321
This proves the equality 3. Now, we will show (4), Now, we are ready to build a new entropy that will allow us to show the exponential decay of the norm H 1 (µ 1/2 ). We define this modified entropy by Proof. Let u ∈ H 1 (µ 1/2 ). Using the Cauchy-Schwarz inequality, we get This implies (23) 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 (4), which will be denoted as f , in the space H 1 (µ 1/2 ) in the sense of an associated semi-group. Using the preceding results, we are able to study the decrease of the modified entropy E(f (t)).
Proof. The time derivatives of the four terms defining E(f (t)) will be calculated separately. For the first term we have

ZEINAB KARAKI
The second term may be calculated as We used the fact that the operators v · ∇ x and (v ∧ B e ) · ∇ v are skew-adjoint in L 2 (µ 1/2 ) by Lemma A.1. According to equalities (1) and (3) of Lemma 3.6, we then obtain d dt The time derivative of the third term can be calculated as follows: We calculate each term of equality (24). For the first term, using equalities (1), (2) and (3) of Lemma 3.6, we obtain For the second term of equality (24), using equality (4) of Lemma 3.6, we have Combining the preceding equalities of the two terms in (24), we get According to Lemma A.1, the operators v · ∇ x and (v ∧ B e ) · ∇ v are skew-adjoint in Using equalities (25)-(26), we obtain

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 323
Finally, the time derivative of the last term takes the following form Tying together all these computations, we get Now, using Lemma A.3, the time derivative of E(f (t)) takes the following form: Now we estimate the scalar products in the previous equality in L 2 (µ 1/2 ). For all η, η and η > 0, we have and using then The last scalar product is bounded by Combining all the previous estimates, we have We notice that We choose η, η , E, D and C such that Under the previous conditions, we get d dt Using the Poincaré inequality in space and velocity variables, we then obtain d dt Which completes Proposition 3.5 with κ = E 8 Proof of Theorem 1.5. Using Lemma 3.7 and Proposition 3.8, we get κ > 0 and 1 < E < D < C such that . This completes the proof of Theorem 1.5.

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 [10].
Notation: Let E be a Banach space.
-We denote by C(E) the space of unbounded, closed operators with dense domains in E.

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 325
-Let a ∈ R. We define the complex half-plane ∆ a = {z ∈ C, Re z > a}.
-Let L ∈ C(E). Σ(L) denote the spectrum of the operator L and Σ d (L) its discrete spectrum. -Let ξ ∈ Σ d (L), for r sufficiently small we define the spectral projection associated with ξ by -Let a ∈ R be such that ∆ a ∩ Σ(L) = {ξ 1 , ξ 2 , ..., ξ k } ⊂ Σ d (L). We define Π L,a as the operator We need the following definition on the convolution of semigroups (corresponding to composition at the level of the resolvent operators).

Definition 4.2. Consider a Banach space (E, · E ) and some operator L ∈ C(E).
We say that L 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 · E on E if for all f ∈ D(L) and f * ∈ E * such that The concept of hypodissipativity will be needed later. We give a practical criterion to prove that an operator L is hypodissipative in E. We refer to the paper [10, Section 2.3] for an introduction to this subject. Now, we recall the crucial Theorem of enlargement of the functional space.
where S B is the semi-group associated to the operator B acting on E. Then for all 0 > a > a, we have the following estimate: where S L is the semi-group associated to the operator L acting on E.
In the application of the previous theorem in the case of the Fokker-Planck operator with magnetic field, the key step of the proof is to obtain decay estimates verifying dissipation and regularization properties. In order to prove the dissipativity of the operator we apply Corollary 4.3. We give a lemma providing a practical criterion to prove hypothesis (H 3 ) of the previous theorem. Then for all a > a, there exist some explicit constants n ∈ N and C a ≥ 1, such that ∀t ≥ 0, Note that, in the application of the previous lemma in the case of the Fokker-Planck operator with magnetic field, only regularization properties of the operator where and where we recall that P 0 was introduced in Section 2 and with and B e is the external magnetic field satisfying B e ∈ W 1,∞ (T 3 ). As mentioned in Section 2, the Maxwellian µ is a solution of the system (2). We will need the

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 327
following modified Poincaré inequality: where λ p > 0 which depends on the dimension (see [22,Lemma 3.6]). See also [24], [2] and [1]. Now we will define the expanded functional space.
Remark 4.7. We note that ∇ v · (v ∧ B e ) = 0 (because B e is independent of v) and for all m which are radial in v and independent of x, we obtain Therefore, we obtain that Ψ m,p is independent of the magnetic field when m is radial in v.
We will show the decay of the semi-group associated with the problem (2) in the spaces L p (m) where p ∈ [1,2], when m verifies the following hypothesis: (W p ) The weight m is defined such that L 2 (µ −1/2 ) ⊂ L p (m) with continuous injection and lim sup |v|→+∞ Ψ m,p =: a m,p < 0.

Remark 4.8.
In the following, we note m 0 = µ −1/2 the exponential weight. By direct computation, (See Lemma 3.7 in [10] for a proof of the previous property). Under the previous hypothesis, by direct computation we obtain that the semi-group S L0 associated to the operator L 0 := −P 0 defined in (27) 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 (W p ).

Proof of
) and p ∈ [1, 2]. We will prove where is such that χ(v) = 1 when |v| ≤ 1. We also denote by A and B the restriction of the operators A and B to the space E. Proof. The proof follows the one given in Lemma 3.8 in [10]. Let F be smooth, rapidly decaying and positive function F . Since of Ψ m,p is independent of the magnetic field (see Remark 4.7), by performing an integration by parts with respect to x and v and using Remark A.2 (specifically the fact that the operator v · ∇x is skew-adjoint in L 2 (m) with m = v k and m is independent of x), we have Let now take a > a m,p . As m satisfies the hypothesis (W p ), there exist M and R two large constants such that and we obtain 1 p This completes the proof of Lemma 4.11.
From now on, a, M and R are fixed. We note that B * is the dual operator of B relative to the pivot space L 2 (T 3 × R 3 ), which is defined as follows:

Lemma 4.12 (Regularization properties).
There exists b ∈ R and C > 0 such that, for all t ≥ 0, where p and q are the conjugates of p and q respectively and m 0 = µ −1/2 .
Proof. We consider F (t) the solution of the evolution equation We introduce the following entropy defined for all t ∈ [0, T ], with T 1 and r > 1 to be fixed later: where B > α > D, β, E < √ βD 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 [14], we choose the constants α, D and E > 0 large enough such that there exist a constant C G > 0 (depending on B e L ∞ (T 3 ) and ∇ x B e L ∞ (T 3 ) ) such that Here, C χ > 0 is a uniform constant in R > 1 but depends on M .
We choose the constants β and T > 0 such that We deduce that Now, the Nash inequality [25] implies that there exists C d > 0 such that We need to have an estimate based on ∇ x,v F L 2 (m0) . Firstly, On the other hand, we use the fact that v m 2 0 = ∇ v (m 2 0 ) to estimate vF L 2 (m0) . We get TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 331 and integrating by parts in v in the previous estimate, we obtain Applying the Cauchy-Schwarz inequality, we get Using the previous estimate and inequality (33), we have Using the previous inequality and the fact that , there exists C d > 0 such that the estimate (32) becomes Using Young's inequality with p = (d + 1) and q = (d + 1)/d , we get, for all ε > 0, Using the previous estimate, we choose ε > 0 small enough that there is a C > 0 According to Remark 4.8 there exists b ∈ R such that ∀p ∈ [1, 2] Finally, using the previous estimate when p = 1 and choosing r = 3d + 1, we deduce that there exists B > 0 such Thanks to Gronwall's Lemma, there exists B > 0 such that Then, As a consequence, using the continuity of S B (t) on L p (m 0 ) with p = 2, ∀t ∈ (T, +∞), and eventually for all t ∈ (0, +∞) Let us now consider p and q satisfying 1 ≤ p ≤ q ≤ 2. S 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 B (t) : L p (m 0 ) → L q (m 0 ), we get the following estimate: This shows the first estimate. Now we will show the second estimate. According to the first estimate, we have where h = m 0 F 0 . Then by duality, we get where p and q are the conjugates of p and q respectively. Which gives the result by reusing the definition of weighted dual spaces This completes the proof.
Proof. We first prove the second inequality. Let F 0 ∈ L p (m) with m a polynomial weight satisfying Hypothesis 1.8. For all 1 ≤ p ≤ 2 and for all t ∈]0, 1] and v ∈ R 3 ,

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 333
using Lemma 4.12 with q = 2, we get where Θ = (3d To show the first estimate, we proceed step by step.
Step 2: Of the inequality (36), it follows that for g = AF 0 , we get by a duality argument and noting that A * = A, we get Finally, according to our definition of weighted dual spaces and replacing h by mF 0 , we obtain To obtain the result, we notice that and we combine the previous estimate with the estimate (37), which completes the proof of the first estimate.
Now we prove Theorem 1.7.
Proof of Theorem 1.7. For p ∈ [1,2]. We consider E = L p (m), E = L 2 (m 0 ), and denote L 0 and L 0 the Fokker-planck operator considered respectively on E and E (defined in (27)). We split the operator as L 0 = A + B as in (30). Let us proceed step by step: Fokker-Planck operator defined in (27) on the space L 2 (m 0 ) where m 0 = µ −1/2 and the constants κ and c > 0, for which, for all F 0 ∈ L 2 (m 0 ) such that F 0 = 0, Which implies the dissipativity of the operator L 0 − a on E, for all 0 > a > −κ.
Then for all a > a, there exist constructible constants n ∈ N and C a ≥ 1 , such that • Step 4: End of the Proof All the hypotheses of Theorem 4.4 are satisfied. We deduce that L 0 − a is a dissipative operator on E for all a > max(a m,p , −κ), with the semi-group S L0 (t) satisfying estimate (8).
Allong with the definition of H, we obtain for all t ∈ (0, T ] Following in the proof of Lemma 4.12, we obtain ∀t ∈ (0, +∞), By interpolation, we get Therefore, using the same techniques as in the proof of Corollary 4. 13 One could try to use Lemma 2.17 in [10] to prove the hypothesis (H 3 ) in Theorem 4.4. But the presence of the magnetic field creates a lot of difficulties in adapting the proof of this lemma. We note also that in the hypothesis of Lemma 2.17 include an estimate of type where T n (t) = (AS B (t)) ( * n) for some n with α ∈ (0, 1] whereas our estimates are of the form α = 3d + 1 > 1 and n = 1.

4.2.2.
Proof of Theorem 1.9. 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 inW 1,p (m) defined in (9).
For the proof of Theorem 1.9, we consider the space E =W 1,p (m) and E = H 1 (m 0 ).

Definition 4.15.
We split operator L 0 into two pieces and define for all R, M > 0 where We also denote A and B the restriction of operators A and B on the space E respectively.  58)) such that operator B − a is dissipative iñ W 1,p (m) where p ∈ [1,2]. In other words, the semi-group S B satisfies the following estimate: Proof. Let F 0 ∈W 1,p (m). We consider F the solution of the evolution equation Recall that the norm on the spaceW 1,p (m) is given by Differentiating the previous equality with respect to t, we get d dt We now estimate each term of the equality (41). For the first term in (41), we apply Lemma 4.11 and get 1 p Secondly, we differentiate the equation (40) with respect to v, and then we use the equalities of Lemma 3.6. We get the following equation (recall d = 3): This gives d dt Then, proceeding exactly as in the proof of Lemma 4.11 and applying Young's inequality, we obtain for all η 1 > 0 d dt Finally, we estimate the last term of the equality (41). We treat two cases, and then we use an interpolation argument to complete the proof. • Case 1: p = 1. We differentiate the equation (40) 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: Using the previous equation, we obtain

TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 337
Using the computations made in Lemma 4.11 for p = 1, using Lemma B.1 in the appendix B, and performing an integration by parts with respect to v, we get d dt where, we used the fact that (v ∧ ∂ xi B e ) · ∇ v m = 0. Then, defining the norm and using the previous definition, we have d dt Collecting all the estimates, we obtain d dt We define then (for M and R to be fixed below).
• Case 2: p = 2. Again, we differentiate the equation (40) with respect to x, and we use the equalities of Lemma 3.6 to obtain the following equation: Using the calculations made in Lemma 4.11 and the previous equation, we obtain d dt Then, by integration by parts with respect to v, we get d dt According to the Cauchy-Schwarz inequality, for every ε > 0, there is a C ε > 0 such that d dt We choose ε = 1 4 , and we finally get d dt Collecting all the estimates, we thus obtain d dt  TREND TO THE EQUILIBRIUM FOR THE FOKKER-PLANCK-MAGNETIC SYSTEM 339 Again, we define then, for M and R to be fixed in the next paragraph Again, we denote Assuming k satisfies we obtain that a i m,2 < 0 for all i = 1, 2, 3. Consequently, we may find M, R > 0 large enough so that for all 0 > a > max(a 1 m,2 , a 2 m,2 , a 3 m,2 ) d dt Hence the operator B − a is dissipative onW 1,2 (m) for such M and R. For the general case 1 ≤ p ≤ 2: The cases 1 and 2 show us that the operator S B (t) is continuous onW 1,1 (m) (onW 1,2 (m)) with the operator B is given by 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 1.8, we obtain that the operator S B (t) is continuous onW 1,p (m) for all 1 ≤ p ≤ 2, with the following dissipative estimate: ∀0 > a > max(a i m,1 , a i m,2 , i = 1, 2, 3), S B (t) F 0 W 1,p (m) ≤ Ce at F 0 W 1,p (m) .
This completes the proof.

Remark 4.17.
We note that the only step where we needed to suppose F ∈ L 2 ( v m) and ∇ v F, ∇ x F ∈ L 2 (m), was in the estimate of But this problem is not encountered in Sobolev space with exponential weight µ −1/2 , because we have the equality that allowed us to estimate the term (60) with m = m 0 = µ −1/2 without the need to add a weight. (See Lemma A.3). This is no longer true with a polymonial weight because From now on, M and R are fixed as in Lemma 4.16.

340
ZEINAB KARAKI Lemma 4.18 (Property of regularization). There exist b and C > 0 such that, for all p, q with 1 ≤ p ≤ q ≤ 2, we have Here 2 ≤ q ≤ p ≤ +∞ are the conjugates of p and q respectively.
Proof. Let F be the solution of the evolution equation In to the proof of Lemma 4.12, the following relative entropy has been introduced . We have shown, for constants α, D, E and β > 0 well chosen, that there exist C > 0 and r = 3d + 1 such that Finally, to complete the proof, we use the Riesz-Thorin Interpolation Theorem in the real case on the operator S B (t) . We obtain the continuity of S B (t) fromW 1,p (m 0 ) toW 1,q (m 0 ), with S B satisfying the estimate (61