Global classical solutions for a compressible fluid-particle interaction model

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1 Global classical solutions for a compressible flui-particle interaction moel Myeongju Chae, Kyungkeun Kang an Jihoon Lee Abstract We consier a system coupling the compressible Navier-Stokes equations to the Vlasov- Fokker-Planck equation on three imensional torus. The coupling arises from a rag force exerte by each other. We establish the existence of the global classical solutions close to an equilibrium, an further prove that the solutions converge to the equilibrium exponentially fast AMS Subject Classification: 35Q30, 35Q83, 35Q84, 76N10 Keywors: Navier-Stokes-Vlasov-Fokker-Planck, compressible flui 1 Introuction In this paper we consier the motion of particles isperse in compressible viscous flows in three imensional torus T 3. Such a moel was first introuce by Williams in the context of combustion theory [22], an also foun in [3]. The particles are escribe by a probability ensity function in the phase space F (t, x, v) 0, which is governe by a kinetic transport equation with a friction force F, t F + v x F + v (F F σ v F ) = 0. (1.1) Here σ is a iffusive coefficient that is non-negative constant. We remark that a iffusion effect is taken into account in (1.1) while a collision effect of particles is ignore. We consier the case that the particles are isperse in a flui flow, which is governe by the following compressible isentropic Navier-Stokes equations: t + iv (u) = 0, (1.2) t (u) + iv (u u) + p κ u κ iv u = F F v, R 3 (1.3) where u(t, x) is the velocity, (t, x) is ensity of the flui, an κ, κ are numbers with κ > 0 an κ + κ > 0. Here we assume that the pressure follows a γ-law, i.e. p() = γ, γ > 1. The equations (1.1) an (1.3) are couple via the frictional force F (t, x), which acts on particles exerte by the flui. In this note, we stuy the case that F = F 0 (u v) with some friction constant F 0 > 0. Such a force is mathematically formulate by a thin spray moel [20], in which the volume fraction of particle is not consiere as a flui-kinetic coupling an the force is reuce to be friction force proportional to the relative velocity F 0 > 0. Therefore, in our consieration, the external force term in the flui equation is given as F F v = F 0 R 3 (u v)f v. R 3 1

2 If constants σ, κ, κ an F 0 are, for simplicity, assume to be 1, we then have the following compressible Navier-Stokes-Vlasov-Fokker-Planck equations: t + iv (u) = 0, t (u) + iv (u u) + p u iv u = (v u)f v, R 3 t F + v x F + iv v ((u v)f v F ) = 0. (1.4) Existence of global weak solution of (1.4) was establishe by Mellet-Vasseur [18] on boune omains with Dirichlet or reflection bounary conitions. For other relate works regaring flui-particle interaction systems, see [1, 2, 4, 5, 9, 13] an references cite therein. The solution (, u, F ) of (1.4) has a steay state (, 0, (2π) 3 2 e v 2 2 ), where is a constant. We enote µ := (2π) 3 2 e v 2 2 which is often referre to the global maxwellian. In this paper we pursuit the global classical solution of (1.4) when (, u, F ) is near equilibrium (, 0, µ). Let us introuce the unknown σ an f such that σ =, F = µ + µf, (1.5) What it follows, we set = 1 without loss of generality. We then rewrite (1.4) in terms of σ an f as follows: t + iv (u) = 0, t u u iv u + p () + 1 = σ ( u + iv u) u u, σ + 1 ( u v ) µfv + u µfv R 3 R 3 t f + v x f + u ( v f v 2 f) u v µ = v f v 2 4 f f. The mass an the momentum conservation hol a priori for the system (1.4): t F vx = 0, T 3 R 3 ) t ux + vf vx = 0. (T 3 T 3 R 3 Due to mass conservation it hols that µfvx = T 3 R 3 We will impose (f 0, u 0, 0 ) to satisfy 0 u 0 x + T 3 thus it hols that T 3 ux + T 3 T 3 T 3 R 3 µf0 vx. (1.6) R 3 v µf 0 vx = 0, (1.7) 2 R 3 vf vx = 0 (1.8)

3 s a priori. For later use we erive the equation for the average of u, let u(t) := ux. T 3 Integrating over T 3 the secon equation of (1.4) an by (1.8), we have t (u) + u + ux + u µfvx = 0. T 3 T 3 R 3 Since = σ, we rewrite the above as follows: t (u) + 2u = σux + T 3 v; T 3 R 3 u µfvx. (1.9) For notational convenience, we enote by, the L 2 inner prouct in (x, v) an,, v in f, g = T 3 fḡ vx, R 3 f, g v = fḡ v, R 3 an the corresponing L 2 norms by f 2 = T 3 f 2 vx, R 3 f 2 v = f 2 v. R 3 We introuce, for simplicity, a multi-inexe ifferential operator efine as follows: α β α 1 x 1 α 2 x 2 α 3 x 3 β 1 v 1 β 2 v 2 β 3 v 3, α = [α 1, α 2, α 3 ], β = [β 1, β 2, β 3 ]. For a multi-inex α = [α 1, α 2, α 3 ] we write α = α 1 + α 2 + α 3. In case that each component of α is not greater than that of α, we enote α α. The notation k stans for all the k-th orer spatial erivatives, i.e. k u 2 = α =k α u 2, k f 2 = α =k α f 2. We use f k to inicate L 2 -norm over T 3 R 3 of spatial erivatives of a function f up to orer k an, on the other han, f k enotes L 2 -norm incluing phase an spatial erivatives up to orer k, i.e., f 2 k = α k α f 2 L 2 (T 3 R 3 ), f 2 k = For an u we use k to inicate L 2 -norm over T 3, i.e. 2 k = α k α 2 L 2 (T 3 ), α + β k u 2 k = α k α β f 2 L 2 (T 3 R 3 ). α u 2 L 2 (T 3 ). We also, for simplicity, use a notational convention (f, u, σ) H k, which stans for (f, σ, u) 2 H k = f 2 k + u 2 k + σ 2 k. We enote by L the linear Fokker-Planck operator via the perturbation (1.6): Lf := v f + v 2 4 f 3 f. (1.10) 2 3

4 v The linear operator L is non-negative since it hols that Lf, f v = f + v 2 f 2 v. We note that + v 2 4 is the well known harmonic oscillator having the least eigenvalue 3 2, for which µ is the unique eigenfunction in three imension (see the section 3). Thus, for fixe (t, x) the null space of L, enote by N, is one imensional space spanne by µ, i.e. N = span { µ}. The operator L is the counterpart of the linearize collision operator in case we escribe collisional particles by Boltzmann equations. The full coercivity of these positive operators are often crucial to have a global classical solution of nonlinear kinetic equations near equllibrium. To settle this issue the iea of macro-micro ecomposition of f an a nonlinear energy estimate was eveloppe in [11, 12] an [15]. In [6] the macro-micro ecomposition was applie for the Vlasov-Maxwell-Fokker-Planck system. For any fixe (t, x), an any function g(t, x, v), we efine P 0 as its v-projection in L 2 (R 3 v) to the null space of L, P 0 : L 2 (R 3 v) N, P 0 f = f, µ v µ. (1.11) We then ecompose g uniquely as g = P 0 g + (I P 0 )g. The ecomposition is orthogonal in L 2 (R 3 v) an the orthogonality is preserve if a spatial erivative is taken; α P 0 g, α (I P 0 )g v = 0. Here P 0 g is calle the hyroynamic or macro part of g, an (I P 0 )g the microscopic part. In general L is coercive with respect to the microscopic part of f (Lemma 2) so that C v (I P 0 )f 2 v Lf, f v. (1.12) R 3 For the system (1.6) the above partial coercivity is enough to have the global solution on T 3 (Proposition 2). It is nicely combine with classical energy estimates of Matsumura-Nishia ([16, 17]) for three imensional compressible Navier-Stokes equations. However, the full coercivity is require for the solution to converge to equilibrium (Proposition 3). To compensate the lack of coercivity we aapt the formulation of the so calle Kawashima compensating function. In [14] Kawashima obtaine a time ecay estimate for the nonlinear Boltzmann equation near maxwellian by introucing a compensating operator S(ω) to have full coercivity. Yang- Yu [23] recalle this iea in the context of the Vlasov-Fokker-Planck equation to construct the global classical solution for the Vlasov-Fokker-Planck-Maxwell system with time ecay. The Vlasov-Fokker-Planck system couple with incompressible Navier-Stokes near equilibrium was first stuie by Gouon et al. ([9]). We refer to [4] by Carrillo et al. for the global classical solution for the Vlasov-Fokker-Planck-incompressible Euler system. For the Vlasov-Fokker- Planck system couple with incompressible Navier-Stokes near vaccum, a global weak solution on two or three imensions an the global classical solution on two imensions were establishe in [7]. The main motivation of the paper is to obtain global existence an asymptotic stability of solution near an equilibrium for the compressible Navier-Stokes-Vlasov-Fokker-Planck equations (1.4) (or (1.6)) in case initial ata is sufficiently close to an equilibrium. To be precise, our main result reas as follows: Theorem 1 There exists M 0 > 0 such that if (f 0, σ 0, u 0 ) 2 H 3 M 0, an (f 0, σ 0, u 0 ) satisfies (1.7), then (1.6) amits a unique solution (f, σ, u) globally in time such that (f, σ, u) L (0, ; H 3 ), (f, σ, u) L 2 (0, ; H 3 ), t u L 2 (0, ; H 2 ). 4

5 Moreover, (f, σ, u) ecays exponentially fast in time if the mean zero conitions of σ 0, f 0 µ hol; σ 0 x = 0, µf0 vx = 0. (1.13) T 3 T 3 R 3 There exists A = A(M 0 ) > 0 such that (f, σ, u)(t) H 3 e At (f, σ, u)(0) H 3. (1.14) Remark 1 In Theorem 1 only spatial regularity of f is mentione but we can observe that f instantly becomes regular in phase variables, too. More precisely, for any given t 0 > 0, we have sup ( f(t) 3 + vf(t) 2 ) C(t 0, M 0 ), t t 0. t t 0 The argument is rather straightforwar an thus the etails are omitte (compare to [9]). This paper is organize as follows. In Section 2 we introuce the local existence result of the system (1.6) an some preliminary lemmas. In Section 3 we prove a uniform estimate inepenent of time with a partial issipation of f. The global existence part of Theorem 1 can be obtaine from this stage. In Section 4 we prove a uniform estimate inepenent of time with a full issipation of f, an complete the proof of Theorem 1. 2 Preliminaries In this section we introuce the local existence result of solution for the system (1.6) an some preliminary result on the partial coercivity of L. The straightforwar energy esimates are inclue as well. What it follows, we enote by C = C(α, β,...) a constant epening on the prescribe quantities α, β,..., which may change from line to line. We first state a result on the local existence of solutions for the system (1.6). Since its proof is rather stanar the usual iterating metho, we just state the result without its etails. Proposition 1 (Local existence) There exist M, T > 0 such that if (f 0, u 0, σ 0 ) H 3 M 0 for any M 0 < M, then the system (1.6) with initial ata (f 0, u 0, σ 0 ) has a unique solution (f, u, σ) satisfying (f, σ, u) L (0, T ; H 3 ), u L 2 (0, T ; H 3 ). Moreover it hols that In aition, if µ + µf 0 0, then F = µ + µf 0 on [0, T ]. sup (f, σ, u) H 3 2M 0 (2.1) [0,T ] We remark that for the compressible Navier-Stokes equations, the local existence of classical solution was shown in [17] an one can refer to, for example, [21] or [4] for the case of the linear Fokker-Planck equation with a given potential. The nonnegativity of F in Proposition 1 can be shown by following the similar argument as in [8]. Next, we show that the linear ifferential operator L efine in (1.10) has a partial coercivity in L 2 v(r 3 ). 5

6 Lemma 2 Let Ω be either R 3 or T 3. Suppose that g(x, v) L 2 (Ω R 3 ). Then Lg = 0 if an only if g = c µ for any c L 2 x(ω). Moreover, there is a C 1, C 2 > 0 such that Lg, g C 1 v (I P 0 )g 2 + C 2 v (I P 0 )g 2 (2.2) where inner prouct an the norm are over Ω R 3. Proof. By stanar elliptic theory the solution of Lg = 0 exists as a Schwartz function. Then the first part is obvious from the ientity v g + v 2 g 2 = Lg, g. The coercivity part follows immeiately if we prove Lg, g C g 2, g N. (2.3) Inee, noting that Lg, g = v g 2 + v 2 4 g2 3 2 g2 xv, we have v g v g 2 = Lg, g g 2 ( 7 + 1) Lg, g. 4C It suffices to show (2.3). A irect way to see it is to expan g using the orthonormal eigenfunction of L; Let ψ klm is the normalize eigenfunction for the harmonic oscillator satisfying ( v + v )ψ klm = (2k + l)ψ klm, k, l {0, 1, 2,... }, 0 m l. It is well-known that {ψ klm } provies a complete orthonormal basis of L 2 (R 3 ) 1. For ψ 000 = µ, we express g N by g(x, v) = (k,l) 0 m l c klm(x)ψ klm (v) an we then obtain Lg, g = (k,l) 0 m l (2k + l) c 2 klm (x)x g 2. The equality hols for g = v i µ, i = 1, 2, 3, so C = 1 in (2.3). Another way to show (2.3) is to use a Poincaré type inequality with respect to the gaussian measure µv as in [23]; It hols that for any suitable function h(v) ( h 2 µv + 2 hµv) h 2 µv. The above inequality was proven in cf. [10]. Now plugging h = (I P 0 )fµ 1 2 Lg, g g 2. This completes the proof. Due to Lemma 2 it hols that in, we have 1 The explicit formula of ψ klm is C v (I P 0 ) α f 2 v α f + v 2 α f 2 = L α f, α f. (2.4) ψ klm (r, θ, φ) = N kl r l e 4 1 r2 L l+ 2 1 k (r 2 /2)Y lm (θ, φ), where N kl is the normalizing constant, L α k (x) is the generalize Laguerre polynomial, an Y lm (θ, φ) is the spherical Harmonic function (pp in [19] setting M = ω/ = 1). The orthogonality is immeiate from those of L k, Y lm. 6

7 What it follows we compute preliminary energy estimates, which become useful in our analysis. Using p() = γ, we have α f t, α f + α (u ( v f v 2 f)), α f α u v µ, α f + v α f + v 2 α f 2 = 0. (2.5) On the other han, from the equation of u α u, α u t α u, α u + iv α u + α u, p () α + (γ 1) α u, α β γ 2 β + α u, α (u v µfv) µfv + u β<α + ( 1 α u, α β β<α ) β (u v µfv + u µfv) (2.6) = α u, α σ ( σ + 1 u + σ σ + 1 iv u) α u, α (u u). As observe in [9] sum of unerline terms in (2.5) an (2.6) are simplifie as follows: v α f + v 2 α f 2 2 α u v µ, α f + α u 2 = α u µ v α f v 2 α f 2. We estimate unerwave terms in (2.5) an (2.6) in the following manner: α (u ( v f v 2 f)), α f + α u, α (u µfv) α (uf) ( α u µ v α f v 2 α f). To cancel out α u, p () x α in (2.6), multiplying the equation for with γ γ 2 α σ an using integrate by parts, we have γ γ 2 α σ, α σ t = γ ( γ 2 α σ), α (u) = γ ( γ 2 α σ), ( α β β<α ( α β = γ ( γ 2 α σ), β<α ) α β σ β u + α u ) α β σ β u (2.7) Using the equation of, it hols that + γ γ 2 x α σ, α u +γ }{{} x γ 2 α σ, α u. α u,p () x α α u, α u t = 1 2 t α u iv (u) α u, α u, γ 2 α σ, α σ t = 1 2 t γ 2 α σ (γ 2) iv (u) α σ, α σ. We also note that 1 2 iv (u) α u, α u + α u, u α u = 0. 7

8 Finally we estimate the mean flui-momentum, u from the equation (1.9). We have 1 2 t u u 2 u σux + u u µfvx. We boun σux = σ u σ/ ( (u) + u ) (2.8) by Poincaré inequality. Using this, we have 1 2 t u 2 + (2 σ/ ) u 2 σ/ (u) 2 + u Summing up all of above, we have the following lemma. u µfvx. Lemma 3 Let (f, σ, u) be the local solution in Proposition 1. For 0 α 3 it hols that 1 ( α f 2 + α u 2 + (γ 1) γ 2 α σ 2 + u 2) 2 t + α u 2 + iv α u 2 + α u µ v α f v (2.9) 2 α f 2 + (2 σ/ ) u 2 σ/ (u) 2 + u u µfvx (:= I 0 ) + α (uf) α u µ v α f v 2 α f (:= I 1 ) +C α u σ α u x (:= I 2 ) +C α u, α β u β u (:= I 3 ) β<α +C (γ 2 α σ), α β β u + C iv (u) α σ, α σ (:= I 4,1 + I 4,2 ) β<α +C α u, α β γ 2 β x (:= I 5 ) β<α +C ( γ 2 ) α σ, α u (:= I 6 ) +C ( ) 1 α u, α β β (u v µfv) µfv + u (:= I 7 ) β<α + α u, α σ ( σ + 1 u + σ iv u σ + 1 (:= I 8 ) := I 0 + I 1 + I 2 + I 3 + I 4,1 + I 4,2 + I 5 + I 6 + I 7 + I 8. (2.10) In the above, the sums β<α are set to be zero when α = 0. We also remin the equivalence of norms u k an k u because of perioic bounary conitions. The terms I 0 +I 1 + +I 8 in (2.10) will be estimate further in the following sections. 8

9 3 Uniform in time estimates with a partial coercive property of L In this section we show that if (f, σ, u)(t) H 3 for the system (1.6) is sufficiently small for some perio of time, then certain type of regularity of solutions is controlle only by initial ata. More precisely, we prove the following: Proposition 2 Let (f, σ, u) be the local solution in Proposition 1. Then there exists ɛ such that if sup [0,T ] (f, σ, u) H 3 < ɛ, then for any s [0, T ] (f, σ, u)(s) 2 H 3 + C s where C is inepenent of T. 0 u t u σ v (I P 0 )f u 2 τ C (f, σ, u)(0) 2 H 3, We will carry out several energy estimates moifying estimates in [9] an [16]. Proof of Proposition 2 follows through Lemma 4 - Lemma 6, an the inequality (3.18), which will be shown below. Lemma 4 Uner the same assumption in Proposition 2 we have following energy estimates: 1 2 t ( f 2 + u 2 + γ 2 σ 2 + u 2) + u 2 + u µ v f v 2 f 2 (3.1) 1 2 t + u 2 Cɛ σ 2, ( k f 2 + k u 2 + γ 2 k σ 2) + k (u µ v f v 2 f) 2 Proof. First we collect some estimates on f: sup x + u 2 k Cɛ σ 2 k 1, k = 1, 2, 3. (3.2) ( f 2 (x, v)v sup x sup x v µfv C 2 f(x, v)) v C v µ( γ 2 γ 2 γ f(x, v) 2 xv, (3.3) γ f 2 x) 1 2 v C f 2, (3.4) v µ β fv L 3 C C β f L 3 x v µv (3.5) ( x β f L 2 x + β f L 2 x ) v µv C β f 1. Using the equation (3.3), we have α uf 2 α u 2 sup x f 2 (x, v)v C α u 2 f

10 When β 0, we estimate α β u β f 2 C α β u 2 L 6 β f 2 L 3 x v C α β u 2 1 β f x β f C u 2 α f 2 α. Thus we have α (uf) 2 C u 2 α f 2 α. (3.6) When α = 0, we rather use uf 2 C u 2 L 6 f 2 L 3 C u 2 L 6 f 2 L 3 x (v)v C u 2 1 f 2 1. (3.7) by Minkowski s inequality for the secon inequality. For L 2 estimates of the equations of (1.6), we have 1 2 t ( f 2 + u 2 + (γ 1) γ 2 σ 2 + u 2 ) + u 2 + u µ v f v 2 f 2 + u 2 C uf u µ v f v 2 f + C ( γ 2 )σ, u + C iv (u)σ, σ + I 0 C u 2 1 f u µ v f v 2 f 2 + C( σ 1, L ) u 1 σ 2 + I 0 (3.8) ue to (3.7) an Höler s inequality. For I 0 first we estimate σ/ (u) 2 ɛ( 2 L u 2 + σ 2 u 2 L ). Using (2.8) an the above we have ū 2 = ( σ)u 2 2( u 2 + σu 2 ) C u 2 + ɛ( u 2 + σ 2 ) u 2 C u 2 + C u 2 + ɛ σ 2. (3.9) Next we estimate So u u µfvx f u u ɛ u ( u + u + ɛ σ ), I 0 ɛ u u 2 + ɛ σ 2. Since all the terms of (3.8) except ɛ σ 2 can be absorbe in the left han sie, we obtain (3.1). To obtain higher orer estimates (3.2), we estimate I 1,..., I 8 of (2.9) as follows: we have I 1 C u 2 α f 2 α α u µ v α f v 2 α f 2 10

11 by (3.6), an I 2 = C α u σ α u x 1 8 α u 2 + C σ 2 α 1 u 2 2. More precisely, Next we have 1 8 α u 2 + C σ 2 α u 2 L if α = 1, I α u 2 + C σ 2 L α u 2 6 L if α = 2, α u 2 + C σ 2 L α u 2 L if α = 3. 2 I 3 = β<α α u, α β u β u Cɛ u 2 α 1. More precisely, L u L u 2 if α = 1, I 3 L u L 2 u 2 if α = 2, L u L 3 u 2 + L 2 u u if α = 3. If α = 1, then I 4,1 C σ 3 L 3 u L + C σ 2 u L. If α 2, then we rewrite Then Similarly Also we have I 4,1 = x ( γ 2 α σ), α σu + x ( γ 2 α σ), β<α,β 0 α β β u. I 4,1 C α σ 2 ( σ 2 u 2 + u 2 ) + C α σ σ α u 2. I 4,2 C u 3 (1 + σ 3 ) α σ 2. I 5 C u 2 σ 2 α, I 7 an I 8 can be estimate as follows. I 7 = β<α α u, α β ( 1 I 6 C σ 2 α σ α u. ) β (u v µfv + u µfv). We present α = 3 case only. The others are easier. ( I 7 L α u α u v µfv L µfv + u + 0<β<α α β L 6 β (u C( u 3, f 3 ) L α u α. v µfv) L µfv + u 3 11

12 We estimate the terms insie parenthesis by C( u 3, f 3 ) α using (3.4), (3.5). Finally I 8 is boune by C α u σ 2 u 2, if α = 1 C α u σ 2 u 2 + C α u 3 2 σ 2 α u 1 2 I 8 1, if α = 2 C α u σ 2 u 2 + C α u 3 2 σ 2 α u C α u σ 2 u 3, if α = 3. Overall we have I 7 + I 8 Cɛ( α u 2 + α σ 2 ) α u 2 1. Using Young s inequality, we have the estimate (3.2). This completes the proof. On the other han, we note that ue to the triangle inequality on T 3, for any s 0 v f + v 2 f 2 s u 2 s 1 + u µ v f v 2 f 2 s + u 2 Therefore, ue to (2.4) an (3.9), we obtain v (I P 0 )f 2 k u 2 k 1 + u µ v f v 2 f 2 k + C u 2 + ɛ σ 2. (3.10) Next corollary is a irect consequence of Lemma 4 an (3.10). Corollary 1 Uner the same assumption in Proposition 2, we have following energy estimates: ( f 2 + u 2 + γ 2 σ 2 + u 2) + u 2 + v (I P 0 )f 2 (3.11) t + u 2 Cɛ σ 2, ( k f 2 + k u 2 + γ 2 k σ 2 + u 2) + u 2 k t + u 2 + v (I P 0 )f 2 k Cɛ σ 2 k 1, k = 1, 2, 3. (3.12) Next we show control of k σ for k = 1, 2, 3. Lemma 5 Let P 0 be the projection operator introuce in (1.11). Uner the same assumption in Proposition 2 we have following energy estimates: σ 2 C( t u 2 + u (I P 0 )f 2 + u 2 ), k = 1, (3.13) k σ 2 C( t u 2 k 1 + u 2 k + (I P 0)f 2 k 1 ), k = 2, 3. (3.14) Proof. Since P 0 f, v µ v = 0, we write the u-equation as t u u iv u + p () x + 1 (u = σ ( u + iv u) u u. σ v µ(i P 0 )fv + u µfv)

13 Multiplying σ to the equation for u we have (3.13) by integration by parts. We estimate u v µfv µfv L µ(i P 0 )fv + u u (1 + ) + (I P 0 )f (3.15) C u + (I P 0 )f, C( u + u + ɛ σ ) + (I P 0 )f using (3.9). The term µfv L is boune by f 2 by (3.4). For (3.14) we take k 1 an multiply k σ to the u equation, an use inuctions on k 1 σ. Next lemma shows control of time erivative of u. To be more precise, we have the following: Lemma 6 Let P 0 be the projection operator introuce in (1.11). Uner the same assumption in Proposition 2 we have following energy estimates: t u(t), p () σ(t) + tu 2 C ( ɛ σ 2 + u 2 + (I P 0 )f 2), (3.16) t k 1 u(t), p () k σ(t) + k 1 t u 2 C ( ɛ σ 2 k 1 + u 2 k 1 + (I P 0)f 2 k 1), k = 2, 3. (3.17) Proof. For the first item we multiply t u to the u equation with replacing f into (I P 0 )f in the first moment integral of f. Using the equation an p () σ = γ γ 1 (γ 1 ) we have So it hols that t u, p () = t u, p () σ + u, γ (γ 2 (u)) t u, p () σ C u 2 Cɛ σ 2. t u, p () σ + tu 2 C(ɛ σ 2 + u 2 + (I P 0 )f ), where we use (3.15). Taking k 1 an multiplying k 1 t u to the u equation, we have the secon item by integration by parts. We present the k = 3 case. For α = 2, we have t α u, α ( p () ) = t u, p () σ + α u, γ α ( γ 2 (u)) t u, p () σ C u 2 α Cɛ σ 2 α. We estimate α ( 1 (u C α 2 u v ) µfv) µ(i P 0 )fv + u 2 v µfv 2 µ(i P 0 )fv + u L 13

14 + 0<β<α Each terms are boune by α β ( 1 +C α (u ) β (u v µfv) 2 µ(i P 0 )fv + u v µ(i P 0 )fv + u µfv) 2. C α σ 2 ( u 2 L + f u 2 f 2 1) +C α σ 2 ( u 2 α 1 + f 2 α ) +C( u 2 α 1 f 2 α + u 2 α 1 f u 2 α 1 + α (I P 0 )f 2 ) by (3.5), (3.6) an α (u µfv) C α (uf). Also we have the boun Overall we have (3.17). ( ) σ α ( u + iv u 2 Cɛ σ 2 α 1 + σ + Cɛ u 2 α +1. We now give the proof of Proposition 2. Proof of Proposition 2. Let us efine k th energy norm an compensating term by an enote E k (f,, u)(t) = k f 2 + k u 2 + γ 2 k σ 2 + u 2, k = 0, 1, 2, 3, ( p C k (, u)(t) = k 1 u, k 1 ) () σ, k = 1, 2, 3, E(f,, u)(t) = E k (f,, u), C(, u)(t) = k=0 C k (, u). Implementing Lemma 5, we summarize the previous estimates into follows: t E 0(f,, u)(t) + u 2 + v (I P 0 )f 2 + u 2 Cɛ( t u 2 + u 2 1), t E k(f,, u)(t) + u 2 k + v (I P 0)f 2 k + u 2 Cɛ t u 2 k 1, k = 1, 2, 3, t C k(, u)(t) + t u 2 k 1 C( u 2 k + (I P 0)f 2 k 1 ), k = 1, 2, 3. Aing up the above inequalities with C k multiplie by a small parameter λ > 0, we have (E(f,, u)(t) + λc(, u)(t)) + u 2 k t + v (I P 0 )f 2 k + λ t u 2 k 1 + u 2 k=0 k=0 ( ) C ɛ t u 2 k 1 + ɛ u λ u 2 k + λ (I P 0 )f 2 k 1. 14

15 For sufficiently small λ an ɛ such that ɛ < { λ 2C, 1 2C }, we have t (E(f,, u)(t) + λc(, u)(t)) + λ 2 t u 2 k ( u 2 k + v (I P 0)f 2 k ) + u 2 0. k=0 In view of Lemma 5 we have also an issipation term of σ 2 3 ; t (E(f,, u)(t) + ac(, u)(t)) + a( tu σ 2 2) ( u v (I P 0 )f 2 3) + u 2 0 (3.18) for a small constant a > 0. Finally note that correction terms are boune by usual energy norms; ( p k 1 u, k 1 ) () σ C( k 1 u 2 + k σ 2 ) C( k 1 u 2 + γ 2 k σ 2 ) by the assumption of Proposition 2. Thus ecreasing a > 0, we have E(f,, u)(t) + t Due to (3.9), it hols that E(f,, u)(t) + 0 t which proves Proposition 2. 0 a( t u σ 2 2) ( u v (I P 0 )f 2 3) + u 2 s CE(f,, u)(0). a( t u σ 2 2) ( u v (I P 0 )f 2 3) + u 2 s CE(f,, u)(0). (3.19) 4 Proof of Theorem 1 We first improve the estimate of Proposition 2 an then we present the proof of Theorem 1. In the next proposition, we obtain the full coercivity of f, instea of a partial control of f such as (I P 0 )f 2 3 shown in Proposition 2. Proposition 3 Let (f, σ, u) be the local solution in Theorem 1 satisfying the mean zero conition (1.13) hols. Then there exists ɛ such that if sup [0,T ] (f, σ, u) H 3 < ɛ, then for any s [0, T ] (f, σ, u)(s) 2 H 3 + s where C is inepenent of t. 0 u t u σ P 0 f v (I P 0 )f u 2 τ (4.1) C (f, σ, u)(0) 2 H 3 Note that (1.13) is preserve in time. The inequality follows from (4.13) presente at the en of the section. In the below we will briefly explain on the Kawashima compensating function in the context of the Vlasov-Fokker-Planck equation. It is much close to the exposition in 15

16 [23]. Let us efine an orthogonal projection P onto the four imensional subspace of L 2 (R 3 ). Consier the orthonormal vectors {e 1, e 2, e 3, e 4 } =: { µ, v 1 µ, v2 µ, v3 µ}, an enote W the subspace spanne by the four vectors; We efine the projection P by P : L 2 (R 3 v) W, P f = f, e k v e k. We then ecompose f uniquely as f = P f + (I P )f. The ecomposition is again orthogonal in L 2 (R 3 v) an the orthogonality is preserve uner the spatial ifferentiation. Let [f, σ, u] be the solution of the (1.6). Plugging the ecomposition f = P f + (I P )f into the system (1.6) we rewrite the equation by t f + v x P f + L(P f) = v x (I P )f L(I P )f + u v µ (:= R ) u ( v v 2 )f (:= g ). (4.2) Putting w j = f, e j v an taking the inner prouct with e j to (4.2), we have t w j + i f, e k v v i e k, e j v + f, e k v L[e k ], e j v = R, e j v + g, e j v, j = 1, 2, 3, 4. We enote R j := R, e j v an ḡ j := g, e j v an set R = ( R 1, R 2, R 3, R 4 ) T an ḡ = (ḡ 1, ḡ 2, ḡ 3, ḡ 4 ) T. We also efine a vector w = (w 1, w 2, w 3, w 4 ) T an 4 4 matrices V i by (V i ) kj = v i e k, e j v for i = 1, 2, 3 2 Then we have the first orer hyperbolic system with repect to w, t w + V i xi w + Lw = R + ḡ, (4.3) i=1 where L is the 4 4 matrix with ( L) kj = L[e k ], e j v. Note that w 1 2 = P 0 f 2. To fin a hien positivity of L, Kawashima consiere (4.3) from the Fourier transform sies (he ha the thirteen moment equations on (t, x) R R 3 in the Boltzmann case). Consier the Fourier expansion formula for a function h L 2 (T 3 ), h = n Z 3 h n e ix n, 2 By a irect computation we have V 1 = , V 2 = , V 3 =

17 where h n is the Fourier coefficient of h on the n cite, n = (n 1, n 2, n 3 ) Z 3. Taking the n th Fourier coefficient of both han sies of (4.3), we have t (w) n + iv i n i (w) n + L(w) n = ( R) n + (ḡ) n, (4.4) where we enote by (w) n = ((w 1 ) n, (w 2 ) n, (w 3 ) n, (w 4 ) n ) T. Let us enote n i / n by ω, an efine V (ω) = 3 i=1 ω iv i for a vector ω = (ω 1, ω 2, ω 3 ) S 2. Then t (w) n + i n V (ω)(w) n + L(w) n = ( R) n + (ḡ) n, (4.5) 0 ω 1 ω 2 ω 3 V (ω) = ω ω ω Observing that for the skew symmetric matrix R(ω) 0 ω 1 ω 2 ω 3 R(ω) = ω ω , ω we have R(ω)V (ω) = iag(1, ω1 2, ω2 2, ω2 3 ). Thus we have the following algebraic relation hols: Let W C 4. Then Re W, R(ω)V (ω)w = W 1 2 ω 2 1 W 2 2 ω 2 2 W 3 2 ω 2 3 W 4 2, (4.6) where X, Y stans for the complex inner prouct X, Y = X T Ȳ. Multiplying ir(ω) on (4.5) an taking the real part, we have t Re (w) n, ir(ω)(w) n + n (w 1 ) n 2 + Re (w) n, R(ω) L(w) n Re (w) n, R(ω)(( R) n + (ḡ) n ) + n (w i ) n 2 thanks to (4.6). Note that summing over n Z 3, we obtain Z 3 n (w 1) n 2 is 1 2 P 0 f 2, which will be a missing issipation on f in Proposition 2. Proof of Proposition 3. Define a Kawashima compensating function S(ω)f of f in L 2 (R 3 ) by S(ω)f = i=2 R(ω) kl f, e l v e k. k,l=1 Clearly S(ω) is a boune linear operator on L 2 (R 3 v), i.e. S(ω)f v C f v. The efinition gives S(ω)(v ω)f, f v = = R(ω) kl (v ω)f, e l v e k, f v k,l=1 R(ω) kl (v ω)f, e l v f, e k v. k,l=1 17

18 Substituting P f = 4 j=1 f, e j v e j, we have Inee, S(ω)(v ω)p f, f v = R(ω)V (ω)w, w. k,l,j=1 = R(ω) kl (v ω) f, e j v e j, e l v f, e k v k,l,j=1 R(ω) kl (v ω)e j, e l v }{{} (V (ω)) jl f, e j v f, e k v = R(ω)V (ω)w, w ue to V (ω) kj := 3 i=1 ω i v i e k, e j v = (v ω)e k, e j v, an w j := f, e j v. Thus, using the ientity (4.6), we have On the other han, we estimate Re S(ω)(v ω)f, f v (C 1 P 0 f 2 v C 2 (I P 0 )f 2 v). (4.7) S(ω)(v ω)(i P )f, f v = R(ω) kl (v ω)(i P )f, e l v f, e k v k,l=1 C (I P )f v P f v. Now, let (f, σ, u) be the local solution in Proposition 1. Multiplying e ix n to the f equation in (1.6), an integrating on T 3, we have t f n + i n (v ω)f n + Lf n = g n u n v µ, ω = n/ n. We enote g = u ( v f v 2f) previously in (4.2). Taking is(ω) to the both sies, i t S(ω)f n + n S(ω)(v ω)f n is(ω)lf n = is(ω)g n + is(ω)u n v µ. Taking inner prouct with n 1 f n in L 2 (R 3 v), we have Re n 1 i t S(ω)f n, f n v + Re S(ω)(v ω)f n, f n v Re n 1 is(ω)lf n, f n v (4.8) We estimate = Re n 1 is(ω)g n, f n v + Re n 1 is(ω)u n v µ, f n v. S(ω)Lf n, f n = R(ω) kl Lf n, e l v e k, f n v k,l=1 = S(ω)g n, f n v = R(ω) kl L[(I P 0 )f n ], e l v e k, f v C (I P 0 )f n v f n v, k,l=1 R(ω) kl g n, e l v e k, f n v C g n, e l v f n v C (uf) n µ 1 2 v f n, k,l=1 S(ω)u n v µ, f n v C f n v u n, 18

19 using bouneness of S(ω) an g n, e l v = ( g, e l v ) n = ( uf, ( v + v 2 )e l v ) n C (uf) n µ 1 2 v, where we enote by µ 1 2 µ 1 2 ɛ for an arbitrary small ɛ > 0. Due to (4.7), we note that Re S(ω)(v ω)f n, f n v (C 1 P 0 f 2 v C 2 (I P 0 )f 2 v). Summing up the above estimates over n Z 3 \ {0}, we have t ( n Z 3 \{0} Re n 1 is(ω)f n, f n v ) + n Z 3 \{0} P 0 f n 2 v C (I P 0 )f n v f n v + (I P 0 )f 2 v n Z 3 n Z 3 + (uf) n µ 1 2 v f n v + f n v u n v. n Z 3 ( ) µ Note that (P 0 f) 0 (v) = T 3 R µfvx = 0 ue to the conition (1.13). By the 3 Cauchy-Schwartz inequality an Plancherel theorem it follows that t ( Re n 1 is(ω)f n, f n v ) + P 0 f 2 n Z 3 /0 ( ) C (I P 0 )f f + (I P 0 )f 2 + ufµ 1 2 f + f u. Note that n Z 3 \{0} Re n 1 is(ω)f n, f n v is boune by f 2. The above oes compensate the missing coercivity of L by the term P 0 f 2. Next we will procee the same by multiplying n 2k 1 (k = 1, 2, 3) to (4.8) to compensate k P 0 f 2 in the en. (4.9) Re n 2k 1 i t S(ω)f n, f n v + n 2k Re S(ω)(v ω)f n, f n v (4.10) = Re n 2k 1 is(ω)lf n, f n v Re n 2k 1 is(ω)g n, f n v + Re n 2k 1 is(ω)u n v µ, f n v. Similarly we estimate n 2k Re S(ω)(v ω)f n, f n v n 2k (C 1 P 0 f 2 v C 2 (I P 0 )f 2 v), n 2k 1 is(ω)lf n, f n v C n k (I P 0 )f n v n k f n v, n 2k 1 is(ω)g n, f n v C n k 1 (uf) n µ 1 2 v f n v, Summing up the above over n Z 3 we have t ( C n Z 3 \0 n 2k 1 is(ω)u n v µ, f n v C n k u n n k f n v. Re is(ω) n k f n, n k 1 f n v ) + P 0 f 2 k ( (I P 0 )f k f k + (I P 0 )f 2 k + k 1 x (uf)µ 1 2 f k + f k u k ), (4.11) 19

20 for k = 1, 2, 3. By using f k P 0 f k + (I P 0 )f k an then Höler s inequality, we put (4.9) an (4.11) into t C n Z 3,n 0 for k=0 Let us efine the new correction term Re is(ω) n k f n, n k 1 f n v + P 0 f 2 k ( ) (I P 0 )f 2 k + ɛ k (uf) 2 + u 2 k C( (I P 0 )f 2 k + ɛ u 2 k ), k = 0, 1, 2, 3. D(f)(t) = k=0 n Z 3,n 0 for k=0 Re is(ω) n k f n, n k 1 f n v. (4.12) We remin that E(f,, u)(t) = C(, u)(t) = In view of (3.18) we have k f 2 + k u 2 + γ 2 k σ 2 + u 2, k=0 ( p k 1 u, k 1 ) () σ. t (E(f,, u)(t) + ac(, u)(t) + a D(f)(t)) + a( t u σ 2 2) + u 2 +a P 0 f ( 1 4 a C) (I P 0 )f ( 1 4 Cɛa ) u (4.13) for a small parameter a > 0. The new correction term D(f)(t) is boune by the energy norm f 2 k by choosing a small a 1 2. Using (3.9) we replace u 2 3 in (4.13) with u 2 4. We can also replace σ 2 2 with σ 2 3 among the issipation terms ue to the mean zero conition (1.13). Finally we have E(f,, u)(t) + 1 k 1 u, p () t 2 k σ + 1 Re is(ω) n k f n, n k 1 f n v 2 k=0 n Z 3 + C( u t u f σ u 2 ) 0. We complete the proof. We are reay to present the proof of Theorem 1. (4.14) Proof of Theorem 1. Let M, T be same constants in Proposition 1, an ɛ, C be in Proposition 3. We may assume C > 1. Consier the initial ata (f 0, u 0, σ 0 ) satisfying f u ( 0 ) 2 3 M 0 for M 0 = 1 C min{m, ɛ/2}. By Proposition 1 there exist T > 0 an the unique local solution (f, σ, u) such that sup ( f u σ 2 3)(t) 2M 0. t [0,T ] 20

21 Since 2M 0 < ɛ, we have (f, σ, u)(s) 2 H 3 + s 0 u t u σ f u 2 τ CM 0 (4.15) for s [0, T ] by Proposition 3. Again by Proposition 1 the existence time interval is extene by T, an it hols that sup ( f u σ 2 3)(t) 2CM 0. t [0,2T ] Since 2CM 0 < ɛ, by Proposition 3 the solution (f, σ, u) satisfies (f, σ, u)(s) 2 H 3 + s 0 u t u σ f u 2 τ CM 0 for all s [0, 2T ]. Then we return to (4.15) an repeat the argument on t = 2T. In this way we can boost the existence time interval up to [0, kt ] for any k N with the boun sup (f, σ, u)(t) 2 H + 3 t [0, ) 0 u t u σ f u 2 τ C (f, u, σ)(0) 2 H 3. The ecay rate of the solution is obtaine by the inequality (4.13). Note that the above proceure keeps the conition of Proposition 3, sup s [0,T ] (f, u, σ) H 3 ɛ, for any T, so the inequality (4.13) hols for all t. Multiplying e at to the above inequality, we have 0 e at (E(f,, u) + 1 k 1 u, p () t 2 k σ + 1 Re is(ω) n k f n, n k 1 f n v ) 2 k=0 n Z 3 ae at (E(f,, u) + 1 k 1 u, p () 2 k σ + 1 Re is(ω) n k f n, n k 1 f n t v ) 2 k=0 n Z 3 + Ce at ( u t u f σ u 2 ) t (eat (E(f,, u) + 1 k 1 u, p () 2 k σ C 1 e at ( u t u f σ u 2 ). by ecreasing a. The compensating terms are boune as k 1 u, p () k σ u σ 2 3, an k=0 n Z 3,n 0 for k=0 k=0 n Z 3 Re is(ω) n k f n, n k 1 f n v ) Re is(ω) n k f n, n k 1 f n v f 2 3. Integrating the above we have (f, σ, u) go to zero exponentially fast, t ( f u σ u 2 )(t) + Ce at e aτ ( u t u f σ u 2 )(τ)τ This completes the proof. 0 e at ( f u σ u 0 2 ). 21

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