HYPOCOERCIVITY WITHOUT CONFINEMENT. 1. Introduction

Transcription

1 HYPOCOERCIVITY WITHOUT CONFINEMENT EMERIC BOUIN, JEAN DOLBEAULT, STÉPHANE MISCHLER, CLÉMENT MOUHOT, AND CHRISTIAN SCHMEISER Abstract. Hypocoercivity methos are extene to linear kinetic equations with mass conservation an without confinement, such that the long-time behavior has algebraic ecay as in the case of the heat equation. Two alternative approaches are evelope: an analysis base on ecouple Fourier moes an a irect approach where, instea of the Poincaré inequality for the Dirichlet form, Nash s inequality is employe. The former is also use to provie a proof of exponential ecay to equilibrium on the flat torus. Finally, the results are extene to larger function spaces by a factorization metho. 1) ) We consier the Cauchy problem 1. Introuction t f + v x f = Lf, fx, v, 0) = f 0 x, v), for the istribution function fx, v, t), with the position variable in the whole space, x, or in the flat -imensional torus, x T represente by the box [0, π) ), with the velocity variable v, an with time t 0. For the collision operator L two choices will be consiere: a) Fokker-Planck: Lf = v vf + v f), b) Scattering: Lf = σv, v ) fv )Mv) fv)mv ) ) v, with the normalize Maxwellian Mv) = π) / e v /. More general microscopic equilibria coul be consiere up to some technicalities, so we shall only consier the Gaussian case for sake of simplicity. The scattering rate σ is assume to satisfy 3) 1 σv, v ) σ, v, v. With σ = 1, case b) inclues the relaxation operator Lf = ρ f M f, with the position ensity ρ f x, t) = fx, v, t) v. Date: June 1, 017.

2 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER Note that micro-reversibility, i.e. symmetry of σ, is not require. Both cases share the properties that the null space of L is spanne by the Maxwellian M, an that L only acts on the velocity variable. Uner the requirement σv, v ) σv, v) ) Mv ) v = 0, v, in case b), both cases satisfy local mass conservation, i.e. R Lf v = 0. We introuce the Fourier representation with respect to x, fx, v, t) = fξ, v, t) e i x ξ µξ), where µξ) = ξ is the Lesbesgue measure for the case x, whereas µξ) is iscrete for x T. The normalization of f an of µξ) are chosen such that the Parseval ientity reas f, v, t) L x) = f, v, t) L µξ)). In both cases, the problem for the Fourier transform is ecouple in the ξ- irection: 4) t f + i v ξ) f = L f, 5) fξ, v, 0) = f0 ξ, v). We shall exten the approach of [] to hypocoercivity theory see [11]), which has been inspire by [5], but is also close to the Kawashima compensating function, see [8, 3]. In the following section we slightly strengthen the abstract hypocoercivity result of [] by allowing complex Hilbert spaces an by proviing explicit formulas for the coefficients in the ecay rate. This result Theorem 1) is applie in Theorem to the Fourier transforme problem 4), 5). In the case b) of a collision operator of scattering type the result can be extene to larger function spaces by the factorization metho of [4]. A simple, abstract version of this metho is formulate an proven in Section 3 Theorem 3), an again applie to 4), 5): see Theorem 4. For the problem 1), ) on the flat torus, exponential ecay to equilibrium is euce from the Fourier transforme problem in Section 4, recovering a result Theorem 5) which can also be obtaine irectly with the metho of []. Here the factorization metho is applie irectly to 1), ), allowing extension to a larger space. Sections 5 an 6 present two ifferent approaches to the whole space problem. In both cases algebraic ecay to zero is shown with the same rate as for the heat equation. In Section 5 the Fourier moe analysis is use, whereas in Section 6 we evelop a variant of the metho of [], where the so-calle macroscopic coercivity conition, typically taking the form of a Poincaré inequality, is replace by an application of Nash s inequality, which can also be use for showing ecay for the

3 HYPOCOERCIVITY WITHOUT CONFINEMENT 3 heat equation. The result by the Fourier moe analysis is stronger in the sense that it also hols on larger spaces. On the other han, the irect approach by Nash s inequality is also applicable to problems with non-constant coefficients.. Moe-by-moe hypocoercivity We shall use the abstract approach of []. Although the extension of the metho to Hilbert spaces over the complex numbers is straightforwar, we carry it out here for completeness an also erive explicit information on the ecay rate. Theorem 1. Let L an T be close linear operators in the complex Hilbert space H,, ) with inuce norm. Let L be hermitian an T be anti-hermitian. Let Π enote the orthogonal projection to the null space of L an efine A := 1 + TΠ) TΠ ) 1 TΠ), where enotes the ajoint with respect to,. Let positive constants λ m, λ M, an C M exist, such that, for any F H, the following properties are satisfie: 6) 7) 8) 9) microscopic coercivity: LF, F λ m 1 Π)F, macroscopic coercivity: TΠF λ M ΠF, parabolic macroscopic ynamics: ΠTΠF = 0, boune auxiliary operators: AT1 Π)F + ALF C M 1 Π)F. Then there exist positive constants C an λ, such that the semigroup generate by L T satisfies e L T)t Ce λt. A possible choice for the constants is { } λ M 10) C = 3, λ = λ M ) min λ m λ M 1, λ m, 1 + λ M ) CM. Proof. The approach of [] relies on the Lyapunov or moifie entropy) functional: F H[F ] := + δ Re AF, F, with a small positive constant δ. Its time erivative along solutions of is 11) F t + TF = LF H[F ] = LF, F δ ATΠF, F t δ Re AT1 Π)F, F + δ Re TAF, F + δ Re ALF, F =: D[F ].

4 4 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER Since the hermitian operator ATΠ can be interprete as the application of the map z z 1+z to TΠ) TΠ, the conitions 6) an 7) imply that the sum of the first two terms in the entropy issipation 11) is coercive: LF, F + δ ATΠF, F λ m 1 Π)F + δ λ M ΠF. 1 + λ M For δ small enough, these two terms control the remaining three terms, if the operators appearing there are boune an act only on the microscopic part 1 Π)F of the istribution. Whereas the latter is straightforwar for AT1 Π) an for AL, for the operators A an TA it follows from 8). Whereas bouneness of AT1 Π) an of AL is assume in 9), it hols in general for A an TA by [, Lemma 1]: 1) AF 1 1 Π)F, TAF 1 Π)F. The former shows the norm equivalence of H[F ] to F for δ < 1: 1 δ 13) F H[F ] 1 + δ F. In the following, we choose δ 1/. With the abbreviations a := 1 Π)F, b := ΠF, the issipation term can be estimate by D[F ] λ m δ) a + δ λ M b δ C M a b. 1 + λ M In view of the first term we a the requirement δ λ m /. Finally, the choice δ = 1 } {1, min λ m λ M λ m, 1 + λ M ) CM implies coercivity with D[F ] λ m 4 a + δ λ M 1 + λ M ) b 1 3 min where, by δ 1/, F 4 3 H[F ] has been use. { λ m, δ λ } M H[F ] = 1 + λ M δ λ M λ M ) H[F ], This result will be applie to 4) for fixe ξ, with the choices ) H = L v/m), F = F M 1 v, ΠF = F v M, It is easily seen that 7) 8) hol with 14) λ M = ξ. an TF = i v ξ)f. Microscopic coercivity 6) with λ m = 1 hols as a consequence of the Poincaré inequality [10] u M v u M v, if u M v = 0,

5 HYPOCOERCIVITY WITHOUT CONFINEMENT 5 with u = F/M in case a). In case b), by assumption 3), we have LF, F F ΠF, F = 1 Π)F as in [1, Proposition.], an hence λ m = 1. The operator A is efine by AF )v) := i ξ 1 + ξ v F v ) v Mv) an satisfies the estimate 1 AF = A1 Π)F 1 + ξ v ξ 1 Π)F v R ) 1 1/ 1 Π)F v ξ) M v = ξ 1 Π)F. 1 + ξ 1 + ξ For estimating AL we note for case a) that v LF v = v F v, an therefore AL = A, whereas in case b) the collision operator L is boune: LF σ ρ f M f) v M = σ 1 Π)F. For both cases we therefore obtain ALF σ ξ 1 Π)F, 1 + ξ where σ can be set to 1 in case a). The operator ATF )v) = Mv) 1 + ξ v ξ) F v ) v can be estimate similarly: 3 ξ ATF 1 + ξ F, meaning that we have proven 9) with 15) C M = σ ξ + 3 ξ 1 + ξ. Thus, Theorem 1 can be applie, an with 14) an 15), the formula 10) for the exponential ecay rate gives ξ { 16) ξ ) min 1 + ξ } 1, σ + K ξ 3 ξ ) 1 + ξ =: λ 1 ξ, K = σ ), which follows from 3 σ K = min t>0 1 + t 1 + a t) = a with a = 3 σ.

6 6 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER Theorem. Let ξ, f 0 ξ, ) L v/m), an f be the solution of 4), 5). Then, with λ ξ efine in 16), fξ,, t) 3 L v/m) e λ ξt f 0 ξ, ), t 0. L v/m) The result hols for both the Fokker-Planck with σ = 1) an the scattering collision operators satisfying assumption 3). 3. Enlarging the space by factorization Square integrability against the inverse of the Maxwellian is a rather restrictive assumption on the initial ata. In this section it will be relaxe with the help of the abstract factorization metho of [4]. Since this approach requires the splitting of the collision operator in a issipative loss term an a ecay-generating gain term, we only consier the case b) of scattering moels in this section. We only nee a special case of the factorization metho which, for completeness, we state an prove below. Theorem 3. Let B 1, B be Banach spaces an let B be continuously imbee in B 1, i.e. 1 c 1. Let B an A + B be the generators of the strongly continuous semigroups e Bt an, respectively, e A+B)t on B 1. Let there be positive constants c, c 3, c 4, λ 1, an λ, such that, for all t 0, e A+B)t c e λ t, e Bt 1 1 c 3 e λ 1t, A 1 c 4, where i j enotes the operator norm for linear mappings from B i to B j. Then there exists a positive constant C = Cc 1, c, c 3, c 4 ) such that, for all t 0, { e A+B)t 1 1 C 1 + λ1 λ 1) e min{λ 1,λ }t for λ 1 λ, C 1 + t) e λ 1t for λ 1 = λ. Proof. Integrating the ientity s with respect to s [0, t] gives e A+B)s e Bt s)) = e A+B)s A e Bt s) e A+B)t = e Bt + t 0 e A+B)s A e Bt s) s, an therefore t e A+B)t 1 1 c 3 e λ1t + c 1 e A+B)s A e Bt s) 1 s c 3 e λ 1t + c 1 c c 3 c 4 e λ 1t 0 t The proof is complete by a straightforwar computation. 0 e λ 1 λ )s s.

7 HYPOCOERCIVITY WITHOUT CONFINEMENT 7 Theorem 3 can be applie to solutions of 4), 5) with a collision operator of scattering type b) an AF v) = Mv) σv, v )F v ) v, R ) BF v) = iv ξ + σv, v )Mv ) v F v), B = L v/m). For the larger Banach space we consier 17) B 1 = L w k v), with w k v) = 1 + v ) k/, k >. Then the assumptions of Theorem 3 are satisfie with λ = λ ξ / 1 4 efine in 16)), with λ 1 = 1 because of σv, v )Mv ) v 1. The bouneness of A : B 1 B follows from 3) an 1/ 18) AF L v/m) σ F L 1 v) σ w 1 k v) F L w k v). ξ 1+ ξ Theorem 4. Let ξ, let f 0 ξ, ) L w k v) with w k as in 17), an let f be the solution of 4), 5) with a collision operator of scattering type b) satisfying assumption 3). Then, there exists a constant C = Ck,, σ) such that, with λ ξ efine in 16), fξ,, t) C L w k v) e λ ξ t f 0 ξ, ), t 0. L w k v) as 4. Exponential ecay to equilibrium in the perioic case The unique global equilibrium in the case x T is given by 19) f x, v) = ρ Mv), with ρ = 1 T f 0 x v. T If we represent the flat torus T by the box [0, π) with perioic bounary conitions, the Fourier variable satisfies ξ Z. For ξ = 0, the microscopic coercivity with λ m = 1 see Section ) implies f0,, t) f 0, ) L v/m) e t f 0 0, ) f 0, ) L v/m). For all other moes f ξ, v) = 0, ξ 0, an we can use Theorem with λ ξ K/, withe notations of 16). An application of the Parseval ientity then proves f,, t) f L x v/m) 3 e tk/4 f0 f L x v/m).

8 8 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER This result can also be erive by irectly applying Theorem 1 to 1), ), as in []. For collision operators of scattering type b) it can be improve by the factorization metho applie to 1) with Afv) = Mv) σv, v )fv ) v, Bfv) = v x fv) fv) σv, v )Mv ) v, R { } B 1 = f L x; L 1 v)) : f x v = 0, T R { } B = f L x v/m) : f x v = 0. T With the notation of the preceing section, we have λ = 1 48, an λ 1 = 1 follows from the explicit representation ) e Bt f 0 )x, v) = exp t σv, v )Mv ) v f 0 x vt, v) from which we euce From 18) we also have e Bt f 0 1 e t f 0 1. Af σ f 1, an ) T R f f 1 = f v x T M v x = f. Therefore the assumptions of Theorem 3 are satisfie. Theorem 5. For collision operators of scattering type b) satisfying assumption 3) there exists a constant C such that for the solution f of 1), ) in the case x T with f 0 L x; L 1 v)), f,, t) f L x;l 1 v)) C e λt f 0 f L x;l 1 v)) hols with f efine in 19) an λ = σ ). 5. Whole space algebraic ecay by Fourier analysis Since the macroscopic limit of 1) is the heat equation, algebraic ecay to zero can be expecte in the case x. Theorem 6. There exists a constant c > 0 such that for the case x with f 0 L v/m; L 1 x)) L x v/m), the solution f of 1), ) satisfies f,, t) L x v/m) c ) 1 + t) / f 0 L x v/m) + f 0 L v/m;l 1 x)),

9 HYPOCOERCIVITY WITHOUT CONFINEMENT 9 where ) f 0 L v/m;l 1 x)) := v f 0 x M. The result hols for Fokker-Planck an for scattering collision operators satisfying assumption 3). In the latter case the weight M 1 can also be replace by w k, given in 17). Proof. We start with the result of Theorem an integrate with respect to ξ. Then the Parseval ientity implies f,, t) L x v/m) C 1 e λξt f 0 ξ v M for some C > 0. The same hols with the weight w k, if we start from Theorem 4. With the splitting e λξt f 0 ξ = we get the estimate ξ 1 I 1 C f 0, v) L 1 x) e λξt f 0 ξ + e λξt f 0 ξ = I 1 + I, ξ >1 ξ 1 e ξ tk/ ξ C ) π K t f 0, v) L 1 x), where the ecay as t is shown by the coorinate transformation η = ξ t, t 1. The secon part ecays exponentially: I C e tk/ f 0, v) L x), where again the Parseval ientity has been use. An integration with respect to v completes the proof. 6. Whole space application of Nash s inequality A irect application of the hypocoercivity approach of [] to the whole space problem fails by lack of a Poincaré inequality or, in the terminology of [], by lack of macroscopic coercivity. In this section, the approach will be aapte by relaxing macroscopic coercivity using Nash s inequality. We use the abstract setting of Section, applie to 1) with x an the notation Tf = v x f, with the scalar prouct, on L x v/m) an the inuce norm. We use the moifie entropy H[f] := f + δ Af, f, with A := 1 + TΠ) TΠ ) 1 TΠ), an its time erivative 0) H[f] = Lf, f δ ATΠf, f δ AT1 Π)f, f +δ TAf, f +δ ALf, f. t

10 10 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER As in the previous sections, the first term on the right han sie satisfies the microscopic coercivity conition Lf, f λ m 1 Π)f. Although the secon term is not coercive, we shall procee as in the original approach an show that the last three terms can be ominate by the first two for small enough δ > 0. By 1) we have TAf, f = TA1 Π)f, 1 Π)f 1 Π)f. For the following steps, the abstract operators nee to be evaluate for the present situation. A straightforwar computation using v v M v = i gives Therefore TΠ) TΠf = ΠT Πf = M x ρ f. ATΠf = M x w f, with w f x w f = ρ f, implying ATΠf, f = ρ f x w f x = w f x w f ) x w f x 1) = x w f L x) + xw f L x). For the thir term on the right han sie of 0) we use ajoint operators as in []: AT1 Π)f, f = 1 Π)f, TA f. Thus we nee an Therefore an A f = TΠ 1 + TΠ) TΠ ) 1 f = T 1 + TΠ) TΠ ) 1 Πf = Tw f M) = v x w f M TA f = v x ) w f M. TA f 3 xw f L x) AT1 Π)f, f 3 1 Π)f xw f L x) = 3 1 Π)f x w f L x). Consiering the representation 1), we have for any γ > 0: AT1 Π)f, f γ 3 ATΠf, f + 3 γ 1 Π)f. It remains to estimate the last term on the right han sie of 0). As in Section we have either AL = A case a)) or Lf σ f case b)). Therefore in both cases ALf, f σ 1 Π)f A f

11 HYPOCOERCIVITY WITHOUT CONFINEMENT 11 with the convention that σ = 1 in case a). Again 1) implies ALf, f σ 1 Π)f x w f L x) γ σ Using these results in 0) gives t H[f] λ m δ δ 3+σ) with = γ ) 1 Π)f + δ λ m σ) 1 Π)f + ATΠf, f ), γ = 1 3+σ, δ = ATΠf, f + σ γ 1 Π)f. λ m 3+ 3+σ). 1 γ 3+σ) ) ATΠf, f We still nee to show that ATΠf, f controls the macroscopic contribution, i.e. ρ f. We start by representing the norm of the macroscopic contribution in terms of w f : Πf = ρ f L x) = w f L x) + xw f L x) + xw f L x) w f L x) + ATΠf, f. At this point we use Nash s inequality [10], an observe that Consequentially, w f L x) C + Nash w f L 1 x) xw f /+) 4 L x), w f, t) L 1 x) ρ f, t) L 1 x) f 0 L 1 v x) =: m, t 0. Πf ATΠf, f + C Nash m 4 + ATΠf, f + =: Φ 1 ATΠf, f ), The function Φ : [0, ) [0, ) satisfies 0 < Φ < 1, which implies 1 Π)f + ATΠf, f 1 1 Π)f +Φ Πf ) Φ f ) Φ 1+δ H[f]), with the last inequality again a consequence of 13). 0 < z z 0, Φ 1 z z) = z 0 + C Nash m 4 + z + z z 0 + On the other han, for C Nash m 4 ) z +. With the choice z 0 := H[f 0] 1+δ, we have Φ 1+δ H[f]) Φ 1+δ H[f 0] ) H[f 0] 1+δ = z 0, an therefore Φ 1+δ H[f]) c H[f] + H[f 0 ] + + m 4 + ) + c H[f] +, H[f 0 ] + m )

12 1 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER where from now on c is an explicit constant possibly changing from line to line. In the secon inequality we have use a + b) p p 1 a p + b p ), a, b 0, p 1, with p = +. Finally, we obtain the entropy ecay inequality ) t H[f] c λ m which proves algebraic ecay of H[f], { H[f] implying ecay of f. 1 + H[f] σ) c λ m H[f 0 ] 3+ 3+σ) H[f 0 ] + m ) H[f 0 ] + m ) } t, ) H[f 0 ], Theorem 7. Let f 0 L x v/m) with x. Then there exists two constants c 1 > 0 an c > 0 such that the solution f of 1), ) satisfies 3) f,, t) L x v/m) c γ t) / f 0 L x v/m), with γ = c 1 + f0 L 1 v x) / f 0 L x v/m)) /. The result hols for both Fokker-Planck operators an scattering collision operators satisfying assumption 3). Remark 8. In contrast to the Fourier moe analysis, the approach of this section oes not require translation invariance of the problem. In particular, the results also hol for x-epenent scattering rates σ, satisfying the bouns 3), or more general Fokker-Planck operators of the form Lf = v Dx)vf + v f)). Note also that for large t the right han sie of 3) is asymptotically proportional to f 0 L 1 x v) + f 0 L x v/m) ) t, which is similar to the result of Theorem 6, because L 1 v) L v/m). Remark 9. If we consier the scale equation ε F t + TF = 1 ε LF which formally correspons to a parabolic rescaling t ε t, x εx), it is straightforwar to check that in the estimate 10) for λ, the gap constant λ m has to be replace by λ m /ε while, with the notations of Theorem 1, C M can be replace by C M /ε for ε < 1. In the asymptotic regime as ε 0 +, we obtain that ε λ M λ m λ M ε H[F ] D[F ] t λ M ) 1 + λ M ) CM D[F ] which proves that the estimate of Theorem 1 becomes λ m λ M λ λ M ) CM

13 HYPOCOERCIVITY WITHOUT CONFINEMENT 13 for ε > 0, small enough. We observe that this rate is inepenent of ε. In the context of Theorem 7, σ has to be replace by σ/ε an in the limit as ε 0 +, ) becomes which is again inepenent of ε. t H[f] c λ m σ + H[f] H[f 0 ] + m ) 7. Whole space An explicit computation for the kinetic Fokker-Planck equation Let us consier the kinetic Fokker-Planck equation of case a) 4) t f + v x f = v v f + v f) on 0, + ) t, x, v). The characteristics associate with the equations x t = v, v t = v suggest to change variables an consier the istribution function g such that ft, x, v) = e t g t, x + 1 e t) v, e t v ) t, x, v) 0, + ). The kinetic Fokker-Planck equation is change into a heat equation in both variables x an v with t epenent coefficients, which can be written as 5) t g = Ḋ g where g = v g, x g) an Ḋ is the t-erivative of the bloc-matrix ) D = 1 a I b I with a = e b I c I t 1, b = e t 1 e t, c = e t 4 e t + t+3. Here I is the ientity matrix on. We observe that Ḋ is egenerate: it is nonnegative but its lowest eigenvalue is 0. However, the change of variables allows the computation of a Green function. Lemma 10. With the above notations, the Green function associate with 5) is given by Gt, x, v) = a x 1 b x v+c v π a c b )) e a c b ) t, x, v) 0, + ). The metho is stanar an goes back to [9] also see, for instance, [6, 7]). It has been repeately use in the theory of the kinetic Fokker-Planck equation. Proof. By a Fourier transformation in x an v, with associate variables ξ an η, we fin that log C log Ĝt, ξ, η) = η, ξ) Dη, ξ) = 1 a η + b η ξ + c ξ ) = 1 a η + b a ξ + 1 A ξ, A = c b a

14 14 E. BOUIN, J. DOLBEAULT, S. MISCHLER, C. MOUHOT, AND C. SCHMEISER for some constant C > 0 which is etermine by the mass normalization conition Gt, ) L 1 ) = 1. We take the inverse Fourier transform with respect to η, π) e iv η Ĝt, ξ, η) η = C v π a) e a i b a v ξ e 1 A ξ = C v π a) e 1 ξ+i a e A b a A v b a A v, an then the inverse Fourier transform with respect to ξ, so that we obtain ) C Gt, x, v) = π a) π A) e 1+ b v a A a e x b A e a A x v It is easy to check that C = π). C = 4π a A) e 1 A x b a v e v a. Let us consier a solution g of 5) with initial atum g 0 L 1 ). From the representation gt,, = Gt,, ) x,v g 0 we obtain the estimate gt,, ) L ) Gt,, ) L ) g 0 L 1 ) = [ 4π e t 1 ) t ) e t + t + )] t e t as t +. As a consequence, we obtain that the solution of 4) with a nonnegative initial atum f 0 satisfies ft,, ) L ) = f 0 L 1 ) 4π t) 1 + o1)) as t +. Using the simple Höler interpolation inequality f L p ) f 1/p L 1 ) f 1 1/p L ), we obtain the following ecay result. Corollary 11. If f is a solution of 4) with a nonnegative initial atum f 0 L 1 ), then we have the ecay estimate ft,, ) L p ) = f 0 L 1 ) 1 + o1)) as t + 4π t) 1 1/p) for any p 1, + ]. By taking f 0 x, v) = G1, x, v), it is moreover straightforwar to check that this estimate is optimal. With p =, this also proves that the ecay rates obtaine in Theorems 6 an 7 for the Fokker-Planck operator, i.e., case a), are the optimal ones.

15 HYPOCOERCIVITY WITHOUT CONFINEMENT 15 Acknowlegments. This work has been partially supporte by the Projects STAB J.D., ANR- 1-BS ) an Kibor E.B., J.D., ANR-13-BS ) of the French National Research Agency ANR). Support by the Austrian Science Founation grants no. F65 an W145) is acknowlege by C.S., who is also grateful for the hospitality at Université Paris Dauphine. c 017 by the authors. This paper may be reprouce, in its entirety, for non-commercial purposes. References [1] Degon, P., Gouon, T., an Poupau, F. Diffusion limit for nonhomogeneous an non-microreversible processes. Iniana Univ. Math. J ), [] Dolbeault, J., Mouhot, C., an Schmeiser, C. Hypocoercivity for linear kinetic equations conserving mass. Trans. AMS ), [3] Glassey, R. T. The Cauchy problem in kinetic theory. Society for Inustrial an Applie Mathematics SIAM), Philaelphia, PA, [4] Gualani, M., Mischler, S., an Mouhot, C. Factorization of non-symmetric operators an exponential H-theorem. arxiv: v3, to appear as a Mémoire e la Société Mathématique e France 118 pages). [5] Hérau, F. Hypocoercivity an exponential time ecay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46, ), [6] Hörmaner, L. Hypoelliptic secon orer ifferential equations. Acta Math ), [7] Il in, A. M., an Has minskiĭ, R. Z. On the equations of Brownian motion. Teor. Verojatnost. i Primenen ), [8] Kawashima, S. The Boltzmann equation an thirteen moments. Japan J. Appl. Math. 7, 1990), [9] Kolmogoroff, A. Zufällige Bewegungen zur Theorie er Brownschen Bewegung). Ann. of Math. ) 35, ), [10] Nash, J. Continuity of solutions of parabolic an elliptic equations. Amer. J. Math ), [11] Villani, C. Hypocoercivity. Memoirs Amer. Math. Soc. 0, 009. E. Bouin) CEREMADE CNRS UMR n 7534), PSL research university, Université Paris-Dauphine, Place e Lattre e Tassigny, Paris 16, France aress: bouin@ceremae.auphine.fr J. Dolbeault) CEREMADE CNRS UMR n 7534), PSL research university, Université Paris-Dauphine, Place e Lattre e Tassigny, Paris 16, France aress: olbeaul@ceremae.auphine.fr S. Mischler) CEREMADE CNRS UMR n 7534), PSL research university, Université Paris-Dauphine, Place e Lattre e Tassigny, Paris 16, France aress: mischler@ceremae.auphine.fr C. Mouhot) DPMMS, Center for Mathematical Sciences, University of Cambrige, Wilberforce Roa, Cambrige CB3 0WA, UK aress: C.Mouhot@pmms.cam.ac.uk C. Schmeiser) Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern- Platz 1, 1090 Wien, Austria aress: Christian.Schmeiser@univie.ac.at