Nonlocal Stokes-Vlasov system: Existence and deterministic homogenization results
aa r X i v : . [ m a t h . A P ] J u l NONLOCAL STOKES-VLASOV SYSTEM: EXISTENCE AND DETERMINISTICHOMOGENIZATION RESULTS
GABRIEL NGUETSENG, CELESTIN WAFO SOH, AND JEAN LOUIS WOUKENG
Abstract.
Our work deals with the systematic study of the coupling between the nonlocal Stokes systemand the Vlasov equation. The coupling is due to a drag force generated by the fluid-particles interaction.We establish the existence of global weak solutions for the nonlocal Stokes-Vlasov system in dimensionstwo and three without resorting to assumptions on higher-order velocity moments of the initial distributionof particles. We then study by the means of the sigma-convergence method, the asymptotic behavior inthe general deterministic framework, of the sequence of solutions to the nonlocal Stokes-Vlasov system. Inguise of illustration, we provide several physical applications of the homogenization result including periodic,almost-periodic and weakly almost-periodic settings. Introduction
This paper is concerned with the rigorous asymptotic analysis of a system of integro-differential equationsmodeling the evolution of a cloud of particles immersed in an incompressible viscous fluid. We neglectparticle-particle collisions in such a way that at the microscale level, particles’ distribution, f ε , satisfies theVlasov equation ∂f ε ∂t + εv · ∇ f ε + div v (( u ε − v ) f ε ) = 0 in Q × R N (1.1)in which the non-dimensional small parameter ε > u ε ( t, x ) is fluid’svelocity at time t and position x , f ε ( t, x, v ) dv is roughly the odd of finding a particle with velocity v near x at time t , Q = (0 , T ) × Ω, Ω ⊂ R N ( N = 2 ,
3) is a bounded domain with smooth boundary , T is a givenpositive real number representing the final time, the operator ∇ (resp. div v ) denotes the gradient operatorwith respect to x ∈ Ω (resp. the divergence operator in R N with respect to v ∈ R N ). We posit that thecloud of particles is highly diluted in such a way that we may assume that the density of the fluid is constant.Thus, the particles evolve in a Newtonian fluid governed by the Stokes system. The viscoelastic constitutivelaw associated to the momentum balance and the fluid-particles interaction give rise to the following Stokessystem: ∂ u ε ∂t − div (cid:18) A ε ∇ u ε + Z t A ε ( t − τ , x ) ∇ u ε ( τ , x ) dτ (cid:19) + ∇ p ε = − Z R N ( u ε − v ) f ε dv in Q, (1.2)div u ε = 0 in Q, (1.3)where p ε is pressure and the oscillating viscosities A ε and A ε are defined by A εi ( t, x ) = A i (cid:0) t, x, tε , xε (cid:1) (( t, x ) ∈ Q and i = 0 , A i s constrained as follows:( A1 ) A i ∈ C ( Q ; L ∞ ( R N +1 y,τ ) N ) are symmetric matrices with A satisfying the following condition: A ξ · ξ ≥ α | ξ | for all ξ ∈ R N and a.e. in Q × R N +1 y,τ with α > x, t, y, τ and ξ . Date : June, 2014.2000
Mathematics Subject Classification.
Key words and phrases.
Nonlocal Stokes-Vlasov system, Deterministic homogenization, Introverted algebras with meanvalue, sigma-convergence, Convolution.
The system (1.1)-(1.3) is supplemented with the initial data u ε (0 , x ) = u ( x ) , f ε (0 , x, v ) = f ( x, v ) , x ∈ Ω , v ∈ R N , (1.4)and the boundary conditions u ε = 0 on ∂ Ω and f ε ( t, x, v ) = f ε ( t, x, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < , (1.5)where v ∗ = v − v · ν ( x )) ν ( x ) is the specular velocity, ν ( x ) is the outward normal to Ω at x ∈ ∂ Ω and thefunctions u and f are chosen as follows:( A2 ) u ∈ L (Ω) N with div u = 0, f ≥ f ∈ L ∞ (Ω × R N ) ∩ L (Ω × R N ).It is opportune to stress that we have not imposed the constraint | v | f ∈ L (Ω × R N ) as suggestedby Yu [38]. Indeed, we are going to see that the Lemma 2.1 of Hamdache [18] renders such assumptionsuperfluous provided appropriate regularization and truncation are performed. In particular, the truncationof the initial distribution of particles in the v -direction relieves us from the assumption on moments. Contraryto contemporary approaches, ours permeates initial distribution of the form α ( x ) / (1 + | v | ), where α ≥ α ∈ L ∞ (Ω).The system (1.1)-(1.5) arises in several applications comprising the modeling of reactive flows of sprays[1, 26], atmospheric pollution modeling [14], and waste water treatment [11]. When there is no particleevolving in the fluid (i.e. when f ε ≡
0) the asymptotic analysis of Eqs. (1.1)-(1.5) reduces to the study ofthe asymptotics of (1.2)-(1.5) (with of course f ε ≡ v -moments of the initial condition f ; 2) we carry out the homogenizationof (1.1)-(1.5) under suitable structural assumptions on the coefficients of the operators involved in (1.2).These assumptions cover a wide set of concrete behaviors such as the classical periodicity assumption, thealmost periodicity hypothesis, weakly almost periodicity hypothesis and much more. In order to achieve ourgoal, we shall use the concept of sigma-convergence [23, 29] which is roughly a formulation of the well-knowntwo-scale convergence method [22] in the context of algebras with mean value [19, 29, 39]. This is the so-called deterministic homogenization theory which includes the periodic homogenization theory as a specialcase. As far as we know, our results are new in the context of general deterministic homogenization sincethe available results deal with either periodic homogenization [6, 9, 10, 15] or rely on the concept of relativeentropy [12, 13, 17, 25, 28, 37].The remainder of this paper is structured as follows. In Section 2, we state and outline the proof of anexistence result for our ε -problem. We also derive some a priori estimates that will be useful in next sections.Section 3 deals with the concept of Σ-convergence and its relation with convolution. We first recall someuseful tools related to algebras with mean value and define convolution over the spectrum of an algebra withmean value. In Section 4, we state and prove the main homogenization result. In Section 5, we give someconcrete situations in which the result of Section 4 is valid. Finally, we summarize our findings in Section 6.In the sequel, unless otherwise specified, the field of scalars acting on vector spaces is the set of realnumbers and scalar functions are real-valued. If X and F respectively denote a locally compact space anda Banach space, then we respectively write C ( X ; F ) and BUC( X ; F ) for continuous mappings of X into F and bounded uniformly continuous mappings of X into F . We shall always assume that BUC( X ; F ) isequipped with the supremum norm k u k ∞ = sup x ∈ X k u ( x ) k in which k·k stands for the norm of F . In thenotations for functions space, we shall omit the codomain when it is R . To wit, C ( X ) will stand for C ( X ; R )and BUC( X ) will be a shorthand notation for BUC( X ; R ). Likewise, the usual Lebesgue spaces L p ( X ; R )and L p loc( X ; R ) where X is equipped with a positive Radon measure, are respectively abbreviated L p ( X ) ONLOCAL STOKES-VLASOV SYSTEM 3 and L p loc( X ). Finally, it will always be assumed that the Euclidian space R N ( N ≥
1) and its open sets areeach endowed with Lebesgue measure dy = dy . . . dy N .2. Existence result and basic a priori estimates
In this part, we focus on the existence of solutions to our ε -problem. We shall define a regularized problem,solve it and show that the limit (in a sense to be specified) of the solution of this regularized problem solvesour ε -problem. In order to implement our program, we shall establish some a priori estimates in someclassical functional spaces we introduce below. These estimates will be used in compactness arguments atvarious stages of this work.The main classical spaces involved in the mathematical study of incompressible fluid flows are spacesconnected to kinetic energy, entropy, the boundary conditions and the conservation of mass. These spaceswill be denoted V and H and they may be respectively constructed as closure of V = { ϕ ∈ C ∞ (Ω) N :div ϕ = 0 } in H (Ω) N ( H (Ω) the usual Sobolev space on Ω), and L (Ω) N . Since ∂ Ω is smooth, we havethat V = { u ∈ H (Ω) N : div u = 0 } and H = { u ∈ L (Ω) N : div u = 0 and u · ν = 0 on ∂ Ω } , ν being theoutward unit vector normal to ∂ Ω. We denote by ( · , · ) the inner product in H , and by · the scalar product in R N . The associated norm in R N is denoted by |·| . All duality pairing are denoted by h· , ·i without referringto spaces involved. Such spaces will be understood from the context. We setΣ ± = { ( x, v ) ∈ ∂ Ω × R N : ± v · ν ( x ) > } . With the functional framework fixed, we can now specify the type of solutions we will be seeking.
Definition 2.1.
A pair ( u ε , f ε ) (for fixed ε >
0) is called a weak solution to the system (1.1)-(1.5) if thefollowing conditions are satisfied: • u ε ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; V ) ∩ C ([0 , T ]; V ′ ); • f ε ( t, x, v ) ≥ t, x, v ) ∈ Q × R N ; • f ε ∈ L ∞ (0 , T ; L ∞ (Ω × R N ) ∩ L (Ω × R N )); • f ε | v | ∈ L ∞ (0 , T ; L (Ω × R N )); • for all φ ∈ C ([0 , T ] × Ω × R N ) with compact support in v , such that φ ( T, · , · ) = 0 and φ ( t, x, v ) = φ ( t, x, v ∗ ) on (0 , T ) × Σ + , we have Z Q × R N f ε (cid:18) ∂φ∂t + εv · ∇ φ + ( u ε − v ) · ∇ v φ (cid:19) dxdvdt + Z Ω × R N f φ (0 , x, v ) dxdv = 0; (2.1) • for all ψ ∈ C ([0 , T ] × Ω) N with div ψ = 0 and ψ ( T, · ) = 0 , Z Q (cid:18) − u ε · ∂ψ∂t + ( A ε ∇ u ε + A ε ∗ ∇ u ε ) · ∇ ψ (cid:19) dxdt = − Z Q × R N f ε ( u ε − v ) · ψ dxdtdv (2.2)+ Z Ω u · ψ (0 , x ) dx. In Eq. (2.2) A ε ∗ ∇ u ε stands for the function defined by( A ε ∗ ∇ u ε )( t, x ) = Z t A ε ( t − τ , x ) ∇ u ε ( τ , x ) dτ whenever ( t, x ) ∈ Q. The main result of this section is summarized in the following theorem.
Theorem 2.1.
Under assumption ( A1 )-( A2 ) and for any fixed ε > , there exists a weak solution ( u ε , f ε ) of (1.1)-(1.5) in the sense of Definition . There also exists a p ε ∈ L (0 , T ; L (Ω) / R ) such that (1.2) issatisfied. The proof of Theorem 2.1 will be done in several steps described in the subsections that follow. Thegeneral idea is loosely to regularize our problem, solve the regularized problem and take the limit of itssolution to obtain a solution of our problem.
GABRIEL NGUETSENG, CELESTIN WAFO SOH, AND JEAN LOUIS WOUKENG
Regularization and truncation.
We start by fixing notations that will be useful in the sequel. Let( θ λ ) λ> be a mollifying sequence in x i.e. its terms belong to C ∞ ( R Nx ) (the space of compactly supported andsmooth functions) such that θ λ ( x ) = λ − N θ ( x/λ ) for all x ∈ R N and λ > θ ∈ C ∞ ( R Nx ), 0 ≤ θ ≤ θ ) ⊂ B (0 ,
1) = { x ∈ R N : | x | ≤ } and R R N θ ( x ) dx = 1. A regularizing sequence in ( x, v ) ∈ R Nx × R Nv will be denoted (Θ λ ) λ> . We shall use the same notation for convolution in x and ( x, v ). The context shallindicate which convolution is used. For vector-valued functions, convolution is done componentwise. Wealso consider γ ∈ C ∞ ( R Nv ) such that supp( γ ) ⊂ B (0 , γ ( v ) = 1 for all v ∈ B (0 ,
1) and 0 ≤ γ ≤
1. Wedefine the truncating sequence ( γ λ ) λ> by γ λ ( v ) = γ ( λv ). It can be verified that γ λ ( v ) → λ → λ , we shall assume throughout that 0 < λ ≤
1. The latter assumption is used when needed to obtaineduniform estimates in λ .Let w ∈ L (0 , T ; V ). The regularized system associated to our ε -problem takes the following form: ∂f ε,λ ∂t + εv · ∇ f ε,λ + div v (( w ∗ θ λ − v ) f ε,λ ) = 0 in Q × R N (2.3) ∂ u ε,λ ∂t − div (cid:18) A ε ∇ u ε,λ + Z t A ε ( t − τ , x ) ∇ w ( τ , x ) dτ (cid:19) + ∇ p ε,λ = − Z R N ( w ∗ θ λ − v ) γ λ ( v ) f ε,λ dv in Q, (2.4)div u ε,λ = 0 in Q, (2.5)The system (2.3)-(2.5) is supplemented with the following initial and boundary conditions.a) u ε,λ (0 , x ) = u ( x );b) f ε,λ (0 , x, v ) := f λ ( x, v ) = γ λ ( v )( f ∗ Θ λ )( x, v ) , x ∈ Ω , v ∈ R N , (2.6)and the boundary conditionsa) u ε,λ = 0 on ∂ Ω;b) f ε,λ ( t, x, v ) = f ε,λ ( t, x, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < . (2.7)2.2. Existence, regularity and estimates of f ε,λ . In this part, we focus on Eq. (2.3) coupled with theinitial condition (2.6 b) and the boundary condition (2.7 b). In what follows, we use C as a generic namefor positive constants independent of both ε and λ . In all the estimates, we suppose when the need arisesthat ε and λ are sufficiently small. We shall assume throughout that f ∈ L p (Ω × R n ) ∩ L ∞ (Ω × R n ), p ≥ | f λ | ≤ | f ∗ Θ λ | , we have (cid:13)(cid:13) f λ (cid:13)(cid:13) L p (Ω × R N ) ≤ (cid:13)(cid:13) f ∗ Θ λ (cid:13)(cid:13) L p (Ω × R N ) ≤ (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) k Θ λ k L (Ω × R N ) ≤ (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) in such a way that (cid:13)(cid:13) f λ (cid:13)(cid:13) L p (Ω × R N ) ≤ (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) . (2.8)By Theorem 4 of Mischler [21], we infer that f ε,λ uniquely exists and belongs to L ∞ (0 , T, L ∞ (Ω × R N ) ∩ L p (Ω × R N )). Since both f λ and the coefficients of Eq. (2.3) are C ∞ , we can deduce using the method ofcharacteristics that f ε,λ is nonnegative and belongs to C ((0 , T ) × Ω × R N ). Following [18, p. 54], we have ddt (cid:18) e Nt Z Ω × R N (cid:0) e − Nt f ε,λ (cid:1) p dxdv (cid:19) = 0 . (2.9)Then, by integrating both sides of Eq (2.9) from 0 to t , we obtain k f ε,λ ( t ) k L p (Ω × R N ) = e Nt ( − p ) (cid:13)(cid:13) f λ (cid:13)(cid:13) L p (Ω × R N ) (2.10)Thus, by using the inequality (2.8), we arrive at the estimate k f ε,λ k L ∞ (0 ,T ; L p (Ω × R N )) ≤ C ( N, T, p ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) . (2.11) ONLOCAL STOKES-VLASOV SYSTEM 5
Now, we turn our attention to the estimates of v -moments of f ε,λ . Lemma 2.1 of Hamdache [18] will beour workhorse. Let us first observe that f λ ∈ L ∞ (Ω × R N ) ∩ L (Ω × R N ) since f λ is compactly supportedand f ∈ L ∞ (Ω × R N ) ∩ L p (Ω × R N ), p ≥ m ≥ Z Ω × R N | v | m f λ dxdv ≤ C ( N, p ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) . (2.12)Indeed, the following inequalities Z Ω × R N | v | m f λ dxdv ≤ Z Ω ×{| v |≤ } | v | m f λ dxdv + Z Ω ×{| v | > } | v | m f λ dxdv = Z Ω ×{| v |≤ } | v | m f λ dxdv ≤ Z Ω ×{| v |≤ } f λ dxdv ≤ | Ω × {| v | ≤ }| /q (cid:13)(cid:13) f λ (cid:13)(cid:13) L p (Ω × R N ≤ C ( N, p ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) (see (2.8)), (2.13)where q is the conjugate exponent of p , are true. In view of the continuous embedding H (Ω) ֒ → L r (Ω) forany 1 ≤ r ≤
6, and since N ≤
3, we may choose m ≥ N + m ≤ ≤ m ≤ N = 2, and 1 ≤ m ≤ N = 3). Thus, for such an m , the fact that w ∈ L (0 , T ; L N + m (Ω) N ) steems fromboth the above continuous embedding and w ∈ L (0 , T ; V ). With this in mind and taking into account Eq.(2.12), we see that we are within the hypotheses of [18, Lemma 2.1]. Hence, the following estimate holds: Z Ω × R N | v | m f ε,λ dxdv ≤ C ( N, T ) "(cid:18)Z Ω × R N | v | m f λ dxdv (cid:19) N + m + ( (cid:13)(cid:13) f λ (cid:13)(cid:13) L ∞ (Ω × R N ) + 1) k w ∗ θ λ k L (0 ,T ; L N + m (Ω) N ) N + m . (2.14)By employing the inequality (2.12) in conjunction with the estimate k w ( t, · ) ∗ θ λ k L N + m (Ω) N ≤ k w ( t, · ) k L N + m (Ω) N k θ λ k L (Ω) ≤ k w ( t, · ) k L N + m (Ω) N , (2.15)and noting that (cid:13)(cid:13) f λ (cid:13)(cid:13) L ∞ (Ω × R N ) ≤ (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) , we arrive at the inequality Z Ω × R N | v | m f ε,λ dxdv ≤ C ( N, m, p, T ) (cid:20)(cid:13)(cid:13) f (cid:13)(cid:13) N + m L p (Ω × R N ) + ( (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) + 1) k w k L (0 ,T ; L N + m (Ω) N ) (cid:21) N + m for any m ≥ N + m ≤
6. By employing the Sobolev embedding H (Ω) ֒ → L N + m (Ω), the latterinequality leads to Z Ω × R N | v | m f ε,λ dxdv ≤ C (cid:20)(cid:13)(cid:13) f (cid:13)(cid:13) N + m L p (Ω × R N ) + ( (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) + 1) k w k L (0 ,T ; V ) (cid:21) N + m (2.16)for any m ≥ N + m ≤
6, where C = C ( N, m, p, T,
Ω).
Remark 2.1.
We shall discover in the sequel that the estimate (2.16) makes assumptions on higher-order v -moments of the initial distribution redundant. It is opportune to emphasize that besides Lemma 2.1 ofHamdache [18], both regularization and truncation have played a fundamental role in deriving the inequality(2.16).Next, we provide estimates which show among other things that the force field F ε,λ = G ε,λ + H ε,λ , (2.17)where G ε,λ ( t, x ) = − Z R N ( w ∗ θ λ − v ) γ λ ( v ) f ε,λ dv (2.18)and H ε,λ ( t, x ) = div (cid:18)Z t A ε ( t − τ , x ) ∇ w ( τ , x ) dτ (cid:19) , (2.19) GABRIEL NGUETSENG, CELESTIN WAFO SOH, AND JEAN LOUIS WOUKENG belongs to L (0 , T ; H − (Ω) N ). So, let Φ ∈ C ∞ (Ω) N . For almost all t ∈ [0 , T ] we have |h G ε,λ ( t, x ) , Φ i| ≤ Z Ω ×{| v |≤ } (1 + | w ∗ θ λ | ) f ε,λ | Φ | dxdv ≤ C ( N ) (cid:16) k w ( t, · ) k L (Ω) N (cid:17) k f ε,λ ( t, · ) k L ∞ (Ω × R N ) k Φ ( t, · ) k L (Ω) N . (2.20)Thus, for almost all t ∈ [0 , T ] we have k G ε,λ ( t, · ) k L (Ω) N ≤ C ( N ) (cid:16) k w ( t, · ) k L (Ω) N (cid:17) k f ε,λ ( t, · ) k L ∞ (Ω × R N ) . (2.21)Since f λ ∈ L ∞ (Ω × R N ), by the maximum principle applied to the transport equation, we have k f ε,λ k L ∞ (0 ,T ; L ∞ (Ω × R N )) ≤ C ( N, T ) (cid:13)(cid:13) f λ (cid:13)(cid:13) L ∞ (Ω × R N ) ≤ C ( N, T ) (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) . (2.22)Therefore, using Eq. (2.22) in Eq. (2.21), we obtain k G ε,λ ( t, · ) k L (Ω) N ≤ C ( N, T ) (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) (cid:16) k w ( t, · ) k L (Ω) N (cid:17) (2.23)for almost all t ∈ [0 , T ]. Thus, since w ∈ L (0 , T ; L (Ω) N ), so is G ε,λ .Now, we turn our attention to H ε,λ : |h H ε,λ ( t, . ) , Φ i| ≤ N k A k L ∞ ( Q × R N +1 ) N Z t Z Ω |∇ w | |∇ Φ | dxdτ ≤ N k A k L ∞ ( Q × R N +1 ) N T k∇ w k L (0 ,T ; L (Ω) N ) k∇ Φ k L (Ω) N . (2.24)Therefore, H ε,λ ∈ L (0 , T ; H − (Ω) N ) and k H ε,λ k L (0 ,T ; H − (Ω) N ) ≤ N k A k L ∞ ( Q × R N +1 ) N T / k∇ w k L (0 ,T ; L (Ω) N ) . (2.25)2.3. Existence of ( u ε,λ , p ε,λ ) and further estimates. We look for u ε,λ ∈ L (0 , T ; V ) such that ∂ u ε /∂t ∈ L (0 , T ; V ′ ) and, for almost all t ∈ [0 , T ] and all Φ ∈ V , (cid:28) d u ε,λ dt , Φ (cid:29) + a ε ( t ; u ε,λ , Φ ) = h F ε,λ ( t ) , Φ i , (2.26) u ε,λ (0) = u ∈ H, (2.27)where a ε ( t ; u , Φ ) = Z Ω A ε ( t, x ) ∇ u ( x ) · ∇ Φ ( x ) dx (2.28)and F ε,λ is defined in Eq. (2.17). Note that by standard arguments [20], the problem (2.26)-(2.27) makessense. By direct computations using assumptions made on the A i s, one arrives at the following propertiesof a ε . • The function t a ε ( t ; u , Φ ) is mesurable for all u , Φ ∈ V . • For almost every t ∈ [0 , T ] and for all u , Φ ∈ V , | a ε ( t ; u , Φ ) | ≤ N k A k L ∞ ( Q × R N +1 ) N k u k V k Φ k V := M k u k V k Φ k V . (2.29) • For almost every t ∈ [0 , T ] and for all v ∈ V , | a ε ( t ; v , v ) | ≥ α k v k V . (2.30)Thus, by Lions’ theorem [20], there is a unique u ε,λ ∈ L (0 , T ; V ) ∩ C (0 , T ; H ) satisfying Eqs (2.26)-(2.27). Since F ε,λ ∈ L (0 , T ; H − (Ω) N ) and w ∈ L (0 , T, V ), it is a simple matter to check that ∂ u ε,λ ∂t − div ( A ε ∇ u ε,λ ) − F ε,λ ∈ H − (Ω) N ⊂ D ′ (Ω) N ( D ′ (Ω) the usual space of distributions on Ω) for almost all t ∈ [0 , T ]. Thus, thanks to Eq. (2.26) and Propositions 1.1 and 1.2 of [33], there is a unique p ε,λ ( t ) ∈ L (Ω) / R such that Eq. (2.4) holds in the sense of distributions and k p ε,λ ( t ) k L (Ω) / R ≤ C (Ω) k∇ p ε,λ ( t ) k H − (Ω) N (2.31) ONLOCAL STOKES-VLASOV SYSTEM 7 for almost all t ∈ [0 , T ].2.4. Solvability of (2.3)-(2.7) with w = u ε,λ . Here, we prove via Schauder’s fixed point theorem thatby letting w = u ε,λ , the regularized problem is still solvable. In order to do that, we consider the mapping S : L (0 , T ; V ) → L (0 , T ; V ) with w u ε,λ , where u ε,λ is the unique solution of Eqs. (2.26)-(2.27). The mapping S is well-defined because of the previousstep . We need to show that S has a fixed point as asserted in the next result. Proposition 2.1.
There exists a function u ε,λ in L (0 , T ; V ) such that S u ε,λ = u ε,λ .Proof. The mapping S is not linear. However, we can check that it is Lipschitz continuous. Indeed, let w , w ∈ L (0 , T ; V ) and set u iε,λ = S w i ( i = 1 , w = w − w . Let us also denote by p iε,λ ( i = 1 ,
2) theassociated pressures. Then, u ε,λ = u ε,λ − u ε,λ and p ε,λ = p ε,λ − p ε,λ solve the following Stokes system ∂ u ε,λ ∂t − div ( A ε ∇ u ε,λ ) + ∇ p ε,λ = div ( A ε ∗ ∇ w ) − Z R N ( w ∗ θ λ ) γ λ ( v ) f ε,λ dv in Q div u ε,λ = 0 in Q u ε,λ = 0 on (0 , T ) × ∂ Ω u ε,λ (0 , x ) = 0, x ∈ Ω . Multiplying the leading equation above by u ε,λ , we find after integrating over Ω that12 ddt Z Ω | u ε,λ | dx + Z Ω ( A ε ∇ u ε,λ + A ε ∗ ∇ w ) · ∇ u ε,λ dx + Z Ω × R N γ λ ( v ) f ε,λ ( w ∗ θ λ ) · u ε,λ dvdx = 0 . Integrating the above equation over (0 , t ) and using assumption ( A1 ), we arrive at the inequality R Ω | u ε,λ | dx + 2 α R t R Ω |∇ u ε,λ | dxdτ ≤ − R t R Ω ( A ε ∗ ∇ w ) · ∇ u ε,λ dxdτ − R t R Ω × R N γ λ ( v ) f ε,λ ( w ∗ θ λ ) · u ε,λ dvdxdτ . Using Young’s inequality yields2 (cid:12)(cid:12)(cid:12)R t R Ω ( A ε ∗ ∇ w ) · ∇ u ε,λ dxdτ (cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)R t (cid:0)R τ (cid:0)R Ω A ε ( τ − s ) ∇ w ( s ) · ∇ u ε,λ ( τ ) dx (cid:1) ds (cid:1) dτ (cid:12)(cid:12)(cid:12) ≤ R t (cid:16)R τ τα k A ε ( τ − s ) ∇ w ( s ) k L (Ω) ds + R τ ατ k∇ u ε,λ ( τ ) k L (Ω) ds (cid:17) dτ ≤ R t (cid:16)R τ c α τ k∇ w ( s ) k L (Ω) ds + R τ ατ k∇ u ε,λ ( τ ) k L (Ω) ds (cid:17) dτ = α R t k∇ u ε,λ ( τ ) k L (Ω) dτ + R t (cid:16) c α τ R τ k∇ w ( s ) k L (Ω) ds (cid:17) dτ ≤ α R t k∇ u ε,λ ( τ ) k L (Ω) dτ + Ct R t (cid:16)R τ k∇ w ( s ) k L (Ω) ds (cid:17) dτ since 0 ≤ τ ≤ t ≤ α R t k∇ u ε,λ ( τ ) k L (Ω) dτ + CT R t (cid:16)R τ k∇ w ( s ) k L (Ω) ds (cid:17) dτ . Also, it can be verified that2 (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Ω × R N γ λ ( v ) f ε,λ ( w ∗ θ λ ) · u ε,λ dvdxdτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( N, T ) (cid:13)(cid:13) f (cid:13)(cid:13) L ∞ (Ω × R N ) | B (0 , | Z t Z Ω | w ∗ θ λ | | u ε,λ | dxdτ ≤ C Z t Z Ω (cid:16) | w ∗ θ λ | + | u ε,λ | (cid:17) dxdτ ≤ C k w ∗ θ λ k L ( Q ) + C Z t k u ε,λ k L (Ω) dτ ≤ C k w k L ( Q ) + C Z t k u ε,λ k L (Ω) dτ , GABRIEL NGUETSENG, CELESTIN WAFO SOH, AND JEAN LOUIS WOUKENG where | B (0 , | stands for the Lebesgue measure of B (0 , Z Ω | u ε,λ | dx + α Z t k∇ u ε,λ k L (Ω) dτ ≤ C k w k L ( Q ) + C Z t (cid:18)Z τ k∇ w ( s ) k L (Ω) ds (cid:19) dτ + C Z t k u ε,λ k L (Ω) dτ and C k w k L ( Q ) + C Z t (cid:18)Z τ k∇ w ( s ) k L (Ω) ds (cid:19) dτ ≤ C k w k L ( Q ) + C Z T Z T k∇ w ( s ) k L (Ω) ds ! dτ ≤ C k w k L ( Q ) + CT Z T k∇ w ( s ) k L (Ω) ds ≤ C k w k L (0 ,T ; V ) . By conflating the previous estimates, we obtain the inequality Z Ω | u ε,λ | dx + α Z t k∇ u ε,λ k L (Ω) dτ ≤ C k w k L (0 ,T ; V ) + C Z t k u ε,λ k L (Ω) dτ . Then, Gronwall’s Lemma implies the inequality Z t k u ε,λ k L (Ω) dτ ≤ C k w k L (0 ,T ; V ) for all 0 ≤ t ≤ T, from which we infer that Z T k∇ u ε,λ k L (Ω) dτ ≤ C k w k L (0 ,T ; V ) or equivalently, k u ε,λ k L (0 ,T ; V ) ≤ C k w k L (0 ,T ; V ) . (2.32)Now, let ϕ ∈ C ∞ (0 , T ) ⊗ V . Then, (cid:28) ∂ u ε,λ ∂t , ϕ (cid:29) = − Z Q A ε ∇ u ε,λ · ∇ ϕdxdt − Z Q ( A ε ∗ ∇ w ) · ∇ ϕdxdt − Z Q (cid:18)Z R N f ε,λ γ λ ( v ) dv (cid:19) ( w ∗ θ λ ) · ϕdxdt and (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ∂ u ε,λ ∂t , ϕ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ u ε,λ k L ( Q ) k∇ ϕ k L ( Q ) + k A ε ∗ ∇ w k L ( Q ) k∇ ϕ k L ( Q ) + C k w ∗ θ λ k L ( Q ) k ϕ k L ( Q ) ≤ C k w k L (0 ,T ; V ) k ϕ k L (0 ,T ; V ) because of (2.32), C being a positive constant that does not depend on ϕ . Therefore, it follows from the density of C ∞ (0 , T ) ⊗V in L (0 , T ; V ) that ∂ u ε,λ /∂t ∈ L (0 , T ; V ′ ) with (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ε,λ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; V ′ ) ≤ C k w k L (0 ,T ; V ) . (2.33)The inequality (2.32) implies that S sends continuously L (0 , T ; V ) into itself. Moreover (2.32) and (2.33)show that S transforms bounded sets in L (0 , T ; V ) into bounded sets in W (0 , T ) = { w ∈ L (0 , T ; V ) : ∂ w /∂t ∈ L (0 , T ; V ′ ) } ( W (0 , T ) being endowed with the norm k w k W (0 ,T ) = [ k w k L (0 ,T ; V ) + k ∂ w /∂t k L (0 ,T ; V ′ ) ] / which makes it a Hilbert space). Furthermore, the range of S is contained in W (0 , T ) which is compact in L (0 , T ; H ) because of the Aubin-Lions lemma. Thus, the range of S is relatively compact in L (0 , T ; H )and hence in L (0 , T ; V ) since the latter space is closed in the former. Hence, by Schauder’s fixed pointtheorem, S admits a fixed point. (cid:3) ONLOCAL STOKES-VLASOV SYSTEM 9
We have just proved the following result.
Proposition 2.2.
For any fixed ε > and λ > , the problem (2.34)-(2.38) below ∂f ε,λ ∂t + εv · ∇ f ε,λ + div v (( w ∗ θ λ − v ) f ε,λ ) = 0 in Q × R N (2.34) ∂ u ε,λ ∂t − div (cid:18) A ε ∇ u ε,λ + Z t A ε ( t − τ , x ) ∇ u ε,λ ( τ , x ) dτ (cid:19) + ∇ p ε,λ = − Z R N ( u ε,λ ∗ θ λ − v ) γ λ ( v ) f ε,λ dv in Q, (2.35)div u ε,λ = 0 in Q, (2.36) u ε,λ (0 , x ) = u ( x ) , f ε,λ (0 , x, v ) := f λ ( x, v ) = γ λ ( v )( f ∗ Θ λ )( x, v ) , x ∈ Ω , v ∈ R N , (2.37) u ε,λ = 0 on ∂ Ω and f ε,λ ( t, x, v ) = f ε,λ ( t, x, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < admits a unique solution ( u ε,λ , f ε,λ , p ε,λ ) such that u ε,λ ∈ L (0 , T ; V ) with ∂ u ε,λ /∂t ∈ L (0 , T ; V ′ ) , f ε,λ ∈C ( Q × R N ) and p ε,λ ∈ L ∞ (0 , T ; L (Ω) / R ) . The following uniform estimates hold true.
Lemma 2.1.
Let ( u ε,λ , f ε,λ , p ε,λ ) be the solution to (2.34)-(2.38) . Then, Z Ω × R N (1+ | v | ) f ε,λ dxdv + Z Ω | u ε,λ | dx +2 Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ + Z t k∇ u ε,λ ( τ ) k L (Ω) dτ ≤ C (2.39) for any ≤ t ≤ T , ε > and λ > , where C > is independent of both λ and ε . Moreover if f ∈ L p (Ω × R N ) , (1 ≤ p ≤ ∞ ) , then k f ε,λ k L ∞ (0 ,T ; L p (Ω × R N )) ≤ exp( N T ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) for any λ, ε > . (2.40) It also holds that (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ε,λ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H − (Ω) N ) ≤ C (2.41) and sup λ,ε> k p ε,λ k L (0 ,T ; L (Ω)) ≤ C. (2.42) Proof.
The inequality (2.40) has already been obtained (see Eq. (2.11)). Let us now check (2.39). Wemultiply (1.1) by | v | and (1.2) by u ε,λ , and we get12 ddt (cid:20)Z Ω × R N | v | f ε,λ dxdv + Z Ω | u ε,λ | dx (cid:21) + Z Ω ( A ε ∇ u ε,λ + A ε ∗ ∇ u ε,λ ) · ∇ u ε,λ dx + Z Ω E ε,λ ( t, x ) dx = 0 , (2.43)where we set E ε,λ ( t, x ) = Z R N f ε,λ ( u ε,λ ∗ θ λ − v ) · u ε,λ dv + ε Z R N ( v · ∇ f ε,λ ) | v | dv + 12 Z R N | v | div v (( u ε,λ ∗ θ λ − v ) f ε,λ ) dv. But Z Ω Z R N ( v · ∇ f ε,λ ) | v | dvdx = Z R N Z ∂ Ω f ε,λ | v | ( v · ν ) dσdv = Z { v · ν> } f ε,λ | v | ( v · ν ) dσdv + Z { v · ν< } f ε,λ | v | ( v · ν ) dσdv. Since v ∗ = v − v · ν ) ν , it holds that v ∗ · ν = − v · ν , | v ∗ | = | v | and dv ∗ = dv . Thus, because of thereflection condition (2.38) on f ε,λ , we have Z { v · ν< } f ε,λ | v | ( v · ν ) dσdv = − Z { v ∗ · ν> } f ε,λ ( t, x, v ∗ ) | v ∗ | ( v ∗ · ν ) dσdv ∗ , so that Z R N Z ∂ Ω f ε,λ | v | ( v · ν ) dσdv = 0 . Also, the following identity holds Z R N | v | div v (( u ε,λ ∗ θ λ − v ) f ε,λ ) dv = − Z R N f ε,λ ( u ε,λ ∗ θ λ − v ) · vdv. It therefore follows that Z Ω E ε,λ ( t, x ) dx = Z Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdv. Integrating Eq.(2.43) over (0 , t ) and using the assumption ( A1 ), we are lead to R Ω × R N | v | f ε,λ dxdv + 2 R t R Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ + R Ω | u ε,λ | dx + 2 α R t R Ω |∇ u ε,λ | dxdτ ≤ − R t R Ω ( A ε ∗ ∇ u ε,λ ) · ∇ u ε,λ dxdτ + R Ω × R N | v | f λ dxdv + R Ω (cid:12)(cid:12) u (cid:12)(cid:12) dx. Now, using Young’s inequality, we infer that2 R t R Ω ( A ε ∗ ∇ u ε,λ ) · ∇ u ε,λ dxdτ = 2 R t (cid:0)R τ (cid:0)R Ω A ε ( τ − s ) ∇ u ε,λ ( s ) · ∇ u ε,λ ( τ ) dx ) (cid:1) ds (cid:1) dτ ≤ R t (cid:16)R τ τα k A ε ( τ − s ) ∇ u ε,λ ( s ) k L (Ω) ds + R τ ατ k∇ u ε,λ ( τ ) k L (Ω) ds (cid:17) dτ ≤ R t (cid:16)R τ c α τ k∇ u ε,λ ( s ) k L (Ω) ds + R τ ατ k∇ u ε,λ ( τ ) k L (Ω) ds (cid:17) dτ = α R t k∇ u ε,λ ( τ ) k L (Ω) dτ + R t (cid:16) c α τ R τ k∇ u ε,λ ( s ) k L (Ω) ds (cid:17) dτ ≤ α R t k∇ u ε,λ ( τ ) k L (Ω) dτ + c α t R t (cid:16)R τ k∇ u ε,λ ( s ) k L (Ω) ds (cid:17) dτ since 0 ≤ τ ≤ t, where c = sup ( t,x ) Q k A ( t, x, · , · ) k L ∞ ( R N +1 y,τ ) N < ∞ . Thus R Ω × R N | v | f ε,λ dxdv + 2 R t R Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ + R Ω | u ε,λ | dx + α R t k∇ u ε,λ k L (Ω) dτ ≤ R Ω × R N | v | f λ dxdv + R Ω (cid:12)(cid:12) u (cid:12)(cid:12) dx + c α t R t (cid:16)R τ k∇ u ε,λ ( s ) k L (Ω) ds (cid:17) dτ . (2.44)We infer from Eq.(2.44) that α Z t k∇ u ε,λ k L (Ω) dτ ≤ c + c α t Z t (cid:18)Z τ k∇ u ε,λ ( s ) k L (Ω) ds (cid:19) dτ where c = Z Ω × R N | v | f λ dxdv + Z Ω (cid:12)(cid:12) u (cid:12)(cid:12) dx ≤ C ( N, p ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) + Z Ω (cid:12)(cid:12) u (cid:12)(cid:12) dx < ∞ . It readily follows from Gronwall’s inequality that Z t k∇ u ε,λ k L (Ω) dτ ≤ exp (cid:18)Z t c τα dτ (cid:19) (cid:20)Z t c α exp (cid:18) − Z s c τα dτ (cid:19) ds (cid:21) = exp (cid:18) c t α (cid:19) (cid:20) c α Z t exp (cid:16) − c α s (cid:17) ds (cid:21) ≤ c α exp (cid:18) c t α (cid:19) Z ∞ exp (cid:16) − c α s (cid:17) ds = c r πc exp (cid:18) c t α (cid:19) for all 0 ≤ t ≤ T. Setting c = c r πc exp (cid:18) c T α (cid:19) , ONLOCAL STOKES-VLASOV SYSTEM 11 we get Z t k∇ u ε,λ k L (Ω) dτ ≤ c , and the above inequality entails Z Ω × R N | v | f ε,λ dxdv + Z Ω | u ε,λ | dx + 2 Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ ≤ c + c T. We deduce Eq. (2.39) by letting C = c + c T .The uniform estimate (2.41) is obtained as Eq. (2.33), and Eq. (2.42) follows in a trivial manner (see e.g.[7]). This concludes the proof. (cid:3) Passing to the limit λ → . We wish to pass to the limit as λ → u ε,λ , f ε,λ ) in order to prove the existence of the solution to our initial problem (1.1)-(1.5). Owing toLemma 2.1, we have k f ε,λ k L ∞ (0 ,T ; L p (Ω × R N )) ≤ C for all 1 ≤ p ≤ ∞ , k u ε,λ k L ∞ (0 ,T ; L (Ω) N ) ≤ C, k∇ u ε,λ k L ( Q ) ≤ C and (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ε,λ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H − (Ω) N ) ≤ C. Using the above uniform estimates (in λ ), we deduce that, given an ordinary sequence λ = ( λ n ) n (with0 < λ n ≤ λ n → n → ∞ , which we denote by λ → λ (still denotedby λ ), functions u ε ∈ L (0 , T ; V ) ∩ L ∞ (0 , T ; H ), f ε ∈ L ∞ (0 , T ; L p (Ω × R N )) and p ε ∈ L (0 , T ; L (Ω) / R )such that, as λ → f ε,λ → f ε in L ∞ (0 , T ; L p (Ω × R N ))-weak ∗ , (2.45) u ε,λ → u ε in L (0 , T ; V )-weak , (2.46) u ε,λ → u ε in L (0 , T ; H )-strong , (2.47) ∂ u ε,λ ∂t → ∂ u ε ∂t in L (0 , T ; H − (Ω) N )-weak (2.48)and p ε,λ → p ε in L (0 , T ; L (Ω) / R )-weak. (2.49)Let φ ∈ C ∞ ( O ) where O = Q × R Nv . We multiply the Vlasov equation (2.34) by φ and integrate by parts toget − Z O f ε,λ (cid:20) ∂φ∂t + εv · ∇ φ + ( u ε,λ ∗ θ λ − v ) · ∇ v φ (cid:21) dxdtdv = 0 . (2.50)We consider the terms in (2.50) respectively. It is easy to see that, as λ → Z O f ε,λ ∂φ∂t dxdtdv → Z O f ε ∂φ∂t dxdtdv. For the second and fourth terms, since the functions ( t, x, v ) v · ∇ φ and ( t, x, v ) v · ∇ v φ belong to C ∞ ( O ), we use them as test functions to get, as λ → Z O f ε,λ [ εv · ∇ φ − v · ∇ v φ ] dxdtdv → Z O f ε [ εv · ∇ φ − v · ∇ v φ ] dxdtdv. Now, as for the term R O f ε,λ ( u ε,λ ∗ θ λ ) · ∇ v φdxdtdv , we claim that Z O f ε,λ ( u ε,λ ∗ θ λ ) · ∇ v φdxdtdv → Z O f ε u ε · ∇ v φdxdtdv. (2.51)Indeed it is sufficient to prove that, under the convergence result (2.47) and for any ψ ∈ C ∞ ( O ) N , Z O f ε,λ u ε,λ · ψdxdtdv → Z O f ε u ε · ψdxdtdv (2.52)and to apply it with ψ = ∇ v φ . For the proof of (2.52), we refer to the proof of a more involved result,Lemma 4.1 in Section 4. Returning to (2.51), we have Z O f ε,λ ( u ε,λ ∗ θ λ ) · ∇ v φdxdtdv = Z O f ε,λ [( u ε,λ − u ε ) ∗ θ λ ] · ∇ v φdxdtdv + Z O f ε,λ ( u ε ∗ θ λ ) · ∇ v φdxdtdv = ( I ) + ( II ) . For the term ( I ), we have the estimate | ( I ) | ≤ k f ε,λ k L ∞ ( O ) k∇ v φ k ∞ k u ε,λ − u ε k L ( Q ) k θ λ k L ( Q ) ≤ C k u ε,λ − u ε k L ( Q ) , in which C is a positive constant independent of ε and λ . Thus, it follows from (2.47) that ( I ) → II ), we have that u ε ∗ θ λ → u ε in L ( Q )-strong (use once again (2.47)), so that by (2.52) wearrive at ( II ) → R O f ε u ε · ∇ v φ dxdtdv . (2.51) follows thereby.Taking into account all the above convergence results and passing to the limit in (2.50) as λ →
0, weobtain − Z O f ε (cid:20) ∂φ∂t + εv · ∇ φ + ( u ε − v ) · ∇ v φ (cid:21) dxdtdv = 0 , which amounts to ∂f ε ∂t + εv · ∇ f ε + div v (( u ε − v ) f ε ) = 0 in D ′ ( O ) . Proceeding as in [21, Section 4] we recover the reflection boundary condition f ε ( t, x, v ) = f ε ( t, x, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < − γ λ ( v ) ≤ {| v |≥ / λ } , we get (cid:12)(cid:12)(cid:12)(cid:12)Z Ω × R N (1 − γ λ ( v ))( f ∗ Θ λ ) dxdv (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω × R N {| v |≥ / λ } ( f ∗ Θ λ ) dxdv (2.53) ≤ λ Z Ω × R N | v | ( f ∗ Θ λ ) dxdv ≤ Cλ (cid:13)(cid:13) f (cid:13)(cid:13) L (Ω × R N ) ; see (2.12) . So, the functions f ∗ Θ λ and γ λ ( v )( f ∗ Θ λ ) have the same L -limit f as λ →
0. Hence, letting λ →
0, wearrive at f ε (0 , x, v ) = f ( x, v ) for ( x, v ) ∈ Ω × R N . Let us now deal with the Stokes system (2.35). We choose Φ ∈ C ∞ ( Q ) N and multiply (2.35) by ψ andintegrate over Q ; − R Q u ε,λ · ∂ Φ ∂t dxdt + R Q A ε ∇ u ε,λ · ∇ Φ dxdt + R Q ( A ε ∗ ∇ u ε,λ ) · ∇ Φ dxdt − R Q p ε,λ div Φ dxdt = − R O γ λ ( v ) f ε,λ ( u ε,λ − v ) · Φ dxdtdv. (2.54)In Eq.(2.54), only the right-hand side is more involved. However, proceeding as in (2.53), one can check that Z O γ λ ( v ) f ε,λ ( u ε,λ − v ) · Φ dxdtdv and Z O f ε,λ ( u ε,λ − v ) · Φ dxdtdv have the same limit, which is, using (2.52) and the convergence results (2.45)-(2.49), nothing else but Z O f ε ( u ε − v ) · Φ dxdtdv. ONLOCAL STOKES-VLASOV SYSTEM 13
Thus, passing to the limit in (2.54), we realize that u ε solves the equation ∂ u ε ∂t − div (cid:18) A ε ∇ u ε + Z t A ε ( t − τ , x ) ∇ u ε ( τ , x ) dτ (cid:19) + ∇ p ε = − Z R N ( u ε − v ) f ε dv in Q. We also obtain the initial condition u ε (0 , x ) = u ( x ), x ∈ Ω.We have just shown that ( u ε , f ε , p ε ) solves the system (1.1)-(1.5). This concludes the proof of Theorem2.1It remains to check that the above triple verifies the same estimates as in Lemma 2.1. As we are going tosee below, this is a mere consequence of the following well known result: • If B is a Banach space with norm k·k and f n → f in B -weak or weak ∗ , then k f k ≤ lim inf k f n k .We can therefore state the counterpart of Lemma 2.1. Lemma 2.2.
Let ( u ε , f ε , p ε ) be the solution to (1.1)-(1.5) constructed in Subsection . Then, Z Ω × R N (1 + | v | ) f ε dxdv + Z Ω | u ε | dx + 2 Z t Z Ω × R N f ε | u ε − v | dxdvdτ + Z t k∇ u ε ( τ ) k L (Ω) dτ ≤ C (2.55) for any ≤ t ≤ T and ε > , where C > is independent of ε . Moreover if f ∈ L p (Ω × R N ) , (1 ≤ p ≤ ∞ ) ,then k f ε k L ∞ (0 ,T ; L p (Ω × R N )) ≤ exp( N T ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω × R N ) for any ε > . (2.56) It also holds that (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ε ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H − (Ω) N ) ≤ C (2.57) and sup ε> k p ε k L (0 ,T ; L (Ω)) ≤ C. (2.58) Proof.
We follow arguments similar to those in [4]. First and foremost, we have by (2.39) that Z Ω × R N (1+ | v | ) f ε,λ dxdv + Z Ω | u ε,λ | dx +2 Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ + Z t k∇ u ε,λ ( τ ) k L (Ω) dτ ≤ C. (2.59)The only term to deal with is actually R t R Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ which we write as Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ − v | dxdvdτ = Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ | dxdvdτ − Z t Z Ω × R N f ε,λ ( u ε,λ ∗ θ λ ) · vdxdvdτ + Z t Z Ω × R N f ε,λ | v | dxdvdτ = ( I ) − II ) + ( III ) . Concerning (
III ), we know that f ε,λ → f ε in L ∞ (0 , T ; L ∞ (Ω × R N ))-weak ∗ . Let 0 < η <
1. Then becauseof (2.39) we have Z Ω × R N | v | f ε,λ γ η ( v ) dxdv ≤ Z Ω × R N | v | f ε,λ dxdv ≤ C. Hence there exists a function g η ∈ L ∞ ([0 , t ]) such that, up to a subsequence of λ →
0, setting M ( f ε,λ γ η )( τ ) = R Ω × R N | v | f ε,λ γ η ( v ) dxdv , M ( f ε,λ γ η ) → g η in L ∞ ([0 , t ])-weak ∗ ;thus k g η k L ∞ ([0 ,t ]) ≤ lim inf λ → Z t M ( f ε,λ γ η )( τ ) dτ ≤ lim inf λ → Z t Z Ω × R N | v | f ε,λ dxdvdτ . On the other hand, the weak ∗ convergence f ε,λ → f ε in L ∞ (0 , T ; L ∞ (Ω × R N )) implies M ( f ε,λ γ η ) → M ( f ε,λ γ η ) = Z Ω × R N | v | f ε γ η ( v ) dxdv in L ∞ ([0 , t ])-weak ∗ since the product of a function χ ∈ L ([0 , t ]) by | v | γ η lies in L ((0 , t ) × Ω × R N ). The uniqueness of theweak ∗ -limit yields M ( f ε,λ γ η )( τ ) = g η ( τ ) a.e. τ ∈ (0 , t ). It therefore follows from the Fatou’s lemma andfrom the fact that | v | f ε γ η ( v ) → | v | f ε as η →
0, that Z Ω × R N | v | f ε dxdv ≤ lim inf η → M ( f ε,λ γ η )( τ ) = lim inf η → g η ( τ ) ≤ lim inf λ → k M ( f ε,λ ) k L ∞ (0 ,t ) , i.e. Z Ω × R N | v | f ε dxdv ≤ lim inf λ → Z Ω × R N | v | f ε,λ dxdv, whence Z t Z Ω × R N | v | f ε dxdvdτ ≤ lim inf λ → Z t Z Ω × R N | v | f ε,λ dxdvdτ . As for the first term ( I ), one has Z t Z Ω × R N f ε,λ | u ε,λ ∗ θ λ | dxdvdτ = Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ (1 − γ η ( v )) dxdvdτ + Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ γ η ( v ) dxdvdτ ≥ Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ γ η ( v ) dxdvdτ . For any fixed η , Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ γ η ( v ) dxdvdτ → Z t Z Ω × R N | u ε | f ε γ η ( v ) dxdvdτ when λ →
0. Indeed, it is easy to see that | u ε,λ ∗ θ λ | γ η → | u ε | f ε γ η in L ((0 , t ) × Ω × R N )-strong as λ →
0, so that combining this with (2.45) (for p = ∞ ) we get our result. Thus, using once again Fatou’slemma, Z t Z Ω × R N | u ε | f ε dxdvdτ ≤ lim inf η → Z t Z Ω × R N | u ε | f ε γ η ( v ) dxdvdτ = lim inf η → lim inf λ → Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ γ η ( v ) dxdvdτ ≤ lim inf λ → Z t Z Ω × R N | u ε,λ ∗ θ λ | f ε,λ dxdvdτ . Finally, for ( II ), we have Z t Z Ω × R N f ε,λ ( u ε,λ ∗ θ λ ) · vdxdvdτ = Z t Z Ω × R N f ε,λ ( u ε,λ ∗ θ λ − u ε ) · vdxdvdτ + Z t Z Ω × R N f ε,λ u ε · vdxdvdτ = ( A ) + ( B ) . Dealing with ( A ), we have, by setting v ε,λ = u ε,λ ∗ θ λ − u ε ,( A ) = Z t Z Ω × R N f ε,λ (1 − γ η ( v )) v ε,λ · vdxdvdτ + Z t Z Ω × R N f ε,λ γ η ( v ) v ε,λ · vdxdvdτ , ONLOCAL STOKES-VLASOV SYSTEM 15 and using the inequality 1 − γ η ( v ) ≤ {| v |≥ / η } , (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Ω × R N f ε,λ (1 − γ η ( v )) v ε,λ · vdxdvdτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t Z Ω × R N f ε,λ {| v |≥ / η } | v ε,λ | | v | dxdvdτ ≤ Z t Z Ω (cid:20)Z R N | v | f ε,λ {| v |≥ / η } dv (cid:21) | v ε,λ | dxdτ ≤ Z t Z Ω (cid:18)Z R N | v | f ε,λ {| v |≥ / η } dv (cid:19) dx ! (cid:18)Z Ω | v ε,λ | dx (cid:19) dτ . But Z Ω (cid:18)Z R N | v | f ε,λ {| v |≥ / η } dv (cid:19) dx ≤ C (Ω) (cid:18)Z Ω Z R N | v | f ε,λ {| v |≥ / η } dvdx (cid:19) ≤ C (Ω) λ (cid:18)Z Ω × R N | v | f ε,λ {| v |≥ / η } dvdx (cid:19) ≤ Cλ because of (2.39).Recalling that v ε,λ ∈ L (0 , T ; H (Ω) N ) ֒ → L (0 , T ; L (Ω) N ), it follows from (2.39) that Z t (cid:18)Z Ω | v ε,λ | dx (cid:19) dτ ≤ C, so that (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Ω × R N f ε,λ (1 − γ η ( v )) v ε,λ · vdxdvdτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ.
It follows that R t R Ω × R N f ε,λ (1 − γ η ( v )) v ε,λ · vdxdvdτ → λ → Z t Z Ω × R N f ε,λ γ η ( v ) v ε,λ · vdxdvdτ → λ → . Indeed (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Ω × R N f ε,λ γ η ( v ) v ε,λ · vdxdvdτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f ε,λ k L ∞ ( O ) Z t Z Ω Z B (0 , | v | | v ε,λ | dvdxdτ ≤ C Z t Z Ω Z B (0 , | v | | v ε,λ | dvdxdτ and Z t Z Ω Z B (0 , | v | | v ε,λ | dvdxdτ ≤ | B (0 , | | Ω | k v ε,λ k L ( Q ) → λ → v ε,λ → L ( Q ) as λ →
0. It follows that ( A ) → λ →
0. We use the same kind of arguments toshow that ( B ) → R t R Ω × R N f ε u ε · vdxdvdτ , that is,( II ) → Z t Z Ω × R N f ε u ε · vdxdvdτ . Coming back to (2.59) and taking there the lim inf as λ →
0, we get at once (2.55). The lemma followsthereby. (cid:3)
Remark 2.2.
We observe that the sequence (cid:0)R R N f ε ( u ε − v ) dv (cid:1) ε> is bounded in L ( Q ) N . Indeed Z Q (cid:12)(cid:12)(cid:12)(cid:12)Z R N f ε ( u ε − v ) dv (cid:12)(cid:12)(cid:12)(cid:12) dxdt ≤ Z Q Z R N f ε | u ε − v | dvdxdt = Z O p f ε (1 + | v | ) p f ε | u ε − v | (1 + | v | ) dvdxdt ≤ √ (cid:18)Z O f ε (1 + | v | ) dxdvdt (cid:19) Z O f ε | u ε − v | (1 + | v | ) dvdxdt ! ≤ C ( see estimate (2.55) of Lemma 2.2.)3. Brief introduction to Σ -convergence This section is far from being a comprehensive introduction to Σ-convergence. It is rather a pretext forfixing notations and recalling fundamental results pertaining to Σ-convergence. We shall restrict ourselvesto concepts relevant to our context.3.1.
Algebras with mean value - An overview.
We refer the reader to [5, 23, 36, 39] for an extensivepresentation of the concept of algebras with mean value (algebras wmv, in short).Let A be an algebra wmv on R N , that is, a closed subalgebra of the C ∗ -algebra of bounded uniformlycontinuous functions on R N , BUC( R N ), which contains the constants, is translation invariant and is suchthat any of its elements possesses a mean value in the following sense: for any u ∈ A , the sequence ( u ε ) ε> (defined by u ε ( x ) = u ( x/ε ), x ∈ R N ) weakly ∗ -converges in L ∞ ( R N ) to some constant real function M ( u )(called the mean value of u ) as ε →
0. We denote by ∆( A ) the spectrum of A and by G the Gelfandtransformation on A . Let B pA ( R N ) (1 ≤ p < ∞ ) denote the Besicovitch space associated to A , that is, theclosure of A with respect to the Besicovitch seminorm k u k p = (cid:18) lim sup r → + ∞ | B r | Z B r | u ( y ) | p dy (cid:19) /p where B r is the open ball of R N centered at the origin and of radius r >
0. We set B ∞ A ( R N ) = { f ∈ ∩ ≤ p< ∞ B pA ( R N ) : sup ≤ p< ∞ k f k p < ∞} and we endow it with the seminorm [ f ] ∞ = sup ≤ p< ∞ k f k p . So topologized, the spaces B pA ( R N ) (1 ≤ p ≤ ∞ )are complete seminormed vector spaces which are not in general Fr´echet spaces since they are not separatedin general. We denote by B pA ( R N ) the completion of B pA ( R N ) with respect to k·k p for 1 ≤ p < ∞ , and withrespect to [ · ] ∞ for p = ∞ . The following hold true [24, 29]:( The Gelfand transformation G : A → C (∆( A )) extends by continuity to a unique continuous linearmapping (still denoted by G ) of B pA ( R N ) into L p (∆( A )), which in turn induces an isometric iso-morphism G of B pA ( R N ) / N = B pA ( R N ) onto L p (∆( A )) (where N = { u ∈ B pA ( R N ) : G ( u ) = 0 } ).Moreover if u ∈ B pA ( R N ) ∩ L ∞ ( R N ) then G ( u ) ∈ L ∞ (∆( A )) and kG ( u ) k L ∞ (∆( A )) ≤ k u k L ∞ ( R N ) .( The mean value M defined on A , extends by continuity to a positive continuous linear form (stilldenoted by M ) on B pA ( R N ) satisfying M ( u ) = R ∆( A ) G ( u ) dβ ( u ∈ B pA ( R N )). Furthermore, M ( τ a u ) = M ( u ) for each u ∈ B pA ( R N ) and all a ∈ R N , where τ a u = u ( · + a ). Moreover for u ∈ B pA ( R N ) wehave k u k p = [ M ( | u | p )] /p , and for u + N ∈ B pA ( R N ) we may still define its mean value once againdenoted by M , as M ( u + N ) = M ( u ).For u = v + N ∈ B pA ( R N ) (1 ≤ p ≤ ∞ ) and y ∈ R N , we define in a natural way the translate τ y u = v ( · + y ) + N of u , and as it can be seen in [34, 36], this is well defined and induces a strongly continuous N -parameter group of isometries T ( y ) : B pA ( R N ) → B pA ( R N ) defined by T ( y ) u = τ y u . We denote by ∂/∂y i (1 ≤ i ≤ N ) the infinitesimal generator of T ( y ) along the i th coordinate direction. We refer the reader to ONLOCAL STOKES-VLASOV SYSTEM 17 [34, 36] for the properties of ∂/∂y i as well as for those of the associated Sobolev-type spaces B ,pA ( R N ) and B ,p A ( R N ).Now, let A be an algebra wmv on R N . For µ ∈ ∆( A ) and f ∈ A , define T µ f by T µ f ( y ) = µ ( τ y f ), y ∈ R N . T µ f is well defined as an element of BUC( R N ) since A is translation invariant. Whence a bounded linearoperator T µ : A → BUC( R N ). Definition 3.1.
The algebra wmv A is said to be introverted if T µ ( A ) ⊂ A for any µ ∈ ∆( A ).Let A be an introverted algebra wmv on R N . Then [34, Theorem 3.2] its spectrum ∆( A ) is a compacttopological semigroup. In order to simplify the notations, the semigroup operation in ∆( A ) is additivelywritten. With this in mind, set K ( A ) = ∩ s ∈ ∆( A ) ( s + ∆( A )), the kernel of ∆( A ) . The following result provides us with the structure of K ( A ). Theorem 3.1 ([34, Theorem 3.4]) . Let A be an introverted algebra wmv on R N . Then (i) K ( A ) is a compact topological group. (ii) The mean value M on A can be identified as the Haar integral over K ( A ) . With the help of Theorem 3.1, we can define the convolution over ∆( A ) in terms of its kernel K ( A ).Indeed, as proved in [34], we have r + s ∈ K ( A ) whenever r ∈ ∆( A ) and s ∈ K ( A ). Thus, let p, q, m ≥ p + q = 1 + m . For u ∈ L p (∆( A )) and v ∈ L q (∆( A )) we define the convolutionproduct u b ∗ v as follows: ( u b ∗ v )( s ) = Z K ( A ) u ( r ) v ( s − r ) dβ ( r ), a.e. s ∈ ∆( A ) , where − r stands for the inverse of r ∈ K ( A ) (recall that K ( A ) is an Abelian group). Then b ∗ is well definedsince K ( A ) is an ideal of ∆( A ), and we have that R K ( A ) u ( r ) v ( s − r ) dβ ( r ) = R ∆( A ) u ( r ) v ( s − r ) dβ ( r ) since β is supported by K ( A ). Indeed for s ∈ ∆( A ) and r ∈ K ( A ), − r exists in K ( A ) and s − r ∈ K (∆( A )). Itholds that u b ∗ v ∈ L m (∆( A )) and further: k u b ∗ v k L m (∆( A )) ≤ k u k L p (∆( A )) k v k L q (∆( A )) . Now let u ∈ L p ( R N ; L p (∆( A ))) and v ∈ L q ( R N ; L q (∆( A ))). We define the double convolution u ∗ ∗ v asfollows: ( u ∗ ∗ v )( x, s ) = Z R N [( u ( t, · ) b ∗ v ( x − t, · )) ( s )] dt ≡ Z R N Z K ( A ) u ( t, r ) v ( x − t, s − r ) dβ ( r ) dt , a.e. ( x, s ) ∈ R N × ∆( A ) . Then ∗∗ is well defined as an element of L m ( R N × ∆( A )) and satisfies k u ∗ ∗ v k L m ( R N × ∆( A )) ≤ k u k L p ( R N × ∆( A )) k v k L q ( R N × ∆( A )) . It is to be noted that if u ∈ L p (Ω; L p (∆( A ))), and v ∈ L q ( R N ; L q (∆( A ))), we still define u ∗ ∗ v by replacing u by its zero extension over R N .Finally, for u ∈ L p ( R N ; B pA ( R N )) and v ∈ L q ( R N ; B qA ( R N )) we define the double convolution still denotedby ∗∗ as follows: u ∗ ∗ v is that element of L m ( R N ; B mA ( R N )) defined by G ( u ∗ ∗ v ) = b u ∗ ∗ b v. -convergence method. Throughout this section, Ω is an open subset of R N , and unless otherwisespecified, A is an algebra with mean value on R N . Definition 3.2. (1) A sequence ( u ε ) ε> ⊂ L p (Ω) (1 ≤ p < ∞ ) is said to weakly Σ -converge in L p (Ω) tosome u ∈ L p (Ω; B pA ( R N )) if as ε → Z Ω u ε ( x ) f ε ( x ) dx → Z Z Ω × ∆( A ) b u ( x, s ) b f ( x, s ) dxdβ ( s ) (3.1)for all f ∈ L p ′ (Ω; A ) (1 /p ′ = 1 − /p ) where f ε ( x ) = f ( x, x/ε ) and b f ( x, · ) = G ( f ( x, · )) a.e. in x ∈ Ω. Wedenote this by u ε → u in L p (Ω)-weak Σ.(2) A sequence ( u ε ) ε> ⊂ L p (Ω) (1 ≤ p < ∞ ) is said to strongly Σ -converge in L p (Ω) to some u ∈ L p (Ω; B pA ( R N )) if it is weakly Σ-convergent and further satisfies the following condition: k u ε k L p (Ω) → k b u k L p (Ω × ∆( A )) . We denote this by u ε → u in L p (Ω)-strong Σ.We recall here that b u = G ◦ u and b f = G ◦ f , G being the isometric isomorphism sending B pA ( R N ) onto L p (∆( A )) and G , the Gelfand transformation on A .In the sequel the letter E will throughout denote any ordinary sequence ( ε n ) n (integers n ≥
0) with0 < ε n ≤ ε n → n → ∞ . The following two results hold (see e.g. [5, 24, 29] for their justification). Theorem 3.2. (i)
Any bounded sequence ( u ε ) ε ∈ E in L p (Ω) (for < p < ∞ ) admits a subsequence which isweakly Σ -convergent in L p (Ω) . (ii) Any uniformly integrable sequence ( u ε ) ε ∈ E in L (Ω) admits a subsequence which is weakly Σ -convergentin L (Ω) . Theorem 3.3.
Let < p < ∞ . Let ( u ε ) ε ∈ E be a bounded sequence in W ,p (Ω) . Then there exist asubsequence E ′ of E , and a couple ( u , u ) ∈ W ,p (Ω; I pA ( R N )) × L p (Ω; B ,pA ( R N )) such that, as E ′ ∋ ε → , u ε → u in L p (Ω) -weak Σ ; ∂u ε ∂x i → ∂u ∂x i + ∂u ∂y i in L p (Ω) -weak Σ , ≤ i ≤ N. Remark 3.1.
In the above result, I pA ( R N ) stands for the space of invariant functions in B pA ( R N ) under thegroup of transformation T ( y ) of the preceding subsection: u ∈ I pA ( R N ) if and only if ∇ y u = 0. If we assumethe algebra A to be ergodic, then I pA ( R N ) consists of constant functions, so that the function u in Theorem3.3 does not depend on y , that is, u ∈ W ,p (Ω). We thus recover the already known result proved in [29]in the case of ergodic algebras.The next result deals with the Σ-convergence of a product of sequences. Theorem 3.4 ([30, Theorem 6]) . Let < p, q < ∞ and r ≥ be such that /r = 1 /p + 1 /q ≤ . Assume ( u ε ) ε ∈ E ⊂ L q (Ω) is weakly Σ -convergent in L q (Ω) to some u ∈ L q (Ω; B qA ( R N )) , and ( v ε ) ε ∈ E ⊂ L p (Ω) is strongly Σ -convergent in L p (Ω) to some v ∈ L p (Ω; B pA ( R N )) . Then the sequence ( u ε v ε ) ε ∈ E is weakly Σ -convergent in L r (Ω) to u v . As a consequence of the above theorem the following holds.
Corollary 3.1.
Let ( u ε ) ε ∈ E ⊂ L p (Ω) and ( v ε ) ε ∈ E ⊂ L p ′ (Ω) ∩ L ∞ (Ω) ( < p < ∞ and p ′ = p/ ( p − ) betwo sequences such that: (i) u ε → u in L p (Ω) -weak Σ ; (ii) v ε → v in L p ′ (Ω) -strong Σ ; (iii) ( v ε ) ε ∈ E isbounded in L ∞ (Ω) . Then u ε v ε → u v in L p (Ω) -weak Σ . Now, assume that the algebra A is introverted. Then its spectrum is a compact topological semigroupwhose kernel is a compact topological group, so that we can define, as in the preceding subsection, theconvolution over ∆( A ). Our aim in the next result is to link the Σ-convergence concept to the convolutionover the spectrum ∆( A ) of A . To see this, let p, q, m ≥ p + q = 1 + m . Let ONLOCAL STOKES-VLASOV SYSTEM 19 ( u ε ) ε> ⊂ L p (Ω) and ( v ε ) ε> ⊂ L q ( R N ) be two sequences. One may view u ε as defined in the whole R N bytaking its extension by zero outside Ω. Define( u ε ∗ v ε )( x ) = Z R N u ε ( t ) v ε ( x − t ) dt ( x ∈ R N ) , which lies in L m ( R N ). Then Theorem 3.5 ([34, Theorem 6.2]) . Let ( u ε ) ε> and ( v ε ) ε> be as above. Assume that, as ε → , u ε → u in L p (Ω) -weak Σ and v ε → v in L q ( R N ) -strong Σ , where u and v are in L p (Ω; B pA ( R N )) and L q ( R N ; B qA ( R N )) respectively. Assume further that the algebra wmv A is introverted. Then, as ε → , u ε ∗ v ε → u ∗ ∗ v in L m (Ω) -weak Σ . In practice, one deals with the evolutionary version of the concept of Σ-convergence. Such concept requiressome further notions such as those related to the product of algebras with mean value. Let A y (resp. A τ )be an algebra with mean value on R Ny (resp. R τ ). We define their product denoted by A = A τ ⊙ A y as theclosure in BU C ( R N +1 y,τ ) of the tensor product A τ ⊗ A y = { P finite u i ⊗ v i : u i ∈ A τ , v i ∈ A y } . It is a wellknown fact that A y ⊙ A τ is an algebra with mean value on R N +1 ( see e.g. [23, 24]).With this in mind, let A = A τ ⊙ A y be as above. The same letter G will denote the Gelfand transformationon A y , A τ and A , as well. Points in ∆( A y ) (resp. ∆( A τ )) are denoted by s (resp. s ). The compactspace ∆( A y ) (resp. ∆( A τ )) is equipped with the M -measure β y (resp. β τ ), for A y (resp. A τ ). We have∆( A ) = ∆( A τ ) × ∆( A y ) (Cartesian product) and the M -measure for A , with which ∆( A ) is equipped, isprecisely the product measure β = β τ ⊗ β y (see [23]). Finally, let 0 < T < ∞ . We set Q = (0 , T ) × Ω as inSection 1 (an open cylinder in R N +1 ) and O = Q × R N .This being so, a sequence ( u ε ) ε> ⊂ L p ( Q ) (1 ≤ p < ∞ ) is said to weakly Σ-converge in L p ( Q ) to some u ∈ L p ( Q ; B pA ( R N +1 )) if as ε → Z Q u ε ( t, x ) f (cid:18) t, x, tε , xε (cid:19) dxdt → Z Z Q × ∆( A ) b u ( t, x, s , s ) b f ( t, x, s , s ) dxdtdβ for all f ∈ L p ′ ( Q ; A ). We may also define the weak Σ-convergence in L p ( O ) as follows: ( u ε ) ε> ⊂ L p ( O )weakly Σ-converges to u ∈ L p ( O ; B pA ( R N +1 )) if Z O u ε ( t, x, v ) f (cid:18) t, x, tε , xε , v (cid:19) dxdtdv → Z Z O× ∆( A ) b u ( t, x, s , s, v ) b f ( t, x, s , s, v ) dxdtdvdβ for any f ∈ L p ′ ( O ; A ). Remark 3.2.
The conclusions of Theorems 3.2-3.5 are still valid mutatis mutandis in the present context(change Ω into Q in Theorem 3.2, W ,p (Ω) into L p (0 , T ; W ,p (Ω)), W ,p (Ω; I pA ( R N )) × L p (Ω; B ,pA ( R N )) into L p (0 , T ; W ,p (Ω; I pA ( R N ))) × L p ( Q ; B pA τ ( R τ ; B ,pA y ( R N )))), provided A is introverted in Theorem 3.5.4. Homogenization results
Throughout this section, we consider the algebras wmv A y and A τ to be as in the end of the precedingsection. We further assume that A τ is introverted.With this in mind, let ( u ε , f ε ) ε> be the sequence of solutions to (1.1)-(1.5). In view of Lemma 2.2, thereis a positive constant C independent of ε > ε> (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ε ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; V ′ ) ≤ C. This, together with the inequality (2.55) in Lemma 2.2 entail the precompactness of the sequence ( u ε ) ε> in L (0 , T ; H ). Thus, given an ordinary sequence E , there are a subsequence E ′ of E and a function u ∈ L ( Q ) N such that, as E ′ ∋ ε → u ε → u in L ( Q ) N . (4.1) In view of (2.55) and by the diagonal process, one can find a subsequence of ( u ε ) ε ∈ E ′ (not relabeled) whichweakly converges in L (0 , T ; V ) to the function u (this means that u ∈ L (0 , T ; V )). From Theorem 3.3, weinfer the existence of a function u = ( u k ) ≤ k ≤ N ∈ L ( Q ; B A τ ( R τ ; B , A y ( R Ny )) N ) such that the convergenceresult ∂ u ε ∂x i → ∂ u ∂x i + ∂ u ∂y i in L ( Q ) N -weak Σ (1 ≤ i ≤ N ) (4.2)holds when E ′ ∋ ε →
0. Still from Lemma 2.2 (see (2.56) for m = 2 and (2.58) therein) there exist asubsequence of E ′ (still denoted by E ′ ) and two functions f ∈ L ∞ (0 , T ; L (Ω × R N ; B A ( R N +1 ))), p ∈ L ( Q ; B A ( R N +1 )) with R Ω pdx = 0 such that, as E ′ ∋ ε → f ε → f in L ( Q × R N )-weak Σ (4.3)and p ε → p in L ( Q )-weak Σ . (4.4)We recall that ∂ u ∂x i = (cid:16) ∂u k ∂x i (cid:17) ≤ k ≤ N ( u = ( u k ) ≤ k ≤ N ) and ∂ u ∂y i = (cid:16) ∂u k ∂y i (cid:17) ≤ k ≤ N .Our goal in this section is the study of the asymptotics (as ε →
0) of ( u ε , f ε , p ε ) ε> under the followingadditional assumption( A3 ) A i ( t, x, · , · ) ∈ (cid:2) B A ( R N +1 y,τ ) (cid:3) N for i = 0 , t, x ) ∈ Q .4.1. Passing to the limit ε → . Let us first find the equation satisfied by f . To that end, let φ ∈C ∞ ( O ) ⊗ A ∞ (where we recall that O = Q × R Nv with Q = (0 , T ) × Ω) and define φ ε ∈ C ∞ ( O ) by φ ε ( t, x, v ) = φ ( t, x, t/ε, x/ε, v ) for ( t, x, v ) ∈ O . Multiplying the Vlasov equation (1.1) by φ ε and integratingby parts, − Z O f ε (cid:20) ∂φ ε ∂t + εv · ∇ φ ε + ( u ε − v ) · ∇ v φ ε (cid:21) dxdtdv = 0 . The above equation is equivalent to the following one − Z O f ε (cid:20)(cid:18) ∂φ∂t (cid:19) ε + 1 ε (cid:18) ∂φ∂τ (cid:19) ε + εv · ( ∇ φ ) ε + v · ( ∇ y φ ) ε + ( u ε − v ) · ( ∇ v φ ) ε (cid:21) dxdtdv = 0 . (4.5)Multiplying the above equation by ε and letting E ′ ∋ ε → E ′ is as above), we end up with (usingthe fact that the sequence ( f ε ( u ε − v )) ε> is bounded in L ( O ); see Remark 2.2) Z Z O× ∆( A ) b f ∂ b φ dxdtdvdβ = 0where ∂ = G ◦ ∂∂τ . It follows that ∂f ∂τ = 0, which amounts to say that f does not depends on τ . Indeedthis is equivalent to f ∈ L ∞ (0 , T ; L (Ω × R N ; B A y ( R N ; I A τ ( R τ )))), and since A τ is introverted, it is ergodic[34, Remark 3.3], so that I A τ ( R τ ) consists of constants. This means that the test functions φ may be chosenindependent of τ ∈ R , that is, φ ∈ C ∞ ( O ) ⊗ A ∞ y and so, φ ε ( t, x, v ) = φ ( t, x, x/ε, v ) for ( t, x, v ) ∈ O . Beforewe can pass to the limit in (4.5), we notice that the function ( t, x, v, y ) v · ∇ v φ lies in L ∞ ( O ; A y ) since ittrivially lies in C ∞ ( O ) ⊗ A ∞ y . Thus, Z O f ε v · ( ∇ v φ ) ε dxdtdv → Z Z O× ∆( A y ) b f v · ∇ v b φ dxdtdvdβ y . (4.6)In order to pass to the limit in the term R O f ε u ε · ( ∇ v φ ) ε dxdtdv , we need the following Lemma 4.1.
Let u and f be as in (4.1) and (4.3) , respectively. Then for any ψ ∈ ( C ∞ ( O ) ⊗ A ∞ y ) N , Z O f ε u ε · ψ ε dxdtdv → Z Z O× ∆( A y ) b f u · b ψ dxdtdvdβ y (4.7) as E ′ ∋ ε → . ONLOCAL STOKES-VLASOV SYSTEM 21
Proof.
First assume ψ = ( ψ i ) ≤ i ≤ N with ψ i = ϕ i ⊗ χ i ⊗ w i with ϕ i ∈ C ∞ ( Q ), χ i ∈ C ∞ ( R Nv ) and w i ∈ A ∞ y .Then Z O f ε u ε · ψ ε dxdtdv = N X i =1 Z O f ε ( t, x, v ) u iε ( t, x ) χ i ( v ) ϕ i ( t, x ) w i (cid:16) xε (cid:17) dxdtdv. Set U iε ( t, x, v ) = u iε ( t, x ) χ i ( v ) for ( t, x, v ) ∈ O . Then U iε → U i ≡ u i ⊗ χ i in L ( O )-strong as E ′ ∋ ε → . Indeed, since Z O (cid:12)(cid:12) U iε − U i (cid:12)(cid:12) dxdtdv ≤ Z R Nv | χ i ( v ) | dv Z Q (cid:12)(cid:12) u iε − u i (cid:12)(cid:12) dxdt, the claim follows from Eq. (4.1). Thus, we infer from Eq. (4.3) and Theorem 3.4 that f ε U iε → f U i in L ( O )-weak Σ as E ′ ∋ ε → . Hence, by choosing the special test function ϕ i ⊗ w i ⊗ R Nv ∈ L ∞ ( O ; A y ), we are led to Z O f ε ( t, x, v ) u iε ( t, x ) χ i ( v ) ϕ i ( t, x ) w i (cid:16) xε (cid:17) dxdtdv → Z Z O× ∆( A y ) b f u i χ i ϕ i b w i dxdtdvdβ y , or, Z O f ε u ε · ψ ε dxdtdv → Z Z O× ∆( A y ) b f u · b ψ dxdtdvdβ y . Now, by some routine computations, the result follows at once from the density of C ∞ ( Q ) ⊗ C ∞ ( R Nv ) ⊗ A ∞ y in C ∞ ( O ) ⊗ A ∞ y . (cid:3) As a consequence of Lemma 4.1, we have Z O f ε u ε · ( ∇ v φ ) ε dxdtdv → Z Z O× ∆( A y ) b f u · d ∇ v φ dxdtdvdβ y . Returning to (4.5) and taking the limit when E ′ ∋ ε →
0, we arrive at − Z Z O× ∆( A y ) b f " ∂ b φ∂t + v · ∂ b φ + ( u − v ) · ∇ v b φ dxdtdvdβ y = 0 , where ∂ b φ = G ◦ ∇ y φ . This gives rise to the following equation satisfied by f : ∂f ∂t + v · ∇ y f + div v (( u − v ) f ) = 0 in O × R Ny . (4.8)Following the lines of [21, Section 4] we prove, by choosing suitable test functions, that the function f satisfies the following reflection boundary and initial conditions f ( t, x, y, v ) = f ( t, x, y, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < v ∗ = v − v · ν ( x )) ν ( x ), f (0 , x, y, v ) = f ( x, v ) for ( x, y, v ) ∈ Ω × R Ny × R Nv (4.10)We consider now the Stokes system (1.2)-(1.3). Choosing ψ = ( ψ k ) ≤ k ≤ N ∈ C ∞ ( Q ) N and ψ =( ψ k ) ≤ k ≤ N ∈ [ C ∞ ( Q ) ⊗ A ∞ ] N , we set Φ = ( ψ , ψ ) and define Φ ε = ψ + ε ψ ε by Φ ε ( t, x ) = ψ ( t, x ) + ε ψ (cid:18) t, x, tε , xε (cid:19) for ( t, x ) ∈ Q .It can be checked that Φ ε ∈ C ∞ ( Q ) N . By plugging Φ ε into the variational formulation of (1.2), we obtain − R Q u ε · ∂ Φ ε ∂t dxdt + R Q A ε ∇ u ε · ∇ Φ ε dxdt + R Q ( A ε ∗ ∇ u ε ) · ∇ Φ ε dxdt − R Q p ε div Φ ε dxdt = − R O f ε ( u ε − v ) · Φ ε dxdtdv. (4.11) Our immediate goal is to pass to the limit in the above equation. We deal with its constituents in turn.Owing to Eq. (4.1), as E ′ ∋ ε → Z Q u ε · ∂ Φ ε ∂t dxdt → Z Q u · ∂ ψ ∂t dxdt. For the next term, one easily shows that as ε → ∂ Φ ε ∂x i → ∂ ψ ∂x i + ∂ ψ ∂y i in L ( Q ) N -strong Σ (1 ≤ i ≤ N ) . Combining the above convergence result with Eq. (4.2), we deduce from Corollary 3.1 that, as E ′ ∋ ε → ∂ u ε ∂x j · ∂ Φ ε ∂x i → (cid:18) ∂ u ∂x j + ∂ u ∂y j (cid:19) · (cid:18) ∂ ψ ∂x i + ∂ ψ ∂y i (cid:19) in L ( Q )-weak Σ . Passing to the limit in the above mentioned term using A as a test function (recall that A ∈ C ( Q ; [ B A ( R N +1 ) ∩ L ∞ ( R N +1 )] N ) by assumption ( A3 ) so that in view of [30, Proposition 8], it is an admissible test functionin the sense of [30, Definition 5]), we get Z Q A ε ∇ u ε · ∇ Φ ε dxdt → Z Z Q ×K b A D u · D Φ dxdtdβ as E ′ ∋ ε → u = ( u , u ), we have D u = ( D j u ) ≤ j ≤ N with D j u = ( D j u k ) ≤ k ≤ N and D j u k = ∂u k ∂x j + ∂ j b u k ( ∂ j b u k = G (cid:0) ∂u k /∂y j (cid:1) ), and the same definition for D Φ .Let us now tackle the term involving convolution. First we know that A ε → A in L ( R N +1 )-strong Σand ∇ u ε → ∇ u + ∇ y u in L ( Q ) N -weak Σ; hence by virtue of Theorem 3.5, we conclude that A ε ∗ ∇ u ε → A ∗ ∗ ( ∇ u + ∇ y u ) in L ( Q ) N -weak Σ as E ′ ∋ ε → (cid:16) b A ∗ ∗ ( \ ∇ u + ∇ y u ) (cid:17) ( t, x, s , s ) = Z t dτ Z K ( A τ ) b A ( τ , x, r , s )( ∇ u + ∂ b u )( t − τ , x, s − r , s ) dβ τ ( r ) , in which the function ∇ u is assumed to be defined on the whole of R N by taking its zero-extension off Ω.Therefore, repeating the same reasoning as for the preceding term, we arrive at (as E ′ ∋ ε → Z Q ( A ε ∗ ∇ u ε ) · ∇ Φ ε dxdt → Z Z Q × ∆( A y ) × K ( A τ ) ( c A ∗ ∗ D u ) · D Φ dxdtdβ, or, as β τ is supported by K ( A τ ), Z Q ( A ε ∗ ∇ u ε ) · ∇ Φ ε dxdt → Z Z Q × ∆( A ) ( c A ∗ ∗ D u ) · D Φ dxdtdβ. As for the term with the pressure, we have Z Q p ε div Φ ε dxdt = Z Q p ε div ψ dxdt + Z Q p ε (div y ψ ) ε dxdt (4.12)+ ε Z Q p ε (div ψ ) ε dxdt. Set p ( x, t ) = R ∆( A ) b p ( t, x, s , s ) dβ for a.e. ( t, x ) ∈ Q , where p is as in (4.4). Then we know that p ε → p in L ( Q )-weak, and, passing to the limit in (4.12) as E ′ ∋ ε → Z Q p ε div Φ ε dxdt → Z Q p div ψ dxdt + Z Z Q × ∆( A ) b p \ div y ψ dxdtdβ. ONLOCAL STOKES-VLASOV SYSTEM 23
Finally, for the term in the right-hand side of (4.11), reasoning as in the proof of (4.6) and (4.7), we get Z O f ε ( u ε − v ) · Φ ε dxdtdv → Z Z O× ∆( A ) b f ( u − v ) · ψ dxdtdβdv. Putting together the previous convergence results, we are led to the fact that the quadruple ( u , u , f , p )determined by (4.1)-(4.4) solves the system consisting of equation (4.8) and − R Q u · ψ ′ dxdt + RR Q × ∆( A ) (cid:16) b A D u + b A ∗ ∗ D u (cid:17) · D Φ dxdtdβ − R Q p div ψ dxdt − RR Q × ∆( A ) b p \ div y ψ dxdtdβ = − RR O× ∆( A ) b f ( u − v ) · ψ dxdtdβdv, for all Φ = ( ψ , ψ ) ∈ C ∞ ( Q ) N × [ C ∞ ( Q ) ⊗ A ∞ ] N . (4.13)From the equality div u ε = 0 we easily deduce that div y u = 0. Next, we need to uncouple Eq. (4.13),which is equivalent to the system (4.14)-(4.15) below: ( RR Q × ∆( A ) (cid:16) b A D u + b A ∗ ∗ D u (cid:17) · ∂ b ψ dxdtdβ − RR Q × ∆( A ) b p \ div y ψ dxdtdβ = 0for all ψ ∈ [ C ∞ ( Q ) ⊗ A ∞ ] N (4.14)and ( − R Q u · ψ ′ dxdt + RR Q × ∆( A ) (cid:16) b A D u + b A ∗ ∗ D u (cid:17) · ∇ ψ dxdtdβ − R Q p div ψ dxdt = − RR O× ∆( A ) b f ( u − v ) · ψ dxdtdβdv for all ψ ∈ C ∞ ( Q ) N . (4.15)For Eq. (4.14), we choose ψ ( x, t ) = ϕ ( x, t ) w with ϕ ∈ C ∞ ( Q ) and w ∈ ( A ∞ ) N . Then (4.14) becomes ( R ∆( A ) (cid:16) b A D u + b A ∗ ∗ D u (cid:17) · ∂ b w dβ − R ∆( A ) b p c div b w dβ = 0for all w ∈ ( A ∞ ) N . (4.16)Now, fix ξ ∈ R N × N and consider the following cell problem: Find u ξ ∈ B A τ ( R τ ; B , ( R Ny )), p ξ ∈ B A τ ( R τ ; B A y ( R Ny ) / R ) such that R ∆( A ) (cid:16) b A ( ξ + ∂ b u ξ ) + b A ∗ ∗ ( ξ + ∂ b u ξ ) (cid:17) · ∂ b w dβ − R ∆( A ) b p ξ c div b w dβ = 0for all w ∈ ( A ∞ ) N (4.17)where B , ( R Ny ) = { v ∈ B , A y ( R Ny ) N : div y v = 0 } . Then Eq. (4.17) is the variational formulation of theproblem (cid:26) − div y (cid:0) A ∇ y u ξ + A ∗ ∗∇ y u ξ (cid:1) + ∇ y p ξ = div y ( A ξ + A ∗ ∗ ξ ) in R N +1 y,τ div y u ξ = 0 . (4.18)Thanks to the properties of the functions A i ( i = 0 , u ξ , p ξ ) ∈ B A τ ( R τ ; B , ( R Ny )) × B A τ ( R τ ; B A y ( R Ny ) / R ).Now, Returning to Eq. (4.17) where we set ξ = ∇ u ( t, x ), we find out that Eqs. (4.16) and (4.17) are thevariational formulation of the same problem (say Eq. (4.18)). Owing to the uniqueness of the solution of Eq.(4.17), we have u = u ∇ u and p = p ∇ u where u ∇ u (resp. p ∇ u ) denotes the function ( t, x ) u ∇ u ( x,t ) (resp. ( t, x ) p ∇ u ( x,t ) ) defined from Q into B A τ ( R τ ; B , ( R Ny )) (resp. B A τ ( R τ ; B A y ( R Ny ) / R )).4.2. Homogenization result.
Let us first define the effective coefficients. Let the matrices C k = ( c kij ) ≤ i,j ≤ N ( k = 0 ,
1) be defined as follows: for any ξ = ( ξ ij ) ≤ i,j ≤ N , C ξ = Z ∆( A ) b A ( ξ + ∂ b u ξ ) dβ, C ξ = Z ∆( A ) (cid:16) b A ∗ ∗ ( ξ + ∂ b u ξ ) (cid:17) dβ (4.19)Then, thanks to the uniqueness of u ξ (for a given ξ ), the matrices C k are well defined and are symmetric.It is obvious that the c kij are obtained by choosing in Eq. (4.19) ξ = ( δ ij ) ≤ i,j ≤ N (the identity matrix), δ ij being the Kronecker delta.The matrix C k are the effective homogenized viscosities which depend continuously on ( t, x ) ∈ Q as seenin the next result whose classical proof is omitted. Proposition 4.1.
It holds that (i) C i ( i = 0 , are symmetric and further C i ∈ C ( Q ) N ; (ii) C λ · λ ≥ α | λ | for all ( x, t ) ∈ Q and all λ ∈ R N , where α is the same as in assumption ( A1 ) . We can now formulate the homogenized problem. To this end, consider Eq. (4.15) in which we take u = u ∇ u . We get − R Q u · ψ ′ dxdt + R Q hR ∆( A ) (cid:16)c A ( ∇ u + ∂ b u ∇ u ) + b A ∗ ∗ ( ∇ u + ∂ b u ∇ u ) (cid:17) dβ i · ∇ ψ dxdt − R Q p div ψ dxdt = − R Q hR R N (cid:16)R ∆( A ) b f dβ (cid:17) ( u − v ) dv i · ψ dxdt for all ψ ∈ C ∞ ( Q ) N ,which is just the variational formulation (where accounting of div u = 0) of the following anisotropicnonlocal Stokes system ∂ u ∂t − div( C ∇ u + R t C ( t − τ , x ) ∇ u ( x, τ ) dτ ) + ∇ p = − R R N f ( u − v ) dv in Q div u = 0 in Q u = 0 on ∂ Ω × (0 , T ) u ( x,
0) = u ( x ) in Ω (4.20)where f = R ∆( A ) b f dβ . Finally, to be more concise, let us put together Eq. (4.8)-(4.10): ∂f ∂t + v · ∇ y f + div v (( u − v ) f ) = 0 in O × R Ny f ( t, x, y, v ) = f ( t, x, y, v ∗ ) for x ∈ ∂ Ω with v · ν ( x ) < f (0 , x, y, v ) = f ( x, v ) for ( x, y, v ) ∈ Ω × R Ny × R Nv (4.21)where v ∗ = v − v · ν ( x )) ν ( x ).In view of what has been done above, we see that the system (4.20)-(4.21) possesses at least solution( u , f , p ) such that u ∈ L (0 , T ; V ) ∩ C ([0 , T ]; H ), f ∈ L ∞ (0 , T ; L ∞ (Ω × R N ) ∩ L (Ω × R N )) and p ∈ L (0 , T ; L (Ω) / R ). We are therefore led to the following homogenization result which is the second mainresult of this work. Theorem 4.1.
Assume that ( A1 )-( A3 ) hold. For each ε > , let ( u ε , f ε , p ε ) be a solution to (1.1)-(1.5) .Then up to a subsequence, the sequence ( u ε ) ε> strongly converges in L ( Q ) N to u , the sequence ( f ε ) ε> weakly Σ -converges in L ( Q × R Nv ) towards f and the sequence ( p ε ) ε> weakly converges in L (0 , T ; L (Ω) / R ) towards p , where ( u , f , p ) is a solution to the system (4.20)-(4.21) . Moreover any weak Σ -limit point ( u , f , p ) in L (0 , T ; V ) ∩ C ([0 , T ]; H ) × L ∞ (0 , T ; L ∞ (Ω × R N ) ∩ L (Ω × R N )) × L (0 , T ; L (Ω) / R ) of ( u ε , f ε , p ε ) ε> is a solution to Problem (4.20)-(4.21) . Some applications
A look at the previous section reveals that the homogenization process has been made possible thanks toAssumption ( A3 ). This assumption is formulated in a general manner encompassing a variety of concretebehaviors of the coefficients of the operator involved in (1.2). We aim at providing in this section somenatural situations leading to the homogenization of (1.1)-(1.5). First and foremost, it is an easy task (using[34]) to see that all the algebras wmv involved in the following problems are introverted.5.1. Problem 1 (Periodic homogenization).
The homogenization of (1.1)-(1.5) can be achieved underthe periodicity assumption( A3 ) The functions A i ( t, x, · , · ) ( i = 0 ,
1) are periodic of period 1 in each scalar coordinate.This leads to ( A3 ) with A = C per( Z × Y ) = C per( Z ) ⊙ C per( Y ) (the product algebra, with Y = (0 , N and Z = (0 , B A ( R N +1 y,τ ) = L per( Z × Y ).5.2. Problem 2 (Almost periodic homogenization).
The above functions in ( A3 ) are both almostperiodic in ( τ , y ) in the sense of Besicovitch [2]. This amounts to ( A3 ) with A = AP ( R N +1 y,τ ) = AP ( R τ ) ⊙ AP ( R Ny ) ( AP ( R Ny ) the Bohr almost periodic functions on R Ny [3]). ONLOCAL STOKES-VLASOV SYSTEM 25
Problem 3 (Weak almost periodic homogenization).
The homogenization problem for (1.1)-(1.5)may also be considered under the assumption( A3 ) A i ( t, x, · , · ) ( i = 0 ,
1) is weakly almost periodic [8]. This leads to ( A3 ) with A = W AP ( R τ ) ⊙ W AP ( R Ny ) ( W AP ( R Ny ), the algebra of continuous weakly almost periodic functions on R Ny ; see e.g.,[8]).5.4. Problem 4.
Let F be a Banach subalgebra of BUC( R m ). Let B ∞ ( R d ; F ) denote the space of allcontinuous functions ψ ∈ C ( R d ; F ) such that ψ ( ζ ) has a limit in F as | ζ | → ∞ . In particular, it is knownthat B ∞ ( R d ; R ) ≡ B ∞ ( R d ).With this in mind, our goal here is to study the homogenization for problem (1.1)-(1.5) under the hy-pothesis( A3 ) A i ( t, x, · , · ) ∈ B ∞ ( R τ ; L per( Y )) for any ( t, x ) ∈ Q , where Y = (0 , N .It is an easy task to see that the appropriate algebra here is the product algebra A = B ∞ ( R τ ) ⊙ C per( Y ).6. Conclusion
In this work, we have constructed weak solutions of a nonlocal Stokes-Vlasov system without any as-sumptions on high-order velocity moments of the initial distribution of particles. Our approach consistedin applying Schauder’s fixed point theorem to a carefully regularized version of the Stokes-Vlasov systemand passing to the limit by means of compactness arguments. Our investigation culminated with the ho-mogenization of Stokes-Vlasov system under generous structural assumptions on coefficients encompassingvarious forms of classical behaviors.
References [1] A.A. Amsden, P.J. O’Rourke, T.D. Butler, KIVA-2: a computer program for chemically reactive flows with sprays, NASASTI/recon technical report N 89 (1989): 27975.[2] A.S. Besicovitch, Almost periodic functions, Cambridge Univ. Press, 1932.[3] H. Bohr, Almost periodic functions, Chelsea, New York, 1947.[4] L. Boudin, L. Desvillettes, C. Grandmont, A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differ. Integral Equ. (2009) 1247-1271.[5] J. Casado Diaz, I. Gayte, The two-scale convergence method applied to generalized Besicovitch spaces, Proc. R. Soc. Lond.A (2002), 2925-2946.[6] N. Crouseilles, E. Fr´enod, S.A. Hirstoaga, A. Mouton, Two-scale macro-micro decomposition of the Vlasov equation witha strong magnetic field, Math. Models Methods Appl. Sci. (2013) 1527-1559.[7] H. Douanla, J.L. Woukeng, Almost periodic homogenization of a generalized Ladyzhenskaya model for incompresibleviscous flow, J. Math. Sci. (N.Y.) (2013), 431-458.[8] W.F. Eberlein, Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. (1949) 217-240.[9] E. Frenod, K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proc. Roy. Soc.Edinburgh Sect. A (1996) 1247-1275.[10] E. Fr´enod, E. Sonnendr¨ucker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strongexternal magnetic field, Asymptot. Anal. (1998) 193-213.[11] P.A. Gokhale, S. Deokattey, V. Kumar, Accelerator driven systems (ADS) for energy production and waste transmutation:International trends in R&D, Progress in Nuclear Energy (2006) 91-102.[12] T. Goudon, P.-E. Jabin, A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime,Indiana Univ. Math. J. (2004) 1495-1515.[13] T. Goudon, P.-E. Jabin, A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime,Indiana Univ. Math. J. (2004) 1517-1536.[14] W. Greenberg, M. Williams, P.F. Zweifel, A report on the seventh international conference on transport theory, TransportTheory and Statistical Physics (1981) 115-130.[15] J.-S. Jiang, C.-K. Lin, Weak turbulence plasma induced by two-scale homogenization, J. Math. Anal. Appl. (2014)585-596.[16] A. Mellet, A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. ModelsMethods Appl. Sci. (2007). 1039-1063.[17] A. Mellet, A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations,Comm. Math. Phys. (2008) 573-596.[18] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Jpn. J. Ind. Appl.Math. (1998) 51-74. [19] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag,Berlin, 1994.[20] J.L. Lions, E. Magenes, Probl`emes aux limites non homog`enes et applications (Vol 1). Paris, Dunod, 1968[21] S. Mischler, On the trace problem for solutions of the Vlasov equation, Comm. Partial Differ. Equ. 25(2000) 1415-1443.[22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal onMathematical Analysis (1989) 608-623.[23] G. Nguetseng, Homogenization structures and applications I, Z. Anal. Anwen. (2003) 73-107.[24] G. Nguetseng, M. Sango, J.L. Woukeng, Reiterated ergodic algebras and applications, Commun. Math. Phys. (2010)835-876.[25] J. Nieto, F. Poupaud, J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal. (2001)1, 29-59.[26] E.S. Oran, J.P. Boris, Numerical simulation of reactive flow, Cambridge University Press, 2005.[27] R. Potthast, P. B. Graben, Existence and properties of solutions for neural field equations, Math. Meth. Appl. Sci. (2010) 935-949.[28] F. Poupaud, J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci. (2000) 1027-1045.[29] M. Sango, N. Svanstedt, J.L. Woukeng, Generalized Besicovitch spaces and application to deterministic homogenization,Nonlin. Anal. TMA (2011) 351–379.[30] M. Sango, J.L. Woukeng, Stochastic sigma-convergence and applications, Dynamics of PDEs (2011) 261-310.[31] J. Simon, On the existence of the pressure for solutions of the variational Navier-Stokes equations, J. Math. Fluid Mech. (1999) 225-234.[32] N. Svanstedt, J.L. Woukeng, Homogenization of a Wilson-Cowan model for neural fields, Nonlin. Anal. RWA (2013)1705-1715.[33] R. Temam, Navier-Stokes equations: Theory and numerical analysis, North-Holland, Amsterdam, 1984.[34] J.L. Woukeng, Introverted algebras with mean value and applications, Nonlin. Anal. (2014) 190-215.[35] J.L. Woukeng, Linearized viscoelastic Oldroyd fluid motion in almost periodic environment, Math. Meth. Appl. Sci., onlinein Wiley Online Library, DOI: 10.1002/mma.3026, 2013.[36] J.L. Woukeng, Homogenization in algebras with mean value, arXiv: 1207.5397v1, 2012 (Submitted).[37] H.-T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. (1991) 63-80.[38] C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl. (2013)275-293.[39] V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki (1983) 571-582(english transl.: Math. Notes, (1983) 294-300). G. Nguetseng, Department of Mathematics, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon
E-mail address : [email protected], [email protected] C. Wafo Soh, Department of Mathematics and Statistical Sciences, College of Science, Engineering and Tech-nology, Jackson State University, JSU Box 17610, 1400 J R Lynch St., Jackson, MS 39217, USA
E-mail address : celestin.wafo [email protected] J. L. Woukeng, Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67,Dschang, Cameroon
E-mail address ::