Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains
aa r X i v : . [ m a t h . A P ] J a n Concentration versus absorption for the Vlasov-Navier-Stokessystem on bounded domains
Lucas Ertzbischoff ∗ , Daniel Han-Kwan † and Ayman Moussa ‡ January 14, 2021
Abstract
We study the large time behavior of small data solutions to the Vlasov-Navier-Stokessystem set on Ω × R , for a smooth bounded domain Ω of R , with homogeneous Dirichletboundary condition for the fluid and absorption boundary condition for the kinetic phase.We prove that the fluid velocity homogenizes to 0 while the distribution function concentratestowards a Dirac mass in velocity centered at 0, with an exponential rate. The proof, whichfollows the methods introduced in [24], requires a careful analysis of the boundary effects.We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviorsfor the kinetic density, from total absorption to no absorption at all. Contents ∗ Centre de Math´ematiques Laurent Schwartz (UMR 7640), Ecole Polytechnique, Institut Polytechnique deParis, 91128 Palaiseau Cedex, France (lucas.ertzbischoff@polytechnique.edu) † Centre de Math´ematiques Laurent Schwartz (UMR 7640), Ecole Polytechnique, Institut Polytechnique deParis, 91128 Palaiseau Cedex, France ([email protected]) ‡ Sorbonne Universit´e, Universit´e Paris-Diderot, CNRS, INRIA, LJLL, F-75005 Paris, France([email protected]) Further description of the asymptotic local density 338 Asymptotic profiles with a prescribed mass 38
A Appendix 46
A.1 Boundary value problem in Ω × R for the kinetic equation . . . . . . . . . . . . 46A.2 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.3 The Wasserstein distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.4 Extension of Lipschitz functions vanishing at the boundary . . . . . . . . . . . . 50A.5 A variant of Gronwall lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.6 Agmon inequality on a bounded domain of R . . . . . . . . . . . . . . . . . . . . 51A.7 Gagliardo-Nirenberg-Sobolev inequality on a bounded domain . . . . . . . . . . . 51A.8 Maximal L p L q regularity for the Stokes system on a bounded domain . . . . . . . 51A.9 Parabolic regularization for the Navier-Stokes system with a source term on abounded domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Let Ω be a smooth connected and bounded open set of R . We consider the Vlasov-Navier-Stokes system in Ω × R ∂ t u + ( u · ∇ ) u − ∆ u + ∇ p = j f − ρ f u, ( t, x ) ∈ R ∗ + × Ω , (1.1)div u = 0 , ( t, x ) ∈ R ∗ + × Ω , (1.2) ∂ t f + v · ∇ x f + div v (( u − v ) f ) = 0 , ( t, x, v ) ∈ R ∗ + × Ω × R . (1.3)where we define ρ f ( t, x ) := Z R f ( t, x, v ) d v,j f ( t, x ) := Z R vf ( t, x, v ) d v. This system of nonlinear PDEs aims at describing the transport of small particles (the dispersed phase ) immerged within a Newtonian viscous and incompressible fluid (the continousphase ). Here, we describe the particles thanks to a kinetic distribution function f ( t, · , · ) onthe phase space Ω × R while the fluid is described thanks to its velocity u ( t, · ) and pressure p ( t, · ). The function f satisfies absorption boundary condition and we impose no-slip boundarycondition for the vector-field u . We refer to Subsection 1.1 for more details.In the large gallery of such fluid-kinetic systems , which stem from the works of O’Rourke[31] and Williams [33], the dispersed phase and the fluid are related through a particular cou-pling (see [14]). Here, we neglect the effect of collisions, coalescence and fragmentation betweenparticles and we work under the assumption of thin sprays , considering that the volume oc-cupied by the droplets is negligeable compared to that occupied by the fluid. This modellingcorresponds to a moderate Reynolds number for the particles and is for instance a prototype forthe description of an aerosol in the air. The coupling is thus made of a drag term in the Vlasovequation (1.3) which appears as a friction term, and of a source term in the Navier-Stokes equa-tions (1.1), called the Brinkman force , which describes the exchange of momentum between the2articles and the fluid. In this system called the Vlasov-Navier-Stokes system, where physicalconstants are all normalized, both unknowns f and u depend on each other.The Vlasov-Navier-Stokes system and its mathematical analysis have received a lot of at-tention in the past twenty years. The question of the global existence of weak solutions (or localexistence of strong solutions) to the Cauchy problem is now well-understood on a large class ofspatial tridimensional domains, like the flat torus T in [5, 12], a fixed bounded domain in [1]and even a time-dependent bounded domain in [6, 7].Concerning the rigorous derivation of theses equations from ”first principles”, little is knownabout the whole system. A first approach consists at looking at fluid-solid equations and their mean-field limits. In a stationary framework, homogenization techniques have been used in[15, 25, 26, 9] to derive the Brinkman force in the fluid equation. The dynamical problem hasonly been adressed when the inertia of the particles is neglected and in some dilute regime(see [27, 29]). An alternative, still partly formal, program has been proposed in [3, 4] wherethe starting point is a coupling between two Boltzmann equations and where both fluid andparticles are considered as dispersed phases.The large time behavior for global solutions to the Vlasov-Navier-Stokes system appears asone of the next important steps in the understanding of such fluid kinetic model. A generalsetting has been highlighted by Jabin in [28] where an asymptotic scenario with concentrationin velocity for the particles (namely, a convergence of the distribution function towards a Diracmass for the velocity part) has been described for some class a kinetic equations (but with adifferent coupling between the fluid and the particles).The asymptotic dynamics of the Vlasov-Navier-Stokes has been studied in a recent series ofpapers [20, 24, 23]: a concentration phenomenon in velocity, leading to monokinetic large timebehavior for the particles, is proven in [24] in the case of the torus T and in [23] in the caseof the whole space R . This asymptotics is obtained for data that are in some sense close toequilibrium (we mention that a first contribution of Choi and Kwon has been made in the samedirection on the torus in [12], but with an a priori assumption on the solutions which is notmade in the two previous articles). We refer to Section 1.2 of this article where we give moredetails about the strategy which is used. When a Fokker-Planck dissipation term (namely, aterm of the form − ∆ v f ) is added in the kinetic equation (1.3), global classical solutions can beconstructed for data close to Maxwellian equilibria and this non-singular steady states locallyattract these solutions (see [21]).Fore more general and physical domains, very few articles deal with the question of thelarge time behavior of the Vlasov-Navier-Stokes system. A seminal article of Hamdache [22]deals with the Vlasov-Stokes system with specular reflexion on a bounded domain of R andshows that, up to a subsequence, the sequence of solutions ( u n ( t ) , f n ( t )) := ( u ( t + t n ) , f ( t + t n ))converges in some sense to a solution of the same system as t n n → + ∞ −→ + ∞ . In [20], the authorsconsider the particular case of the Vlasov-Navier-Stokes system on a rectangle in R , wherethe particular geometry of the domain allows to construct weak solutions around non-singularstationary equilibrium and where a geometric control condition helps to avoid concentration invelocity scenario for the particles (we also refer to Section 1.2 below for more details).In this article, we study the Vlasov-Navier-Stokes system on a bounded domain of R withabsorption boundary condition for the particle distribution and homogeneous Dirichlet bound-ary condition for the fluid velocity. We aim at giving a proof of the large time monokineticbehavior to solutions to the system. This singular behavior is adressed for global weak so-lutions satisfying a natural energy-dissipation inequality and starting at initial data close toequilibrium. Furthermore, there is a competition between concentration and absorption whichdetermine the final dynamics. A broad range of outcomes is possible for the asymptotic spatialprofile as we illustrate by exhibiting examples of initial data, leading to a variety of behaviors,3rom total absorption of the particles to no absorption at all. We denote the phase-space domain by O := Ω × R , In the following, the outer-pointing normal to the boundary at a point x ∈ ∂ Ω will be denotedby n ( x ).We first define the class of admissible initial data for (1.1) - (1.3). Definition 1.1 (Initial condition).
We shall say that a couple ( u , f ) is an admissible initialcondition if u ∈ L (Ω) , div u = 0 ,f ∈ L ∩ L ∞ (Ω × R ) ,f ≥ , Z Ω × R f ( x, v ) d x d v = 1 , ( x, v ) f ( x, v ) | v | ∈ L (Ω × R ) . The system (1.1) - (1.3) is supplemented with the following initial conditions for the fluidvelocity u and the distribution function fu | t =0 = u in Ω ,f | t =0 = f in O . We prescribe the following homogeneous Dirichlet boundary condition for the fluid velocity u ( t, · ) = 0 , on ∂ Ω . (1.4)We also need to introduce the following outgoing/incoming phase-space boundary for the dis-persed phase: Σ ± := (cid:8) ( x, v ) ∈ ∂ Ω × R | ± v · n ( x ) > (cid:9) , Σ := (cid:8) ( x, v ) ∈ ∂ Ω × R | v · n ( x ) = 0 (cid:9) , Σ := Σ + ⊔ Σ − ⊔ Σ = ∂ Ω × R . Then, we prescribe the following absorption boundary condition for the distribution function f : f ( t, · , · ) = 0 , on Σ − , (1.5)meaning that all the particles reaching transversally the physical boundary are deposited.We then define the energy and the dissipation of the whole system. Definition 1.2.
1. The kinetic energy of the Vlasov-Navier-Stokes system is defined forall t ≥ t ) := 12 Z Ω | u ( t, x ) | d x + 12 Z Ω × R f ( t, x, v ) | v | d v d x. (1.6)4. The dissipation of the Vlasov-Navier-Stokes system is defined for all t ≥ t ) := Z Ω × R f ( t, x, v ) | u ( t, x ) − v | d v d x + Z Ω |∇ u ( t, x ) | d x. (1.7)These two functionals naturally appear when looking for a priori estimates satisfied bysolutions to the Vlasov-Navier-Stokes system. One can indeed check that the following energy-dissipation identity formally holds dd t E( t ) + D( t ) = 0 . (1.8)We denote by D div (Ω) the set of smooth R valued divergence free vector-fields havingcompact support in Ω. The closures of D div (Ω) in L (Ω) and in H (Ω) are respectively denotedby L (Ω) and by H (Ω). We write H − (Ω) for the dual of the latter. We then introduce thenotion of weak solution to the system. Definition 1.3 (Weak solution).
Consider an admissible initial condition ( u , f ) in thesense of Definition 1.1. A global weak solution to the Vlasov-Navier-Stokes system with initialcondition ( u , f ) on Ω is a pair ( u, f ) such that u ∈ L ∞ loc ( R + ; L (Ω)) ∩ L ( R + ; H (Ω)) ,f ∈ L ∞ loc ( R + ; L ∩ L ∞ (Ω × R d )) ,j f − ρ f u ∈ L ( R + ; H − (Ω)) ,f ( t, x, v ) ≥ t, x, v ) ∈ R + × Ω × R , with u being a Leray solution to the Navier-Stokes equations (1.1) - (1.2) (with initial condition u ) and f being a renormalized solution in the sense of Di Perna-Lions (see Appendix A.1) tothe Vlasov equation (1.3) (with initial condition f ), and such that the following energy estimateholds for almost all s ≥ s = 0) and all t ≥ s E( t ) + Z ts D( σ )d σ ≤ E( s ) . (1.9) Remark 1.4.
A weak solution ( u, f ) to the Vlasov-Navier-Stokes system satisfies the followingweak formulations. • For all
T >
0, for all φ ∈ D ([0 , T ] × Ω) such that φ ( T ) = 0 and div x φ = 0 Z T Z Ω [ u · ∂ t φ + ( u ⊗ u ) : ∇ x φ − ∇ x u : ∇ x φ ] ( t, x ) d x d t = − Z T h j f − ρ f u, φ i ( t ) d t − Z Ω u ( x ) · φ (0 , x ) d x. • For all
T >
0, for all ψ ∈ D ([0 , T ] × Ω × R ) such that ψ ( T ) = 0 and vanishing on R + × (Σ + ⊔ Σ ) Z T ZZ Ω × R f [ ∂ t ψ + v · ∇ x ψ + ( u − v ) · ∇ v ψ ] ( t, x, v ) d x d v d t = − ZZ Ω × R f ( x, v ) ψ (0 , x, v ) d x d v. Furthermore, for such a weak solution to the Vlasov equation, we can define a trace on the phasespace boundary Σ in the Di Perna-Lions framework for transport equations Ω × R : we refer toSection A.1 of the Appendix for further properties of the solution to this initial boundary valueproblem (see [8, Section 1 - Chap 6] and [30]). 5inally, we introduce the following definitions that will be useful later. Definition 1.5.
For any α >
0, and any measurable function f : R + × Ω × R → R + , we set m α f ( t, x ) := Z R | v | α f ( t, x, v ) d v,M α f ( t, x ) := Z Ω × R | v | α f ( t, x, v ) d v d x = Z Ω m α f ( t, x ) d x. Definition 1.6.
We say that an initial kinetic condition f satisfies the pointwise decay as-sumption of order q > N q ( f ) := sup x ∈ Ω v ∈ R (1 + | v | q ) f ( x, v ) < ∞ . (1.10) As already said, the large-time behavior of the Vlasov-Navier-Stokes has been tackled in thecase of the torus T in [24] and of the whole space in R [23] for which concentration in velocityhappens in some small data regime. We refer to Section 1.2 of the introduction of [24] forheuristics about this monokinetic asymptotic phenomenon when there are no boundaries, relyingon an explicit formula for the solutions to the linearized equations around states of the form( U , f = 0) with U ∈ R .On the contrary, non-singular equilibra have been constructed in the case of a bidimensionalrectangle in [20]. Taking advantage of the specific geometry of the domain and of absorbingboundary conditions for the particles, it has been proven that such stationary solutions arelocally asymptotically stable relatively to compact perturbationsLet us describe more specifically the main strategy that has been employed in each of thesesituations. • On the torus, Choi and Kwon have introduced in [12] a version of the following so-called modulated energy E T ( t ) := 12 Z T × R f ( t, x, v ) | v − h j f ( t ) i| d v d x + 12 Z T | u ( t, x ) − h u ( t ) i| d x + 14 |h j f ( t ) i − h u ( t ) i| , (1.11)where h·i stands for the average on T . They have used this functional to describe the large-timedynamics of the system but under an a priori assumption on the solutions. In [24], the authorshave worked in a small data regime and have provided the first complete description of theasymptotic behavior of the system. Loosely speaking, under a condition of the type E T (0) + k u k ˙H / ( T ) ≪ , (1.12)the authors have showed that the fluid velocity u ( t ) homogenizes when t → + ∞ to the constantvelocity U := h u + j f i /
2, while the kinetic distribution function f ( t ) concentrates in velocityto the Dirac mass supported at U . Moreover, this convergence is exponentially fast and canbe measured thanks to the 1-Wasserstein distance on T × R .The previous modulated energy is at the heart of the proof of this monokinetic large timebehavior for the system on the torus and is linked to the dissipation thanks to the formal identitydd t E T ( t ) + D( t ) = 0 .
6s a matter of fact, this modulated energy essentially captures concentration phenomena sothat controlling such a quantity is the main key to understand the large-time dynamics of thesystem. In short, under the assumption that ρ f ∈ L ∞ ( R + ; L / ( T )), Choi and Kwon proved in[12] that ∀ t ≥ , E T ( t ) . e − λt E (0) , (1.13)for some λ >
0. Thanks to this exponential decay, they deduced that the asymptotics we havementioned above hold for the fluid velocity and for the kinetic distribution.The main strategy of [24] is based on a bootstrap analysis whose aim is to ensure that k ρ f k L ∞ ( R + ;L ∞ ( T )) < ∞ . Thanks to a straigthening change of variable in velocity, it is shown in[24] that this condition is actually implied by an estimate of the type Z ∞ k∇ u ( s ) k L ∞ ( T ) d s ≪ . (1.14)Then, the framework of the Fujita-Kato solutions to the Navier-Stokes system has been leveragedin [24] to ensure that such a control holds under the condition (1.12). • In the case of the euclidean space R , the large-time behavior of the system has beeninvestigated in [23] where it has been shown that concentration in velocity towards a Diracmass supported at 0 occurs also for the kinetic distribution, while the fluid velocity convergesto 0. Because of the unbounded nature of the spatial domain, this convergence is only at apolynomial rate. More precisely, the good quantity to look at to understand the concentrationphenomenon iturns out to be the kinetic energy E. One of the the main results of [23] is aconditional decay of the form ∀ t ≥ , E( t ) ≤ ϕ α (E(0))(1 + t ) α , for all α ∈ ]0 , / , for some function ϕ α , up to an a priori control on the moment ρ f . As in the torus case, abootstrap analysis is used to obtain such a control for small data solutions, but here the decayof the energy is only polynomial. In this unbounded context, the Brinkman force requires amore careful treatment which has led to the derivation of a new family of identities for higherorder dissipation functionals. • In [20], the authors have dealt with the particular case of a bidimensional rectangleΩ := ( − L, L ) × ( − , u p (which is a stationary solution to the Navier-Stokesequations). For the particles, partly absorbing boundary conditions are used with absorptionboundary conditions on the horizontal parts and an injection boundary condition on the verticalleft part. Thanks to a geometric control condition (referred to as the exit geometric condition ),compelling the particles to be absorbed by the boundary before a fixed finite time, one canconstruct non-trivial smooth equilibria ( u, f ) for the system, so that the concentration in velocityscenario doesn’t occur. Furthermore, if ( u, f ) is a smooth stationary solution close to ( u p , u p ), any ( u , f ) which is a smallperturbation of ( u, f ) gives birth to a weak solution ( u, f ) to the system satisfying ∀ t ∈ R + , k f ( t ) − f k L x,v + k u ( t ) − u k L x . e − t . In short, the previous equilibria are asymptotically stable in L , under small localized pertur-bations. 7n the following, we will consider solutions starting at initial data close to the equilibrium(0 ,
0) and prove the existence of an asymptotic profile ρ ∞ ∈ L ∞ (Ω) such that the following weakconvergence holds f ( t ) t → + ∞ ⇀ ρ ∞ ⊗ δ v =0 , where the tensor product is in ( x, v ). As said before, this kind of singular limit was alreadypresent in the case of the torus and of the whole space while geometric control was preventingsuch monokinetic behavior in the case of the rectangle. Once this monokinetic large timebehavior is established, we will be interested in a possible further description of the spatialprofile ρ ∞ : because of the boundary effects, it may indeed exist some particle trajectorieswhich are leaving the domain during the evolution. Therefore, we would like to study differentpossible scenarios, where absorption of the particles can prevail or not, in order to describe theasymptotic local density. The main result of this article is stated in the following Theorem and Corollary.
Theorem 1.7.
There exists a universal constant ε > and a nondecreasing function ϕ : R + → R + such that the following holds. Let ( u , f ) be an admissible initial condition in the sense ofDefinition 1.1 satisfying u ∈ H (Ω) ,M f + N q f < ∞ , for some q > . (1.15) If the initial kinetic energy is small enough in the sense that ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) E(0) < ε , (1.16) where E is defined in (1.6), then for any weak solution ( u, f ) to the Vlasov-Navier-Stokes systemwith initial data ( u , f ) , there exist constants λ, C λ > such that ρ f ∈ L ∞ ( R + ; L ∞ (Ω)) , (1.17)E( t ) ≤ E(0) C λ exp( − λt ) , t ≥ . (1.18) Remark 1.8.
In the case of the torus [24], the required assumption on the initial fluid velocity u was u ∈ ˙H / ( T ) (with a sufficiently small norm) and allowed to rely on some parabolicsmoothing for the solution u . Here, we have preferred to state the result with the assumption(1.15) in order to avoid unnecessary technical developments. Corollary 1.9.
Under the same assumptions of Theorem (1.7), for any weak solution ( u, f ) to the Vlasov-Navier-Stokes system with admissible initial data ( u , f ) , there exist constants λ, C λ > and ρ ∞ ∈ L ∞ (Ω) such that for all t ≥ (cid:16) f ( t ) , ρ f ( t ) ⊗ δ v =0 (cid:17) + k u ( t ) k L (Ω) ≤ E(0) / C λ exp( − λt ) , (1.19) ρ f ( t ) t → + ∞ −→ ρ ∞ in H − (Ω) , (1.20) where W stands for the Wasserstein distance on Ω × R . Moreover, the last convergence alsooccurs with an exponential rate.
8s in [24], it is possible to provide a further description of the limit profile ρ ∞ , using anotion of asymptotic characteristics. Theorem 1.10.
For δ small enough, under the assumptions of Theorem 1.7, and if Z + ∞ k∇ u ( τ ) k L ∞ (Ω) d τ ≤ δ, (1.21) then there exists a vector field R × R −→ R ( x, v ) X ∞ ( x, v ) , belonging to C ( R × R ) and such that the following holds. For all v ∈ R , the mapping X ∞ ,v : x X ∞ ( x, v ) is a C -diffeomorphism from R to itself and we have for almost every x ∈ Ω ρ ∞ ( x ) = Z R U ∞ ( x, v ) f (cid:0) X − ∞ ,v ( x ) , v (cid:1) | det D x X − ∞ ,v ( x ) | d v, (1.22) where the set U ∞ is defined as follows: for ( x, v ) ∈ Ω × R , ( x, v ) ∈ U ∞ ⇔ ∃ ! y ∈ Ω , x = X ∞ ( y, v ) and ∀ t ≥ , X t ( y, v ) ∈ Ω , where X t ( y, v ) = x + (1 − e − t ) v + Z t (1 − e τ − t ) u ( τ, X τ ( x, v )) d τ, (1.23)X ∞ ( y, v ) = x + v + Z ∞ u ( τ, X τ ( y, v )) d τ. (1.24) Remark 1.11.
In the previous statement, we can actually get rid of the assumption (1.21) byonly considering the evolution of the system from time t = 1. Indeed, the proof of Theorem1.7 and Corollary 1.9 will ensure (see the bootstrap procedure in Section 6) that for all ǫ > k∇ u k L ( ε, + ∞ ;L ∞ (Ω)) can be ensured as small as required up to imposing a relevantsmallness assumption (1.16). Nevertheless, for the sake of clarity, we have decided to state theresult from time t = 0. If one wants to get the result without further assumption near the time0, one has to replace f by f | t =1 together with integrals starting at t = 1 and for a set U ∞ defined with the functionX t ( y, v ) = x + (1 − e − t ) v + Z t (1 − e τ − t ) u ( τ, X τ ( x, v )) d τ. As mentioned before, an important difference with the case of the torus is that the particletrajectory may possibly escape the domain Ω because we prescribe absorption boundary condi-tions for the Vlasov equation. This means that, in Corollary 1.9 and Theorem 1.10, we obtainan asymptotic spatial profile ρ ∞ whose total mass is unknown: indeed, part of the initial massof the system may have disappeared throughout the evolution. In the following result, we showthat any fixed mass which is less than or equal to the initial mass can be reached by the systemfor some well-chosen data. We recall that we consider initial distribution functions f such that R Ω × R f ( x, v ) d x d v = 1. 9 roposition 1.12. Let α ∈ [0 , . There exists ( u , f ) an admissible initial condition in thesense of Definition 1.1 such that for any weak solution ( u, f ) to the Vlasov-Navier-Stokes systemstarting at ( u , f ) , their exists ρ ∞ ∈ L ∞ (Ω) which satisfies ρ f ( t ) t → + ∞ ⇀ ρ ∞ in C (Ω) ′ , (1.25) Z Ω ρ ∞ ( x ) d x = α. (1.26)Let us describe how this paper is organized. Following the strategy that has been used inthe case of the torus, we first study the conditional decay of the kinetic energy in Section 2.As explained before, such a decay will essentially be enough for the concentration velocity tohappen. Here, we strongly rely on the Poincar´e inequality which holds on the domain Ω for thefluid velocity. In short, we can hope for an exponential decay provided that the local density ρ f is controlled.In Section 3, we investigate a way to get the desired control on this moment ρ f . Thanks toa straightening change of variable in velocity, this issue actually reduces to an inequality of thetype k∇ u k L ( R + ;L ∞ (Ω)) ≪
1. As said before, this condition was already present in the case of thetorus and the whole space and was based on a representation formula for the kinetic distribution f thanks to the method of characteristics. Here, we rely on a representation formula for weaksolutions to the Vlasov equation that is valid on a bounded domain and that takes into accountthe boundary effects.A third step is made of a bootstrap analysis and aims at obtaining the previous control on k∇ u k L ∞ (Ω) . To do so, we interpolate this quantity by second order derivatives of u togetherwith the kinetic energy of the system, in order to get integrability in time thanks to the expo-nential decay of this energy. This procedure requires some higher regularity estimates for thesolutions to the Navier-Stokes equations which are provided in Sections 4-5. Namely, we usethe framework of strong solutions for the Navier-Stokes equations, which is allowed if the initialcondition and the Brinkman force are small enough. The bootstrap analysis then takes place inSection 6 where the smallness condition (1.16) we have used in our statements plays a key role.We then provide a further description of the spatial density profile ρ ∞ of the particles inSection 7. In order to prove Theorem 1.10, we study precisely the position of the particles alongthe characteristic curves for the Vlasov equation. Again, we have to deal carefully with thepossible exit of these trajectories from the domain Ω. Finally, we provide sufficient conditionson the support of the initial distribution f and on the velocity profile u to ensure scenarios inwhich, the whole mass of the system is preserved throughout the evolution or, on the contrary,vanishes after a certain time (see Propositions 8.1-8.5 in Section 8). We combine these twoexamples to prove Proposition 1.12 . We start with a Lemma which states that the total mass of the system is nonincreasing alongthe evolution.
Lemma 2.1.
For any weak solution ( u, f ) to the Vlasov-Navier-Stokes system and for all t ≥ ,we have Z Ω × R f ( t, x, v ) d x d v ≤ Z Ω × R f ( x, v ) d x d v. roof. First suppose that u , f and f are smooth. We integrate the Vlasov equation on Ω × R and use Green formula to getdd s Z Ω × R f ( s, x, v ) d x d v = − Z ∂ Ω × R ( γf )( s, x, v ) v · n ( x ) d σ ( x ) d v. The absorption boundary condition (1.5) implies that the r.h.s. of this equality is actuallynonpositive, so that we get the result by integrating between 0 and t .In the general case, we use a stability result coming from the Di Perna-Lions theory fortransport equations (see Section A.1 of the Appendix): we consider a sequence of nonnegativedistribution functions ( f n ) n solutions to the Vlasov equation with the absorption boundarycondition, associated to regularized velocity fields ( u n ) n and regularized and truncated initialconditions ( f ,n ) n . We know that f is the strong limit in L ∞ loc ( R + ; L p (Ω × R )) for 1 ≤ p < ∞ and the weak- ⋆ limit in L ∞ ([0 , T ] × Ω × R ) of the sequence ( f n ) n . This is thus enough to passto the limit in the inequality Z Ω × R f n ( t, x, v ) d x d v ≤ Z Ω × R f ,n ( x, v ) d x d v, which concludes the proof because f ∈ C ( R + ; L (Ω × R )) (see A.1 in Appendix).We can now state the following inequality which highlights the role of the kinetic energy forthe study of the asymptotics. Lemma 2.2.
For any weak solution ( u, f ) to the Vlasov-Navier-Stokes system and for all t ≥ ,we have the following inequality W ( f ( t ) , ρ f ( t ) ⊗ δ v =0 ) + k u ( t ) k L (Ω) . E( t ) / , (2.1) where W stands for the Wasserstein distance on Ω × R .Proof. First, we observe that for all times t ≥
0, the measures f ( t ) d x d v and ( ρ f ( t )d x ) ⊗ δ v =0 have the same mass on Ω × R . Therefore we use the Monge-Kantorovich duality for W onΩ × R (see Appendix A.3) to getW ( f ( t ) , ρ f ( t ) ⊗ δ v =0 ) = sup k∇ x,v φ k ∞ ≤ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω × R f ( t, x, v )( φ ( x, v ) − φ ( x, x d v (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≤ Z Ω × R f ( t, x, v ) | v | d x d v. The Cauchy-Schwarz inequality and the previous Lemma 2.1 together with the normalisationfor the density f give usW ( f ( t ) , ρ f ( t ) ⊗ δ v =0 ) ≤ (cid:18)Z Ω × R f ( t, x, v ) | v | d x d v (cid:19) / (cid:18)Z Ω × R f ( t, x, v ) d x d v (cid:19) / ≤ √ t ) / , by definition of the kinetic energy. The inequality k u ( t ) k L (Ω) . E( t ) / is also straigthforward.We now give the following Proposition relating the dissipation and the kinetic energy.11 roposition 2.3. The exists a continuous nondecreasing function ψ : R + → R + such thatthe following holds. Let ( u, f ) be a weak solution to the Vlasov-Navier-Stokes system such that ρ f ∈ L ∞ loc ( R + ; L ∞ (Ω)) . Fix T > and set λ := ψ sup [0 ,T ] k ρ f ( t ) k L ∞ (Ω) ! , then ∀ t ∈ [0 , T ] , λ E( t ) ≤ D( t ) , (2.2) and we have the following exponential decay ∀ t ∈ [0 , T ] , E( t ) . λ E(0) e − λt , (2.3) where . λ refers to a constant only depending on λ .Proof. The proof follows the same arguments as in [24, Lemma 3.4] and was originally obtainedin [12, Theorem 1.2] in the case of the torus. It mainly relies on the fact that the Poincar´einequality is valid for the fluid velocity on Ω. • First, we note that (2.2) implies (2.3). Indeed, if (2.2) holds, the energy-dissipationinequality (1.9) shows that for almost all 0 ≤ s ≤ t ≤ T ,E( t ) + λ Z ts E( τ ) d τ ≤ E( s ) , from which we get λ Z Tt E( τ ) d τ ≤ E( t ) , E( t ) ≤ E( s ) , s ≤ t. We get the desired exponential decay (2.3) thanks to Lemma A.8 (see Appendix A.5). • It remains to get a λ > λ E ≤ D on [0 , T ]. Thanks to the Poincar´e inequalityin H (Ω), the exists a constant c P > t ) ≥ Z Ω × R f ( t ) | u ( t ) − v | d v d x + 12 c P k u ( t ) k (Ω) . Let us denote e E( t ) := E( t ) − k u ( t ) k (Ω) = 12 Z Ω × R f ( t, x, v ) | v | d v d x. We see that it is sufficient to find some γ > β ∈ (0 , c P ) such that Z Ω × R f ( t ) | u ( t ) − v | d v d x ≥ γ e E( t ) − β k u ( t ) k (Ω) . (2.4)Indeed, if this holds, we can define λ := min( γ/ , c P − β ) > t ) ≥ λ E( t ) + 12 (cid:18) c P − β − λ (cid:19) k u ( t ) k (Ω) ≥ λ E( t ) . In order to get the inequality (2.4), we first write Z Ω × R f ( t ) | u ( t ) − v | d v d x = Z Ω × R f | v | d v d x + Z Ω ρ f ( t ) | u ( t ) | d x − Z Ω × R f ( t ) v · u ( t ) d v d x. − Z Ω × R f ( t ) v · u ( t ) d v d x ≥ − α Z Ω × R f ( t ) | v | d v d x − α − Z Ω ρ f ( t ) | u ( t ) | d x, for some α ∈ (0 , Z Ω × R f ( t ) | v − u ( t ) | d v d x ≥ (1 − α ) Z Ω × R f ( t ) | v | d v d x − ( α − − Z Ω ρ f ( t ) | u ( t ) | d x. From the definition of e E( t ), we can deduce that Z Ω × R f ( t ) | u ( t ) − v | d v d x ≥ (1 − α ) e E( t ) − ( α − − Z Ω ρ f | u ( t ) | d x ≥ (1 − α ) e E( t ) − ( α − −
1) sup s ∈ [0 ,T ] k ρ f ( s ) k L ∞ (Ω) k u ( t ) k (Ω) . By our assumptions, the quantity sup s ∈ [0 ,T ] k ρ f ( s ) k L ∞ (Ω) is finite.In order to get the inequality (2.4), we eventually choose α := c P s ∈ [0 ,T ] k ρ f ( s ) k L ∞ (Ω) − ∈ (0 , , and then β := ( α − −
1) sup s ∈ [0 ,T ] k ρ f ( s ) k L ∞ (Ω) = c P / γ := 1 − α . Such an α is a continuousnondecreasing function of the variable sup s ∈ [0 ,T ] k ρ f ( s ) k L ∞ (Ω) and doesn’t vanish. Thus, we get λ := min((1 − α ) / , c P / , which is of the desired form. The proof is therefore complete.The following Proposition highlights the fact that a global bound on the local density ρ f isenough to obtain the convergence towards an asymptotic profile. Proposition 2.4.
For any weak solution ( u, f ) to the Vlasov-Navier-Stokes system for which sup t ≥ k ρ f ( t ) k L ∞ (Ω) < ∞ , E( t ) −→ t → + ∞ , with exponential decay , (2.5) there exists a profile ρ ∞ ∈ L ∞ (Ω) such that ρ f ( t ) t → + ∞ −→ ρ ∞ in H − (Ω) , (2.6) exponentially fast.Proof. It relies strongly on the exponential decay of the kinetic energy: indeed, as a consequenceof Theorem 2.3, we have E( t ) −→ t → + ∞ . We will use Cauchy’scriterion in H − (Ω) in order to obtain the existence of an asymptotic profile.To derive an estimate in H − norm, we fix ψ ∈ C ∞ c (Ω) and we use the weak formulation forthe Vlasov equation (see Appendix A.1) to write Z Ω ψρ f ( t ) − Z Ω ψρ f ( s ) = Z ts Z Ω ∇ ψ · j f ( τ ) d τ − Z ts Z ∂ Ω × R ( γf ) ψ ( x ) v · n ( x ) d σ ( x ) d v d τ, ≤ s ≤ t . Since ψ is compactly supported in Ω, the last term vanishes. For the otherterm, we then use the Cauchy-Schwarz inequality to get, omitting the variable to simplify theformula (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψρ f ( t ) d x − Z Ω ψρ f ( s ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ts (cid:18)Z Ω ρ f |∇ ψ | d x (cid:19) / (cid:18)Z Ω × R f | v | d x d v (cid:19) / d τ, so that we eventually obtain (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψρ f ( t ) d x − Z Ω ψρ f ( s ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ρ f k / ∞ ( R + ;L ∞ (Ω)) k∇ ψ k L (Ω) Z ts E( τ ) / d τ. Thanks to the assumption (2.5), we infer that k ρ f ( t ) − ρ ( s ) k H − (Ω) . k ρ f k / ∞ ( R + ;L ∞ (Ω)) Z ts E( τ ) / d τ. (2.7)By the exponential decay of the energy on R + , the function t E / ( t ) in integrable on R + . Therefore, the Cauchy criterion for the k · k H − (Ω) norm applied to the function ρ f ( t ) when t → + ∞ gives us the convergence ρ f ( t ) → ρ ∞ in H − (Ω) when t → + ∞ , for some ρ ∞ ∈ H − (Ω).But again thanks to (2.5), any sequence ( t n ) n → + ∞ produces a sequence of functions( ρ f ( t n )) n bounded in L ∞ (Ω). We deduce that ρ ∞ ∈ L ∞ (Ω) and it is easy to see that theconvergence towards this asymptotic profile is exponentially fast by looking at the estimate(2.7) and letting t → + ∞ . In this section, we first derive a representation formula for the kinetic distribution thanks tomethod of characteristics for the Vlasov equation, taking into account the absorption boundarycondition. Then, we explain how to use a straightening change of variable in velocity in theintegral formula giving ρ f to get a bound of the formsup ≤ s ≤ t k ρ f ( s ) k L ∞ (Ω) . . This idea has already been used in [24, 23] and can be applied if the change of variable is valid,which is the case if the semi norm k∇ u k L (0 ,t ;L ∞ (Ω)) is small enough. Given a time-dependent vector field u on R + × Ω, a time t ∈ R + and a point ( x, v ) ∈ R × R ,we define the characteristic curves s ∈ R + (X( s ; t, x, v ) , V( s ; t, x, v )) ∈ R × R for theVlasov equation (associated to u ) as the solution of the following system of ordinary differentialequations ˙X( s ; t, x, v ) = V( s ; t, x, v ) , ˙V( s ; t, x, v ) = ( P u )( s, X( s ; t, x, v )) − V( s ; t, x, v ) , X( t ; t, x, v ) = x, V( t ; t, x, v ) = v. (3.1)where the dot means derivative along the first variable. Here, P is the linear extension operatorcontinuous from L ∞ (Ω) to L ∞ ( R ) and from W , ∞ (Ω) to W , ∞ ( R ) defined by ∀ x ∈ R d , ( P w )( x ) := ( w ( x ) , if x ∈ Ω , x ∈ R \ Ω , (3.2)14nd which satisfies ∀ w ∈ L ∞ (Ω) , k P w k L ∞ ( R ) = k w k L ∞ (Ω) , (3.3) ∀ w ∈ W , ∞ (Ω) , k∇ ( P w ) k L ∞ ( R ) ≤ k∇ w k L ∞ (Ω) . (3.4)For the sake of completeness, we refer to Section A.4 of the Appendix for a simple justificationof these facts. Also, we will use the convention( P u )( t, · ) = P ( u ( t, · )) . Now, let
T > u ∈ L (0 , T ; H (Ω)) ∩ L (0 , T ; W , ∞ (Ω)) . We can apply the Cauchy-Lipschitz theorem to show the following Proposition.
Proposition 3.1.
Given ( x, v ) ∈ R × R and a time t ∈ [0 , T ] , the system (3.1) admits aunique solution s Z s,t ( x, v ) ∈ R × R on [0 , T ] and Z s,t : ( R × R −→ R × R ( x, v ) Z s,t ( x, v ) := (X( s ; t, x, v ) , V( s ; t, x, v )) is a diffeomorphism of R × R whose inverse is given by Z − s,t = Z t,s . In what follows, for all ( x, v ) ∈ R × R , we will sometimes use the notationZ s,t ( x, v ) := (X s,t ( x, v ) , V s,t ( x, v )) := (X( s ; t, x, v ) , V( s ; t, x, v )) . (3.5) We keep the same assumptions and notations as in the previous section. For ( x, v ) ∈ Ω × R ,we now define for any t ≥ τ − ( t, x, v ) := inf { s ≤ t | ∀ σ ∈ [ s, t ] , X( σ ; t, x, v ) ∈ Ω } , (3.6)where we use the harmless convention X( σ, t, x, v ) = X(0 , t, x, v ) for all σ < t ≥ t ≥ O t := (cid:8) ( x, v ) ∈ Ω × R | τ − ( t, x, v ) < (cid:9) . (3.7)We now derive an important representation formula for the distribution function wich solvesthe Vlasov equation in the weak sense. Proposition 3.2.
Let f be the weak solution to the initial boundary value problem the Vlasovequation, associated to the velocity field u ∈ L (0 , T ; H (Ω)) ∩ L (0 , T ; W , ∞ (Ω)) with initialcondition f and with absoption boundary condition. There holds f ( t, x, v ) = e t O t ( x, v ) f (Z ,t ( x, v )) a.e. (3.8)This formula seems to be folklore but we have not been able to find a proof in the literature.For the sake of completeness, and because this will be useful in the later Sections 7-8, we givea complete proof of Proposition 3.2 in Section A.2 of the Appendix.15 .3 Change of variable in velocity and bounds on moments In order to get global bounds on the moments ρ f and j f , we rely on a change of variable in veloc-ity (inspired by [2]). As in [24, 23], such a strategy can be used if the quantity k∇ u k L ( R + ;L ∞ (Ω)) is small enough. We use the same notations as before. Lemma 3.3.
Suppose u ∈ L ( R + , H (Ω)) ∩ L ( R + ; L ∞ (Ω)) . Fix δ > satisfying δe δ ≤ / .Then, for all times t ∈ R + satisfiying Z t k∇ u ( s ) k L ∞ (Ω) d s < δ, (3.9) and for all x ∈ Ω , the map Γ t,x : v V(0; t, x, v ) , is a global C -diffeomorphism from R to itself satisfying furthermore ∀ v ∈ R , | det D v Γ t,x ( v ) | ≥ e t . (3.10) Proof.
The proof is a straightforward adaptation of that of [24, Lemma 4.4]: here, the charac-teristic curves Z = (X , V) are defined in R × R but we can use the inequality Z t k∇ ( P u )( s ) k L ∞ ( R ) d s ≤ Z t k∇ u ( s ) k L ∞ (Ω) d s. which follows from (3.4). The rest of the proof is then similar.As a consequence, we obtain the following proposition which is based on the representationformula for the distribution function. Proposition 3.4.
Suppose u ∈ L ( R + , H (Ω)) ∩ L ( R + ; L ∞ (Ω)) . If the assumption (3.9) issatisfied at a time t ≥ , then we have k ρ f ( t ) k L ∞ (Ω) ≤ I q N q ( f ) , k j f ( t ) k L ∞ (Ω) ≤ I q e − t (cid:18)Z t e s k u ( s ) k L ∞ (Ω) d s + 1 (cid:19) N q ( f ) , where we recall that N q ( f ) is defined is (1.10) and where I q := Z R | v | | v | q d v. Proof.
Note that because of the assumption (3.9) on u , we can define the characteristics curvesfor the Vlasov-Navier-Stokes system in a classical sense as in the previous subsection. We usethe representation formula (3.8) to write that for all x ∈ Ω ρ f ( t, x ) = e t Z R O t ( x, v ) f (X ,t ( x, v ) , V ,t ( x, v )) d v. Now, using the change of variable v Γ t,x ( v ) = V(0; t, x, v ), we get ρ f ( t, x ) = e t Z R O t ( x, Γ − t,x ( w )) f (X ,t ( x, Γ − t,x ( w )) , w ) | detD w (Γ t,x ) − ( w ) | d w, ρ f ( t, x ) ≤ Z R O t ( x, Γ − t,x ( w )) f (X ,t ( x, Γ − t,x ( w )) , w ) d w Now, thanks to the definition (3.7) of O t , ( x, Γ − t,x ( w )) ∈ O t if and only if τ − ( t, x, Γ − t,x ( w )) < | w | q ) O t ( x, Γ − t,x ( w )) f (X ,t ( x, Γ − t,x ( w )) , w ) ≤ sup ( x,v ) ∈ Ω × R (1 + | v | q ) f ( x, v ) = N q ( f ) . We thus deduce the desired inequality on ρ f . We proceed in the same way for the bound on j f ,namely we start from the following representation formula j f ( t, x ) = e t Z R Γ − t,x ( w ) O t ( x, Γ − t,x ( w )) f (X ,t ( x, Γ − t,x ( w )) , w ) | detD w (Γ t,x ) − ( w ) | d w. Using the formula V(0; t, x, v ) = e t v − Z t e τ ( P u )( τ, X( τ ; t, x, v )) d τ, we infer that w = e t Γ − t,x ( w ) − Z t e τ ( P u )( τ, X( τ ; t, x, Γ − t,x ( w ))) d τ, which becomes | Γ − t,x ( w ) | ≤ e − t (cid:18) | w | + Z t e τ k ( P u )( τ ) k L ∞ ( R ) d τ (cid:19) ≤ e − t (1 + | w | ) (cid:18) Z t e τ k u ( τ ) k L ∞ (Ω) d τ (cid:19) . Coming back to the representation formula for j f , we can conclude exactly as in the previouscase.In the following Lemma, we study how the pointwise decay condition of Definition 1.6 canbe propagated after times t = 0. Lemma 3.5.
Let t > . If N q ( f ) < ∞ and if u ∈ L ( R + ; H ∩ L ∞ (Ω)) then N q ( f ( t )) < ∞ with N q ( f ( t )) . e t (1 + k u k q L (0 ,t ;L ∞ (Ω)) ) N q ( f ) . Proof.
In this proof, we apply the stability results coming from the DiPerna-Lions theory (seeAppendix A.1) since the vector field u has not enough regularity to define the characteristiccurves in a classical sense. We omit the details and write the proof of the desired inequalitywith only sufficiently regular solutions and data (we refer to the proof of Lemma 2.1 where wehave already explained this regularization argument)We use the representation formula f ( t , x, v ) = e t O t ( x, v ) f (Z ,t ( x, v )) . t , x, v ) = e t v − Z t e s ( P u )( s, X( s ; t , x, v )) d s, to get v = e − t V(0; t , x, v ) + Z t e s − t ( P u )( s, X( s ; t , x, v )) d s, and therefore obtain | v | ≤ | V(0; t , x, v ) | + Z t k u ( s ) k L ∞ (Ω) d s. Thus, we can estimate(1+ | v | q ) f ( t , x, v ) . (cid:16) | V(0; t , x, v ) | q + k u k q L (0 ,t ;L ∞ (Ω)) (cid:17) e t O t ( x, v ) f (Z(0; t , x, v )) , and the very definition of N q ( f ) leads to(1 + | v | q ) f ( t , x, v ) . e t (1 + k u k q L (0 ,t ;L ∞ (Ω)) ) N q ( f ) , which concludes the proof.By performing the same analysis as in the three previous results and by replacing the initialtime t = 0 by t and f by f ( t ), we get the following statement. Lemma 3.6.
Let u ∈ L ( R + , H (Ω)) ∩ L ( R + ; L ∞ (Ω)) . Let δ be fixed such that δe δ ≤ / .For all times t ≥ t ≥ such that Z tt k∇ u ( s ) k L ∞ (Ω) d s ≤ δ, (3.11) we have k ρ f ( t ) k L ∞ (Ω) . e t N q ( f )(1 + k u k q L (0 ,t ;L ∞ (Ω)) ) , k j f ( t ) k L ∞ (Ω) . e t − t (cid:18)Z tt e s k u ( s ) k L ∞ (Ω) d s + 1 (cid:19) N q ( f )(1 + k u k q L (0 ,t ;L ∞ (Ω)) ) . Remark 3.7.
We will actually see later that for any weak solution ( u, f ) to the Vlasov-Navier-Stokes system, we have u ∈ L ( R + ; L ∞ (Ω)): this comes from the local estimates of Proposition4.8 which are independent of this section. This section aims at providing local in time controls on ρ f and j f by using the results of theprevious section and at deriving a L ∞ H ∩ L H estimate for the fluid velocity. These two typesof estimates will be crucial in order to show Theorem 1.7.We first introduce the following useful notations, which allows us to track the dependencyon the initial data in the later estimates. This will be useful at the very end of the our proof ofTheorem 1.7 because of the smallness condition (1.16).18 otation 4.1. The notation A . B means A ≤ ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) B, (4.1)where ϕ : R + → R + is onto, continuous and nondecreasing, and q > u, f ) a weak solution to the Vlasov-Navier-Stokes system in the senseof Definition 1.3 with admissible initial data ( u , f ) in the sense of Definition 1.1. Such a weaksolution has for instance been constructed in [6]. Notation 4.2.
We set F := j f − ρ f u,S := F − ( u · ∇ ) u. First, we have the following estimate on the Brinkman force F . Lemma 4.3.
For all t ≥ , we have Z t k F ( s ) k (Ω) d s ≤ E(0) sup s ∈ [0 ,t ] k ρ f ( s ) k L ∞ (Ω) . Proof.
Using Cauchy-Schwarz inequality, we get by dropping the time variable, | F | = (cid:12)(cid:12)(cid:12)(cid:12)Z R f ( v − u ) d v (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ / f (cid:18)Z R f | v − u | d v (cid:19) / , therefore for almost every s ∈ [0 , t ] k F ( s ) k (Ω) ≤ k ρ f ( s ) k L ∞ (Ω) D( s ) ≤ sup s ∈ [0 ,t ] k ρ f ( s ) k L ∞ (Ω) D( s ) , where D is defined in (1.7). Then, we integrate the last inequality between 0 and t and concludethanks to the energy inequality (1.9). We first recall standard interpolation estimates on the moments (see for instance [22]) for whichwe refer to the notations of Definition 1.5.
Proposition 4.4.
Let k > and let g be a nonnegative function in L ∞ ( R + × Ω × R ) . Thenthe following estimates hold for any ℓ ∈ [0 , k ] and a.e ( t, x ) m ℓ g ( t, x ) . (cid:0) k g k L ∞ ( R + × Ω × R ) + 1 (cid:1) m k g ( t, x ) ℓ +3 k +3 , k m ℓ g ( t ) k L k +3 ℓ +3 (Ω) ≤ C k,ℓ k g ( t ) k k − ℓk +3 L ∞ (Ω × R ) M k g ( t ) ℓ +3 k +3 . Lemma 4.5.
Consider α ≥ such that u ∈ L ( R + ; L α +3 ∩ W , (Ω)) and M α f < ∞ . Then M α f ( t ) < ∞ for all t > and we have M α f ( t ) . (cid:18) M α f + e tα +3 Z t k u ( s ) k L α +3 (Ω) d s (cid:19) α +3 . (4.2)19 roof. The proof basically follows the same arguments as in [24, Lemma 4.2]. As in the proof ofLemma 2.1, we rely on the stability results of Di Perna-Lions theory on Ω × R (see AppendixA.1). We don’t detail the argument and we write the proof as if u and f were smooth.We multiply the Vlasov equation by | v | α and we integrate on Ω × R to getdd t M α f ( t ) + αM α f ( t ) + Z R Z ∂ Ω | v | α ( γf ) n ( x ) · v d σ ( x ) d v = α Z Ω × R f | v | α − v · u d x d v. As the boundary term of the l.h.s is non-negative (because of (1.5)), we obtaindd t M α f ( t ) + αM α f ( t ) ≤ α Z Ω × R f | v | α − v · u d x d v. The rest of the proof is similar to that of [24, Lemma 4.2], by using the interpolation estimateon the moments (Proposition 4.4) and the rough control k f ( t ) k L ∞ (Ω × R ) ≤ e t k f k L ∞ (Ω × R ) , a.e. t ≥ , satisfied by any renormalized solution to the Vlasov equation Lemma 4.6. If M f < + ∞ , we have M f ∈ L ∞ loc ( R + ) . Moreover, there exists a continuousnon-negative and nondecreasing function η such that k ρ f ( t ) k L (Ω) + k j f ( t ) k L / (Ω) . η ( t ) . Proof.
Thanks to the estimate (4.2) with α = 3, we obtain M f ( t ) . (cid:18) M f + e t/ Z t k u ( s ) k L (Ω) d s (cid:19) , as in [24, Lemma 4.2]. Then, the rest of the proof is similar to that of [24, Lemma 4.3]: we useSobolev embedding and the Poincar´e inequality on H (Ω) to write k u ( s ) k L (Ω) . k∇ u ( s ) k L (Ω) so that the Cauchy-Schwarz inequality yields Z t k u ( s ) k L (Ω) d s . t / (cid:18)Z t k∇ u ( s ) k (Ω) d s (cid:19) / ≤ t / E(0) / , where we have used the energy inequality (1.9).In the following lemma, we slightly improve the rough bound on ρ f and j f which was givenin Lemma 4.6. Lemma 4.7.
There exists a continuous nondecreasing function η : R + → R + such that for all s ≥ k ρ f ( s ) k L (Ω) + k j f ( s ) k L / (Ω) . η ( s ) . (4.3) Moreover, j f − ρ f u ∈ L ( R + ; L (Ω)) . roof. In the proof we denote by η a generic positive nondecreasing continuous function whichmay vary from line to line.Recall that in (1.15), we suppose M f < ∞ and M f < ∞ so that M f . M f + M f < ∞ . We first write that the velocity field u is solution of the following Stokes system ∂ t u + Au = P S, div x u = 0 ,u | ∂ Ω = 0 ,u (0) = u , (4.4)with S := j f − ρ f u − ( u · ∇ ) u, where P stands for the Leray projection on vector fields and where A is the Stokes operator (seeSection A.8 of the Appendix). To estimate this source term, we first infer from Lemma 4.6 that k ρ f ( t ) k L (Ω) + k j f ( t ) k L / (Ω) . η ( t ) . The previous estimate, with the H¨older inequality and the energy inequality (1.9), allows us toobtain the following inequalities for all t ≥ Z t k j f ( s ) k / (Ω) d s . η ( t ) , Z t k ρ f ( s ) u ( s ) k / (Ω) d s ≤ Z t k u ( s ) k (Ω) k ρ f ( s ) k (Ω) d s . η ( t ) , where we have again used the Sobolev inequality k u ( s ) k L (Ω) . k∇ u ( s ) k L (Ω) . Thus, we get j f − ρ f u ∈ L ( R + ; L / (Ω)) . Then, since u ∈ L ∞ loc ( R + ; L (Ω)) ∩ L ( R + ; H (Ω)) and again by Sobolev embedding, we get( u · ∇ ) u ∈ L ( R + ; L (Ω)) and ( u · ∇ ) u ∈ L ( R + ; L / (Ω)) , so that ( u · ∇ ) u ∈ L p loc ( R + ; L q (Ω)) with1 p = θ + 12 , q = 3 − θ , ≤ θ ≤ . We choose θ = 2 / u · ∇ ) u ∈ L / ( R + ; L / (Ω)). We end up with P S = j f − ρ f u − ( u · ∇ ) u ∈ L / ( R + ; L / (Ω)) , because of the continuity of the operator P from L / (Ω) to L / (Ω). Thanks to this integrabilityof the source term and the assumption (1.15), we can use the maximal L p − L q regularity resultfor the Stokes system (see Section A.8 of the Appendix) to deduce that u ∈ L / ( R + ; W , / (Ω)) . By the Sobolev embedding W , / (Ω) ֒ → L (Ω), we finally end up with u ∈ L / ( R + ; L (Ω)) .
21n particular, we obtain u ∈ L ( R + ; L (Ω)). With the estimate (4.2) of Lemma 4.5, we firsthave M f ( t ) . (cid:18) M f + e t Z t k u ( s ) k L (Ω) d s (cid:19) . η ( t ) , which can be used together with the interpolation estimate (4.4) for moments with k = 6 and ℓ ∈ { , } to get k ρ f ( t ) k L (Ω) + k j f ( t ) k L / (Ω) . η ( t ) . We can now combine the last estimate with the H¨older inequality and the energy inequality(1.9) to write for all t ≥ Z t k j f ( s ) k (Ω) d s . Z t k j f ( s ) k / (Ω) d s . η ( t ) , and Z t k ρ f ( s ) u ( s ) k (Ω) d s ≤ Z t k u ( s ) k (Ω) k ρ f ( s ) k (Ω) d s . Z t k∇ u ( s ) k (Ω) k ρ f ( s ) k (Ω) d s . η ( t ) . This concludes the proof of Lemma 4.7.The next proposition tells us that any weak solution u to the VNS system belongs toL ( R + ; L ∞ (Ω)). This result has been already proved in [24, Proposition 5.1] in the case of thetorus but the proof seems to be specific to the periodic setting.On a bounded domain, this type of integrability of Leray solutions to the Navier-Stokesequations is actually a general property which can be deduced from [18, p104-106] combinedwith the end of the proof of [32, Lemma 8.15]. At first sight, the result stated in these referencesholds under the general condition that the source term of the Navier-Stokes equations belongsto L ∞ loc ( R + ; L (Ω)). Even this is not (yet) the case for the Brinkman force j f − ρ f u , we willeasily adapt the proof. Proposition 4.8.
We have u ∈ L ( R + ; L ∞ (Ω)) , (4.5) ρ f , j f ∈ L ∞ loc ( R + ; L ∞ (Ω)) . (4.6) More precisely, there exists a continuous nondecreasing function η : R + → R + such that k u k L (0 ,t ;L ∞ (Ω)) . η ( t ) , (4.7) k ρ f k L ∞ (0 ,t ;L ∞ (Ω)) + k j f k L ∞ (0 ,t ;L ∞ (Ω)) . η ( t ) . (4.8) Proof. • Let
T > u ∈ L (0 , T ; L ∞ (Ω)). As already said, someparts of the proof mimic those of [18, p104-106] but we will introduce some modifications. First,thanks to Lemma 4.7, we know that the source term j f − ρ f u of the Navier-Stokes equationssatisfied by u belongs to L (0 , T ; L (Ω)). So, we can apply the theory of epochs of regularity (see[18, II - Section 7 ]) for u on [0 , T ]. We get the existence of a subset σ T ⊂ [0 , T ] of full measurein [0 , T ] with σ T = S i ] a i , b i [ and where u is a strong solution (namely u ∈ L ∞ t H x ∩ L t H x ) oneach ] a i , b i [. On each interval ] a i , b i [, we can take the L (Ω) inner product (denoted by h· , ·i )22f the Navier-Stokes equations with Au , where A stands for the Stokes operator on L (Ω), inorder to obtaindd t k∇ u k (Ω) + 2 k Au k (Ω) + 2 h P ( u · ∇ ) u, Au i = 2 h P ( j f − ρ f u ) , Au i on ] a i , b i [ , where we have used the fact that dd t k∇ u k (Ω) = 2 h ∂ t u, Au i (see [32, Lemma 6.7]). Here and inwhat follows, we omit the time variable for the sake of clarity. Then, we combine the Agmoninequality (see Section A.6 of the Appendix), the Poincar´e inequality and the elliptic estimate k u k H (Ω) . k Au k L (Ω) to obtain |h P ( u · ∇ ) u, Au i| ≤ k u k L ∞ (Ω) k∇ u k L (Ω) k P Au k L (Ω) . k∇ u k / (Ω) k P Au k / (Ω) . using that the operator P is self-adjoint on L (Ω). The Young inequality is now used twice towrite 2 |h P ( u · ∇ ) u, Au i| ≤ C k∇ u k (Ω) + 34 k Au k (Ω) , |h P ( j f − ρ f u ) , Au i| ≤ (cid:18) C k j f − ρ f u k (Ω) + 12 k Au k (Ω) (cid:19) , thanks to the continuity of the operator P on L (Ω). Here, C is a positive constant independentof the time variable. All in all, we end up withdd t k∇ u k (Ω) + k Au k (Ω) . k j f − ρ f u k (Ω) + k∇ u k (Ω) , (4.9)on each interval ] a i , b i [, where . hides a constant independent of the time variable and inde-pendent of i .To deal with the Brinkman force k j f − ρ f u k (Ω) , we invoke the inequality (4.3) of Lemma 4.7according to which there exists a continuous nondecreasing function η : R + → R + (independentof T ) such that for all s ∈ [0 , T ] k ρ f ( s ) k L (Ω) + k j f ( s ) k L / (Ω) . η ( s ) . (4.10)For all s ∈ [0 , T ], we can now estimate the L (Ω) norm of the Brinkman force j f ( s ) − ρ f ( s ) u ( s )in the following way, k j f ( s ) − ρ f ( s ) u ( s ) k (Ω) . k j f ( s ) k (Ω) + k ρ f ( s ) u ( s ) k (Ω) . k j f ( s ) k (Ω) + k ρ f ( s ) k (Ω) k∇ u ( s ) k (Ω) , where we have used the H¨older inequality and the Sobolev embedding H (Ω) ֒ → L (Ω). So,thanks to (4.10), we obtain the following inequality for all s ∈ [0 , T ] k j f ( s ) − ρ f ( s ) u ( s ) k (Ω) . η ( s ) + η ( s ) k∇ u ( s ) k (Ω) ≤ C T (cid:16) k∇ u ( s ) k (Ω) (cid:17) . Coming back to (4.9), we getdd t k∇ u k (Ω) + k Au k (Ω) ≤ C T (cid:16) k∇ u k (Ω) + k∇ u k (Ω) (cid:17) ,
23n each ] a i , b i [, where C T ≥
0. We divide this inequality by (1 + k∇ u k (Ω) ) to find that1(1 + k∇ u k (Ω) ) dd t k∇ u k (Ω) + 1(1 + k∇ u k (Ω) ) k Au k (Ω) ≤ C T k∇ u k (Ω) ) + k∇ u k (Ω) (1 + k∇ u k (Ω) ) ! , which gives1(1 + k∇ u k (Ω) ) dd t k∇ u k (Ω) + 1(1 + k∇ u k (Ω) ) k Au k (Ω) ≤ C T (cid:16) k∇ u k (Ω) (cid:17) , (4.11)on each ] a i , b i [. We now integrate this inequality between a and b , with a i < a < b < b i , to get11 + k∇ u ( a ) k (Ω) −
11 + k∇ u ( b ) k (Ω) + Z ba k Au ( s ) k (Ω) (1 + k∇ u ( s ) k (Ω) ) d s ≤ C T Z ba (cid:16) k∇ u ( s ) k (Ω) (cid:17) d s. As ] a i , b i [ is a maximal interval of strong regularity for u (if b i = T ), we know that k∇ u ( b ) k L (Ω) goes to infinity when b → b i . Hence, if b i = T , we have Z b i a i k Au ( s ) k (Ω) (1 + k∇ u ( s ) k (Ω) ) d s ≤ C T Z b i a i (cid:16) k∇ u ( s ) k (Ω) (cid:17) d s. We sum the inequality on all i (for which b i = T ) and use the fact that σ T is of full measure in[0 , T ] to obtain Z T k Au ( s ) k (Ω) (1 + k∇ u ( s ) k (Ω) ) d s ≤ C T Z T (cid:16) k∇ u ( s ) k (Ω) (cid:17) d s + 11 + k∇ u ( T ) k (Ω) ≤ C T ( T + E(0)) + 1 . We conclude the proof as in [18, p104-106] and [32, Lemma 8.15] by writing Z T k Au ( s ) k / (Ω) d s = Z T k Au ( s ) k / (Ω) (1 + k∇ u ( s ) k (Ω) ) / (1 + k∇ u ( s ) k (Ω) ) / d s ≤ Z T k Au ( s ) k (Ω) (1 + k∇ u ( s ) k (Ω) ) d s ! / (cid:18)Z T (1 + k∇ u ( s ) k (Ω) ) d s (cid:19) / ≤ ( C T ( T + E(0)) + 1) / ( T + E(0)) / < ∞ , and by using again the Agmon inequality (A.6), the Poincar´e inequality, the elliptic estimate k u k H (Ω) . k Au k L (Ω) and the Young inequality, we finally deduce that Z T k u ( s ) k L ∞ (Ω) d s . Z T k∇ u ( s ) k (Ω) d s + Z T k Au ( s ) k / (Ω) d s ≤ E(0) + ( C T ( T + E(0)) + 1) / ( T + E(0)) / < ∞ . To prove the last estimate (4.8), we use Lemma 3.5 to write that for all s ∈ [0 , t ] k ρ f ( s ) k L ∞ (Ω) + k j f ( s ) k L ∞ (Ω) . N q ( f ( s )) . e s (1 + k u k q L (0 ,s ;L ∞ (Ω)) ) N q ( f ) ≤ e t (1 + k u k q L (0 ,t ;L ∞ (Ω)) ) N q ( f ) , so that the conclusion follows. We state here a smoothing property of the Vlasov-Navier-Stokes system for the fluid velocity u .We rely on the parabolic regularization for the Navier-Stokes equations : in short, there is a gainof regularity if the initial data and the source term, that is the Brinkman force F := j f − ρ f u ,are small enough. Proposition 4.9.
There exist universal constants C , C > such that the following holds.Assume that for some T > , one has k∇ u k (Ω) + C Z T k F ( s ) k (Ω) d s ≤ √ C C . (4.12) Then one has for all ≤ t ≤ T k∇ u ( t ) k (Ω) + 12 Z t k Au ( s ) k (Ω) d s . k∇ u k (Ω) + E(0) sup s ∈ [0 ,t ] k ρ f ( s ) k L ∞ (Ω) , (4.13) where . only depends on C and C .Proof. The estimate is a direct consequence of the parabolic regularisation for the Navier-Stokessystem with source F = j f − ρ f u , that we state in Theorem A.12 in Section A.9 of the Appendix,together with the estimate on the Brinkman force of Lemma 4.3. Remark 4.10.
By choosing an appropriate function ϕ in (1.16), we can actually ensure that k∇ u k (Ω) ≤ √ C C . (4.14)Furthermore, again in view of (1.16), we can also reduce k∇ u k (Ω) and E(0) in the sequel ifnecessary. Remark 4.11.
By Proposition 4.8, the r.h.s of (4.13) is finite. Using the elliptic estimate k u k H (Ω) . k Au k L (Ω) , we infer that u ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) , and in particular ∇ u ∈ L (0 , T ; H (Ω)) . Thus, we get the following estimate k u k ∞ (0 ,t ;L (Ω)) . k∇ u k ∞ (0 ,t ;L (Ω)) . k∇ u k (Ω) + E(0) sup s ∈ [0 ,t ] k ρ f ( s ) k L ∞ (Ω) . (4.15)In order to ensure that the smallness condition (4.12) is satisfied for all times, we now intro-duce the following terminology, which has been already used in [24] and [23] to take advantageof the parabolic regulazation for the fluid. 25 efinition 4.12 (Strong existence time). A real number T ≥ strong existence timewhenever (4.12) holds. Lemma 4.13.
The smallness condition (1.16) of Theorem 1.7 ensures that T = 1 is a strongexistence time in the sense of Definition 4.12.Proof. Recall the meaning of the notation . from Definition 4.1. We use Lemma 4.3 and thelocal estimate (4.8) to write Z k F ( s ) k (Ω) d s ≤ E(0) sup s ∈ [0 , k ρ f ( s ) k L ∞ (Ω) . E(0) ≤ C √ C C , where we have used the assumption (1.16). Combining this inequality with (4.14) leads to theresult. In this section, we provide a crucial estimate on the second order derivatives of u : to do so,we look at the convection term ( u · ∇ ) u and at the Brinkman force F = j f − ρ f u as sourceterms for the Navier-Stokes equations. We rely on some maximal L p L q regularity for the Stokesequation on a bounded domain in order to get estimates involving moments of f .We first introduce the following useful notations involving the moments of the kinetic dis-tribution. Definition 5.1.
We set for all t ≥ ρ f ( t ) := sup s ∈ [1 ,t ] k ρ f ( s ) k L ∞ (Ω) , M j f ( t ) := sup s ∈ [1 ,t ] k j f ( s ) k L ∞ (Ω) , M ρ f ,j f ( t ) := M ρ f ( t ) + M j f ( t ) . Remark 5.2.
Note that thanks to Proposition 4.8, we can control ρ f ( s ) and j f ( s ) on [0 ,
1] inL ∞ (Ω). Therefore, if t >
1, we will make a constant use ofsup [0 ,t ] k ρ f ( s ) k L ∞ (Ω) . ρ f ( t ) , sup [0 ,t ] k j f ( s ) k L ∞ (Ω) . j f ( t ) . Proposition 5.3.
Let a, b, r ∈ ]1 , ∞ [ and λ > fixed. For all t ≥ / and all < q ≤ a, b , wehave Z t / e − λs k D u ( s ) k q L r (Ω) d s . Φ( λ ) (cid:16) k ( u · ∇ ) u k q L a (0 ,t ;L r (Ω)) + k F k q L b (0 ,t ;L r (Ω)) (cid:17) , where Φ : R + → R + is nonincreasing.Proof. We introduce w et w which are the unique divergence-free solutions on [0 , + ∞ [ to thefollowing Cauchy problems (cid:26) ∂ t w + Aw = − P ( u · ∇ ) u,w (0) = 0 , and (cid:26) ∂ t w + Aw = P F,w (0) = 0 . (5.1)26here A stands for the Stokes operator on L (Ω) and where P is the Leray projection on vectorfields.Thanks to the maximal regularity for the Stokes system and the continuity of the operator P on L r (Ω) (see Section A.8 of the Appendix), we infer the following estimates for all t ≥ (cid:18)Z t k D w ( s ) k a L r (Ω) d s (cid:19) /a . (cid:18)Z t k ( u · ∇ ) u k a L r (Ω) d s (cid:19) /a , (5.2) (cid:18)Z t k D w ( s ) k b L r (Ω) d s (cid:19) /b . (cid:18)Z t k F ( s ) k b L r (Ω) d s (cid:19) /b . (5.3)On the other hand, if we set e u := u − ( w + w ), we have e u ( t ) = e − tA u ∈ D ( A s ) for all s ≥ a k ) k ≥ be an Hilbertian basis of L (Ω) made up of eigenfunctions of A with associatedpositive eigenvalues ( λ k ) k ≥ such that the sequence ( λ k ) k ≥ is nondecreasing and λ k k → + ∞ −→ + ∞ .We can write in L (Ω) u ( x ) = ∞ X k =0 c k a k , c k := h u , a k i L (Ω) , and the continuity of e − tA on L (Ω) yields e u ( t, x ) = ∞ X k =0 c k e − λ k t a k , t ≥ . Then, we use the fact for all t ≥ ℓ ≥
0, we have k e u ( t ) k H ℓ (Ω) . k A ℓ/ e u ( t ) k L (Ω) , (see e.g: [32, Chapter 2 - p69]) so that we get for all t ≥ ℓ ≥ k e u ( t ) k ℓ (Ω) . ∞ X k =0 | c k | λ ℓk e − λ k t . Note that there exists a constant
C > k and t such that λ ℓk e − λ k t ≤ Ce − λ k t , t ≥ / , ℓ ≥ . Indeed, for all t ≥ / λ ℓk e − λ k t ≤ λ ℓk e − λ k / , and the r.h.s of this inequality tends to 0 as → + ∞ because λ k k → + ∞ −→ + ∞ . Thus, for all t ≥ / ℓ ≥
1, we have the following estimate k e u ( t ) k ℓ (Ω) . ∞ X k =0 | c k | ! e − λ t = k u k (Ω) e − λ t , where we have used the Plancherel-Parseval theorem. We therefore have, for all t ≥ / ℓ ≥ Z t / k e u ( s ) k q H ℓ (Ω) d s . k u k q L (Ω) Z + ∞ / e − qλ s/ d s . k u k q L (Ω) . (5.4)27y using the estimate (5.4) with ℓ large enough, together with Sobolev embedding, we thus get Z t / k D e u ( s ) k q L r (Ω) d s ! /q . . (5.5)We then write u = w + w + e u and it follows that k D u ( s ) k q L r (Ω) . k D w ( s ) k q L r (Ω) + k D w ( s ) k q L r (Ω) + k D e u ( s ) k q L r (Ω) . We have just dealt with the last term. For the other ones, and for a = q, b = q , we use theH¨older inequality, which is justified since aq , bq ≥
1, to write Z t / e − λs k D w ( s ) k q L r (Ω) d s ≤ (cid:18)Z t e − λs aa − q d s (cid:19) − q/a (cid:18)Z t k D w ( s ) k a L r (Ω) d s (cid:19) q/a , (5.6) Z t / e − λs k D w ( s ) k q L r (Ω) d s ≤ (cid:18)Z t e − λs bb − q d s (cid:19) − q/b (cid:18)Z t k D w ( s ) k b L r (Ω) d s (cid:19) q/b . (5.7)The first integral in the r.h.s of (5.6) (resp of (5.7)) is equal to a − qa λ (resp equal to b − qb λ ) andthese expression are indeed nonnegative and nonincreasing in λ . If a = q , we have to to replacethe previous inequality by Z t / e − λs k D w ( s ) k q L r (Ω) d s ≤ e − λ/ (cid:18)Z t k D w ( s ) k a L r (Ω) d s (cid:19) q/a , and in a similar way for the term with w if b = q .Combining the inequalities (5.2), (5.3) et (5.5), we end up with Z t / e − λs k D u ( s ) k q L r (Ω) d s . ( λ − + e − λ/ ) (cid:16) k D w k q L a (0 ,t ;L r (Ω)) + k D w k q L b (0 ,t ;L r (Ω)) (cid:17) + k D e u k q L q (1 / ,t ;L r (Ω)) . Φ( λ ) (cid:16) k ( u · ∇ ) u k q L a (0 ,t ;L r (Ω)) + k F k q L b (0 ,t ;L r (Ω)) (cid:17) , for some nonincreasing function Φ : R + → R + . Remark 5.4.
Note that we can also state a variant of the result of Proposition 5.3 under theform Z t e − λs k D u ( s ) k q L r (Ω) d s . Ψ( λ ) (cid:16) k ( u · ∇ ) u k q L a (1 / ,t ;L r (Ω)) + k F k q L b (1 / ,t ;L r (Ω)) (cid:17) , for all t ≥ R + → R + . To do so, we have tochange the previous proof in the following way: we define w and w as solutions to the Cauchyproblems (5.1) with w (1 /
2) = w (1 /
2) = 0 and we still set e u := u − ( w + w ). We thus have e u ( t ) = e − tA u (1 /
2) and we can write e u ( t ) = ∞ X k =1 γ k e − λ k ( t − / a k , γ k := h u (1 / , a k i L (Ω) . The rest of a the proof is then similar. 28n the two following lemmas, we give L p L q estimates on the convection term and on theBrinkman force F = j f − ρ f u . These results are obtained in the very same way as in thecase of the torus [24, Lemmas 6.2 - 6.3] by using interpolation arguments and the parabolicregularization of (4.13) which is valid for strong times, together with Remark 4.11. Lemma 5.5.
There exist a ∈ (2 , and r a ∈ (2 , such that for all strong existence times t ≥ k ( u · ∇ ) u k L a (0 ,t ;L ra (Ω)) . ρ f ,j f ( t ) . Lemma 5.6.
For all b > , there exists r b > such that for all strong existence times t ≥ k F k L b (0 ,t ;L rb (Ω)) . ρ f ,j f ( t ) − b . We are now able to set up the bootstrap procedure we have mentioned in the end of theIntroduction. In order to get a control on ∇ u , we interpolate the higher regularity estimateswith the poinstwise L (Ω) bound on u provided by the exponential decay of the total kineticenergy. We first obtain the following result on ∇ u , which is non-uniform in time for the moment. Lemma 6.1.
For any strong existence time t ≥ , one has ∇ u ∈ L (1 , t ; L ∞ (Ω)) . Proof.
Let t ≥ u is 0 on ∂ Ω, we use the Gagliardo-Nirenberg-Sobolev inequality (see Theorem A.10 in Appendix) with ( j, m, q ) = (1 , ,
2) to write k∇ u ( s ) k L p (Ω) . k D u ( s ) k α L r (Ω) k u ( s ) k − α L (Ω) , s ≥ / , for all p ∈ [1 , ∞ ] and r ∈ [1 , ∞ ] satisfying the relation1 p = 13 + α (cid:18) r − (cid:19) + 1 − α , (6.1)and where α ∈ [1 / , Z t / k∇ u ( s ) k c L p (Ω) d s ≤ E(0) c − α c − ( e ct − e c ) Z t / e − cs k D u ( s ) k cα L r (Ω) d s, which turns into Z t / k∇ u ( s ) k c L p (Ω) d s . E(0) c − α c − ( e ct − e c )Φ( c ) (cid:16) k ( u · ∇ ) u k cα L a (0 ,t ;L r (Ω)) + k F k cα L b (0 ,t ;L r (Ω)) (cid:17) , thanks to Proposition 5.3, for all c ∈ [1 , + ∞ [ such that cα ≤ a, b and exponents 1 < a, b < ∞ .Now using Lemma 5.5 and Lemma 5.6, we obtain a, b, r a , r b such that b > > a > r := min( r a , r b ) > Z t / k∇ u ( s ) k c L p (Ω) d s . γ ( t ) ,γ ( t ) := E(0) c − α c − ( e ct − e c )Φ( c ) h ρ f ,j f ( t )) cα + (cid:16) ρ f ,j f ( t ) − b (cid:17) cα i , (6.2)29rovided that α ∈ [1 / ,
1[ and p ∈ [1 , ∞ ] satisfy αc ≤ min( a, b ) and the relation (6.1).However, reaching p = ∞ is not possible yet because the relation (6.1) would imply α = 5 (cid:18) − r (cid:19) − , so that the condition α ∈ [1 / ,
1[ is actually equivalent to the condition r >
3. This lastinequality is not a priori satisfied by our choice r = min( r a , r b ). Nevertheless, we will replace r a by a larger exponent. To do so, we first use (6.2) with c = a < b and by carefully looking atthe relation (6.1) when α is close enough to 1, we see there exists α ∈ [1 / ,
1[ related to p > k∇ u k L a (1 / ,t ;L p (Ω)) . γ ( t ) . Since p >
6, we now use the H¨older inequality to find ˜ r a := 6 p/ ( p − > Z t / k ( u · ∇ ) u ( s ) k a L ˜ ra (Ω) d s ! /a ≤ k u k L ∞ (1 / ,t ;L (Ω)) k∇ u k L a (1 / ,t ;L p (Ω)) ≤ γ ( t ) k u k L ∞ (0 ,t ;L (Ω)) ≤ γ ( t ) (cid:16) k∇ u k (Ω) + E(0) η ( t ) (cid:17) , thanks to the estimates (4.15) and (4.8).It allows us to replace Lemma 5.5 by the previous inequality so that we can get ˜ r a > r a . Now, this yields ˜ r := min( r b , ˜ r a ) > α = 5 (cid:18) − r (cid:19) − , we have ˜ α ∈ [1 / , , α (cid:18) r − (cid:19) + 1 − ˜ α . Arguing as in the beginning of the proof, we now use Remark 5.4 with ˜ r > c = 1 and ˜ α andwe get Z t k∇ u ( s ) k L ∞ (Ω) d s . ˜ γ ( t ) , where e γ ( t ) := E(0) − ˜ α ( e t − e )Φ(1) (cid:20) ρ f ,j f ( t )) ˜ α + (cid:16) ρ f ,j f ( t ) − b (cid:17) ˜ α (cid:21) . Because of (4 . e γ ( t ) is finite so that this concludes the proof.In order to set up a bootstrap argument, we naturally introduce the following quantity: t ⋆ := sup (cid:26) strong existence times t ∈ R + such that Z t k∇ u ( s ) k L ∞ (Ω) d s < δ (cid:27) , (6.3)where δ > δe δ ≤ /
9. Our main goal is now to showthat t ⋆ = + ∞ . 30 emma 6.2. We have t ⋆ > and for any t < t ⋆ , the estimate M ρ f ,j f ( t ) . holds.Proof. By reducing E(0) in Lemma 4.3 and by the same proof as in Lemma 4.13 , we observethat t = 1 + ε is still a strong existence time for ε small enough. Thus, thanks to Lemma6.1, we can find ε small enough such that 1 + ε is a strong existence time and such that k∇ u k L (1 , ε ;L ∞ (Ω)) < δ . It therefore implies that t ⋆ > t ∈ [1 , t ⋆ [ and we write it as t = t ⋆ − r with r >
0. By definition of t ⋆ , thereexists a time ˜ t such that t = t ⋆ − r < ˜ t < t ⋆ and such that ˜ t is a strong existence satisfying k∇ u k L (1 , ˜ t ;L ∞ (Ω)) < δ . Now, we can use Lemma 3.6 with t = 1 and the estimate (4.6) for u toget k ρ f ( t ) k L ∞ (Ω) . N q ( f )(1 + k u k q L (0 , ∞ (Ω)) ) . N q ( f )(1 + η (1) q ) , therefore M ρ f ( t ) .
1, uniformly in t . Similarly, for j f , we have k j f ( t ) k L ∞ (Ω) . e − t (cid:18)Z t e s k u ( s ) k L ∞ (Ω) d s + 1 (cid:19) N q ( f )(1 + k u k q L (0 , ∞ (Ω)) ) . e − t (cid:18)Z t e s k u ( s ) k L ∞ (Ω) d s + 1 (cid:19) . The Sobolev embedding H (Ω) ֒ → L ∞ (Ω) and the elliptic estimate k u k H (Ω) . k Au k L (Ω) ensurethat for all t ∈ [1 , t ⋆ [ e − t Z t e s k u ( s ) k L ∞ (Ω) d s . e − t Z t e s k Au ( s ) k L (Ω) d s. Thanks to Cauchy-Schwarz inequality, we get for all t ∈ [1 , t ⋆ [ e − t Z t e s k u ( s ) k L ∞ (Ω) d s ≤ e − t (cid:18)Z t e s d s (cid:19) / (cid:18)Z t k Au ( s ) k (Ω) d s (cid:19) / . (cid:18)Z t k Au ( s ) k (Ω) d s (cid:19) / . s ∈ [0 ,t ] k ρ f ( s ) k L ∞ (Ω) ! / . (cid:0) ρ f ( t ) (cid:1) / , where we have used the parabolic estimate (4.13) (which holds since t < ˜ t ). This concludes theproof as we have already proved that M ρ f ( t ) . Remark 6.3.
Note that the estimate we have just proved is uniform in time. Therefore, byconsidering for example a sequence ( t n ) n of strong existence times with k∇ u k L (1 ,t n ;L ∞ (Ω)) < δ such that t n → t ⋆ and t n ≤ t ⋆ , we get M ρ f ,j f ( t ⋆ ) . Proposition 6.4. If t ⋆ < ∞ , there exists γ > such that the following estimate holds Z t ⋆ k∇ u ( s ) k L ∞ (Ω) d s . E(0) γ . (6.4) Proof.
We modify the proof of Lemma 6.1 and take advantage of the exponential decay ofthe kinetic energy on [1 , t ⋆ ] in order to get uniform in time estimates. We start again using31he Gagliardo-Nirenberg-Sobolev inequality (see Theorem A.10 in Appendix) with ( j, m, q ) =(1 , ,
2) to write k∇ u ( s ) k L p (Ω) . k D u ( s ) k α L r (Ω) k u ( s ) k − α L (Ω) , s ≥ / , for all p ∈ [1 , ∞ ] and r ∈ [1 , ∞ ] satisfying the relation (6.1) and where α ∈ [1 / , t ⋆ < ∞ , we can combine Proposition 2.3 on [0 , t ⋆ ] and Proposition 6.2 to get a rate λ ⋆ such thatE( t ) ≤ e − λ ⋆ t E(0) on [0 , t ⋆ ]. By looking at the definition of the kinetic energy and by setting λ := λ ⋆ (1 − α ) /
2, we have k∇ u ( s ) k L p (Ω) . E(0) − α e − λs k D u ( s ) k α L r (Ω) , s ∈ [1 / , t ⋆ ] . (6.5)This inequality is the key to get uniform in time estimates.Moreover, by taking a sequence ( t n ) n of strong existence times with k∇ u k L (1 ,t n ;L ∞ (Ω)) < δ such that t n → t ⋆ and t n ≤ t ⋆ , we see that estimate (4.13) holds for t ⋆ thanks to Lemma 6.2 sothat the statements of Proposition 5.3 and Lemmas 5.5-5.6 still hold at time t = t ⋆ .We now perfom the same arguments as in the proof of Lemma 6.1: namely, we replace theinequality (6.2) by the inequality (6.5). We eventually end up with a control on ∇ u of the form(6.4) and this concludes the proof.In view of Lemma 6.2, it remains to show the following statement. Proposition 6.5.
We have t ⋆ = + ∞ .Proof. By contradiction, let assume that t ⋆ < + ∞ . We will get a contradiction by proving(provided that E(0) is small enough) the existence of a strong existence time t > t ⋆ such thatthe estimate k∇ u k L (1 ,t ;L ∞ (Ω) < δ holds. • We first prove that there exists a strong existence time larger than t ⋆ . Recall that we workunder the assumption k∇ u k (Ω) ≤ (2 √ C C ) − (see (4.14)). Recall also that the estimate(4.3) yields Z t k F ( s ) k (Ω) d s . E(0) (cid:0) ρ f ,j f ( t ) (cid:1) , for all t ∈ [1 , t ⋆ ], so that Lemma 6.2, Remark 6.3 and the Notation (4.1) for the symbol . provide the existence of a nondecreasing function ϕ : R + → R + which is onto and such that Z t ⋆ k F ( s ) k (Ω) d s ≤ E(0) ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) . We thus infer that k∇ u k (Ω) + C Z t ⋆ k F ( s ) k (Ω) d s ≤ k∇ u k (Ω) + C E(0) ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) ≤ √ C C + C E(0) ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) . Thanks to the smallness condition (1.16), we can choose E(0) small enough so that k∇ u k (Ω) + C Z t ⋆ k F ( s ) k (Ω) d s < √ C C . s
7→ k F ( s ) k (Ω) on [0 , T ] for all T >
0, we obtain bycontinuity a strong existence time (strictly) larger than t ⋆ . • Now, we turn to the existence of a strong existence time larger than t ⋆ which satisfies(6.3). We use the uniform control of Proposition 6.4 to get the existence of a nondecreasingcontinuous and onto function ϕ : R + → R + such that Z t ⋆ k∇ u ( s ) k L ∞ (Ω) ≤ ϕ (cid:16) k u k H (Ω) + N q f + M f + E(0) (cid:17) E(0) γ , with γ >
0. A smallness condition such as (1.16) actually ensures that Z t ⋆ k∇ u ( s ) k L ∞ (Ω) ≤ δ . Thanks to Lemma 6.1, a continuity argument shows that there exists a strong existence time t > t ⋆ such that Z t k∇ u ( s ) k L ∞ (Ω) < δ. This is a contradiction with the definition of t ⋆ . Therefore we must have t ⋆ = + ∞ and theproof of Theorem 1.7 is finally complete. This section aims at providing a further description of the asymptotic behavior of f in the spacevariable, that is to say at proving Theorem 1.10. Indeed, we have obtained the existence of aspatial profile ρ ∞ in an abstract framework in Corollary 1.9. However, a careful study of theparticle trajectory will bring more information about this asymptotic state. We will see that, infull generality, the description of this profile will depend on the whole evolution of the system.We first refer to Section 3 where we have defined the characteristic curves s (X( s ) , V( s ))in R × R for the Vlasov equation (associated to the natural extension P u for the fluid velocity u ). We will make a constant use of the notations used in this section.Suppose that u ∈ L ( R + ; H ∩ W , ∞ (Ω)). The characteristic curves for the Vlasov equationsare given for all t, s ≥ X( s ; t, x, v ) = x + (1 − e − s + t ) v + Z st (1 − e τ − s )( P u )( τ, X( τ ; t, x, v )) d τ, V( s ; t, x, v ) = e − s + t v + Z st e τ − s ( P u )( τ, X( τ ; t, x, v )) d τ. (7.1)We will also use the notation Z s,t defined in (3.5).We then recall the following representation formula for a given weak solution ( u, f ) to theVlasov-Navier-Stokes system (see Proposition 3.8). We have f ( t, x, v ) = e t f (X(0; t, x, v ) , V(0; t, x, v )) O t ( x, v ) , (7.2)where O t is defined in (3.7). 33n order to describe the asymptotic profile ρ ∞ of Corollary 1.9, we take a test function ψ ∈ C ∞ c (Ω) and we look at the following quantity for t ≥ Z Ω ρ f ( t, x ) ψ ( x ) d x. Thanks to the previous representation formula (7.2) for the Vlasov equation, we can write forall t > Z Ω ρ f ( t, x ) ψ ( x ) d x = Z Ω × R f ( t, x, v ) ψ ( x ) d x d v = Z Ω × R e t f (X(0; t, x, v ) , V(0; t, x, v )) O t ( x, v ) ψ ( x ) d x d v. From the previous formula, we use the natural change of variable z = Z ,t ( x, v ), by rememberingthat Z ,t ( O t ) = (cid:8) ( x, v ) ∈ Ω × R | τ + (0 , x, v ) > t (cid:9) , where the forward exit time τ + (0 , x, v ) isdefined by τ + (0 , x, v ) = sup { s ≥ | ∀ σ ∈ [0 , s ] , X( σ ; 0 , x, v ) ∈ Ω } , (see Section A.2 of the Appendix). We thus infer that for all t > Z Ω ρ f ( t, x ) ψ ( x ) d x = Z Ω × R τ + (0 ,x,v ) >t f ( x, v ) ψ (X t, ( x, v )) d x d v, as in the proof of Proposition 3.2 in Section A.2. We now remark that for any t > τ + (0 ,x,v ) >t = 0 if x / ∈ Ω. We thus get for all t > Z Ω ρ f ( t, x ) ψ ( x ) d x = Z R × R τ + (0 ,x,v ) >t f ( x, v ) ψ (X t, ( x, v )) d x d v. (7.3)To go further, we need to understand the behavior of the family of curves X t, ( x, v ) when t → ∞ . This is the aim of the following Lemma, which is in the same spirit as [24, Lemma 8.1]. Lemma 7.1.
There exists δ > such that if u ∈ L ( R + ; H ∩ W , ∞ (Ω)) with Z ∞ k∇ u ( s ) k L ∞ (Ω) d s ≤ δ, (7.4) then the family of functions ( x, v ) X t, ( x, v ) converges in C ( R × R ) , when t → + ∞ ,towards X ∞ : R × R X ∞ ( x, v ) ∈ R with X ∞ ( x, v ) = x + v + Z ∞ ( P u )( τ, X τ, ( x, v )) d τ. Proof.
In order to show that the family of curves (X t, ( x, v )) t has a limit in C ( R × R ) when t → + ∞ , we will show that is satisfies a local Cauchy’s criterion when t → + ∞ .By coming back to the expression (7.1), we first writeX t, ( x, v ) = x + v + Z ∞ τ ≤ t ( P u )( τ, X τ, ( x, v )) d τ + ε ( t, x, v ) , (7.5)where ε ( t, x, v ) := e − t v − Z ∞ e τ − t τ ≤ t ( P u )( τ, X τ, ( x, v )) d τ. τ, ( z ) := X τ, ( x, v ), where z = ( x, v ) × R × R as usual.To show that a Cauchy’s criterion is satisfied, we compute the difference between two ex-pressions (7.5) at t and t with 0 < t < t and we get | X t , ( z ) − X t , ( z ) | ≤ Z t t | ( P u )( τ, X τ, ( z )) | d τ + | ε ( t , x, v ) − ε ( t , x, v ) |≤ Z t t k ( P u )( τ ) k L ∞ ( R ) d τ + | ε ( t , x, v ) − ε ( t , x, v ) |≤ Z t t k u ( τ ) k L ∞ (Ω) d τ + | ε ( t , x, v ) − ε ( t , x, v ) | . We now fix a compact set K ⊂ R × R . The previous computation thus yieldssup z ∈ K | X t , ( z ) − X t , ( z ) | ≤ Z t t k u ( τ ) k L ∞ (Ω) d τ + sup z ∈ K | ε ( t , z ) − ε ( t , z ) | := (I) + (II) . and it remains to see how the terms (I) and (II) behave when min( t , t ) → + ∞ .Thanks to Poincar´e inequality on H (Ω) ∩ W , ∞ (Ω) and to the assumption (7.4), we observethat t
7→ k u ( t ) k L ∞ (Ω) is integrable on R + . We thus infer that the term (I) converges to 0 whenmin( t , t ) → + ∞ .For the term (II), we remark that for all t > z ∈ K | ε ( t, z ) | . K e − t + Z ∞ e τ − t τ ≤ t k ( P u )( τ ) k L ∞ ( R ) d τ. Since e τ − t τ ≤ t k ( P u )( τ ) k L ∞ ( R ) t → + ∞ −→ τ ∈ R + and since it is controlled bythe integrable function τ
7→ k u ( τ ) k L ∞ (Ω) on R + , we can apply the dominated convergencetheorem to show that the previous r.h.s converges to 0 when t → + ∞ . Thanks to the triangularinequality, the term (II) finally tends to 0 when min( t , t ) → + ∞ .Therefore we have shown thatsup z ∈ K | X t , ( z ) − X t , ( z ) | −→ t , t ) −→ + ∞ . This yields the existence of a mapping ( s, z ) X ∞ ( x, v ) ∈ R which is the uniform limitin C ( R × R ) of the maps z X t, ( z ) quand t → + ∞ . Again thanks to the dominatedconvergence theorem, we can pass to the limit when t → + ∞ in the expression (7.1) to get, asannounced X ∞ ( x, v ) = x + v + Z ∞ ( P u )( τ, X τ, ( x, v )) d τ. For the moment, the limit X ∞ ( x, v ) is only continuous in its variables ( x, v ) as a uniformlimit of such functions. A (local) Cauchy’s criterion for the family of derivatives in space-velocity z D z X t, ( z ) will be actually satisfied when t → + ∞ . Indeed, we differentiate the expression(7.1) to get D z X t, ( z ) = 2A + Z ∞ τ ≤ t ∇ ( P u )( τ, X τ, ( z ))D z X τ, ( z ) d τ + e ε ( t, z ) , e ε ( t, z ) := e − t B − Z ∞ e τ − t τ ≤ t ∇ ( P u )( τ, X τ, ( z ))D z X τ, ( z ) d τ, A := (cid:0) I | I (cid:1) ∈ M , ( R ) , B := (cid:0) M , ( R ) | I (cid:1) ∈ M , ( R ) , this identity being valid thanks to the integrability of τ
7→ k∇ ( P u )( τ ) k L ∞ ( R ) on R + . As before,if t < t , we have the following inequality | D z X t , ( z ) − D z X t , ( z ) | ≤ Z t t |∇ ( P u )( τ, X τ, ( z ))D z X τ, ( z ) | d τ + | e ε ( t , z ) − e ε ( t , z ) | . We thus need to derive some bounds on D z X τ, ( z ). We have for all 0 ≤ s ≤ t | D x X s, ( z ) | ≤ Z ∞ τ ≤ t (1 − e τ − t ) |∇ ( P u )( τ, X τ, ( z ) || D x X τ, ( z ) | d τ ≤ ≤ τ ≤ t | D x X τ, ( z ) | Z ∞ k∇ ( P u )( τ ) k L ∞ ( R ) d τ ≤ ≤ τ ≤ t | D x X τ, ( z ) | Z ∞ k∇ u ( τ ) k L ∞ (Ω) d τ. Thanks to the assumption (7.4) on ∇ u , we obtainsup ≤ τ ≤ t | D x X τ, ( z ) | ≤ − δ . (7.6)We observe that the very same procedure leads tosup ≤ τ ≤ t | D v X τ, ( z ) | ≤ − δ . (7.7)Imposing δ ≤ /
2, the two previous r.h.s become smaller that 2 and we will use this uniformbound in what follows. Again, we fix a compact set K ⊂ R × R . We havesup z ∈ K | D z X t , ( z ) − D z X t , ( z ) | ≤ Z t t k∇ ( P u )( τ ) k L ∞ ( R ) d τ + sup z ∈ K | e ε ( t , z ) − e ε ( t , z ) |≤ Z t t k∇ u ( τ ) k L ∞ (Ω) d τ + sup z ∈ K | e ε ( t , z ) − e ε ( t , z ) | := ( e I) + ( e II) . As before, the first term ( e I) tends to 0 when min( t , t ) goes to + ∞ because of the integra-bility assumption (7.4).For the second term ( e II), we proceed as before and we write for all t > z ∈ K | e ε ( t, z ) | ≤ e − t + Z ∞ e τ − t τ ≤ t k∇ ( P u )( τ ) k L ∞ ( R ) d τ. Again, the dominated convergence theorem shows that this expression tends to 0 when t tendsto + ∞ . and this implies that the term ( e II) converges to 0 when min( t , t ) goes to 0.All in all, we have shown thatsup z ∈ K | D z X t , ( z ) − D z X t , ( z ) | −→ t , t ) −→ + ∞ . Cauchy’s criterion then applies for the derivatives and this concludes the proof.36n what follows, we use the notation X t, ,v ( x ) := X( t ; 0 , x, v ) and X ∞ ,v ( x ) := X ∞ ( x, v ). Lemma 7.2.
Under the assumption (7.4), for all v ∈ R and t ∈ R + , the maps X t, ,v : x X t, ,v ( x ) and the map X ∞ ,v : x X ∞ ( x, v ) are C -diffeomorphisms from R to R .Proof. The proof follows the same steps as that of [24, Lemma 8.2]. Indeed, from the expression(7.1), we getD x X t, ,v ( x ) − I = Z ∞ τ ≤ t (1 − e τ − t ) ∇ ( P u )( τ, X τ, ( x, v ))D x X τ, ( x, v ) d τ, and then, thanks to the bound (7.6), we can write k D x X t, ,v − I k ∞ ≤ δ − δ , so that up to taking δ small enough in the assumption (7.4), we can obtain k D x X t, ,v − I k ∞ ≤ . (7.8)A variant of the global inversion theorem about perturbation of the identity mapping (see [24,Lemma 9.4]) leads to the conclusion.In order to pass to the limit in (7.3) when t → + ∞ , we need to determine the limit of τ + (0 ,x,v ) >t when t → + ∞ . Lemma 7.3.
For almost every ( x, v ) ∈ R × R , we have τ + (0 ,x,v ) >t −→ t → + ∞ O ∞ , where O ∞ := (cid:8) ( x, v ) ∈ Ω × R | ∀ t ≥ , X( t ; 0 , x, v ) ∈ Ω (cid:9) . Proof.
First, we note that for every ( x, v ) ∈ Ω × R , we have τ + (0 , x, v ) > x, v ) ∈ Ω c × R , we have τ + (0 , x, v ) = 0 and X (0; 0 , x, v ) ∈ Ω c . We then focus on the followingalternative when ( x, v ) ∈ Ω × R . If τ + (0 , x, v ) = + ∞ , then t τ + (0 ,x,v ) >t is constantequal to 1 for all t > σ ≥
0, X( σ, , x, v ) ∈ Ω. If τ + (0 , x, v ) = + ∞ , then τ + (0 ,x,v ) >t −→ t → + ∞ τ + (0 , x, v ) , , x, v ) ∈ ∂ Ω so that the conclusion holds.We can now conclude the proof of Theorem 1.10 by passing to the limit in the expression(7.3) when t → + ∞ . Indeed, thanks to Lemma 7.1 and 7.3 and since f ∈ L (Ω × R ), we canuse the dominated convergence theorem when t → + ∞ to get Z Ω ρ f ( t, x ) ψ ( x ) d x −→ t → + ∞ Z R × R O ∞ ( x, v ) f ( x, v ) ψ (X ∞ ,v ( x )) d x d v. We then use the reverse change of variable x = X − ∞ ,v ( y ) for all velocity v ∈ R in the previousintegral thanks to Lemma 7.2 in order to obtain Z R × R O ∞ ( x, v ) f ( x, v ) ψ (X ∞ ( x, v )) d x d v = Z R × R O ∞ (X − ∞ ,v ( x ) , v ) f (X − ∞ ,v ( x ) , v ) | det D x X − ∞ ,v ( x ) | ψ ( x ) d x d v = Z Ω Z R O ∞ (X − ∞ ,v ( x ) , v ) f (X − ∞ ,v ( x ) , v ) | det D x X − ∞ ,v ( x ) | d v ψ ( x ) d x, ψ is compactly supported in Ω. Thanks to the convergence ρ f ( t ) t → + ∞ −→ ρ ∞ in H − (Ω)of Corollary 1.9, we thus get by uniqueness of the limit that for a.e x ∈ Ω, ρ ∞ ( x ) = Z R O ∞ (X − ∞ ,v ( x ) , v ) f (X − ∞ ,v ( x ) , v ) | det D x X − ∞ ,v ( x ) | d v. Note that the meaning of the previous indicator function is the following. If ( x, v ) ∈ Ω × R ,(X − ∞ ,v ( x ) , v ) ∈ O ∞ ⇔ X − ∞ ,v ( x ) ∈ Ω , and ∀ t ≥ , X( t, , X − ∞ ,v ( x ) , v ) ∈ Ω ⇔ ∃ ! y ∈ Ω , X ∞ ,v ( y ) = x, and ∀ t ≥ , X( t, , y, v ) ∈ Ω . This concludes the proof of Theorem 1.10.
As already explained in the Introduction, the asymptotic profile ρ ∞ has a total mass which isnot known a priori , because t
7→ k ρ f ( t ) k L (Ω) can be decreasing. We can actually find someclass of initial data for which any prescribed mass (which is less than of equal to the initial mass1) will be achieved, that is Proposition 1.12.In order to do so, we study two scenarios for the support of the initial density, namely weimpose supp f ⊂ K × K where • K is a compact of Ω, at strictly positive distance from ∂ Ω, say included in B x ( a, ε ) ⊂ Ω, • K is a compact of R , say included in B v (0 , R ) for small R > or is an exterior domainof R , say included in ( R \ B v (0 , R )) for large R > t ≥ f ( t ) has a supportin space and velocity which is transported by the flow from the initial support of f . Moreprecisely, we have for all t ≥ f ( t ) ⊂ (cid:16) X( t ; 0 , supp f ) × V( t ; 0 , supp f ) (cid:17) ∩ O t ⊂ R × R . Our main idea is that small initial velocities will lead to a limit profile which is compactlysupported in Ω while high initial velocities will give a profile vanishing in Ω.In this section, we shall use several times the DiPerna-Lions theory for transport equationin the same fashion as in the proof of Lemma 3.5: this allows us to define the characteric curvesin a classical sense. Note that this procedure is actually only required on the interval of time[0 ,
1] because the smallness condition 1.16 will ensure that u ∈ L (1 , + ∞ ; W , ∞ (Ω)) (see theproof of Propositions 8.1-8.5). We first investigate the case in which is included in a small ball, leading to a situation whereparticles stay confined in the domain Ω, and far from the boundary.
Proposition 8.1.
Let ( u , f ) be an admissible initial condition in the sense of Definition 1.1satisfying supp f ⊂ B x ( a, ε ) × B v (0 , R ) ⊂ Ω × R , here ε, R > satisfy the geometric condition d (B x ( a, ε ) , ∂ Ω) > , R < d (B x ( a, ε ) , ∂ Ω) . (8.1) If the smallness condition (1.16) is satisfied, then, for any weak solution ( u, f ) to the Vlasov-Navier-Stokes system with initial data ( u , f ) , there exists d = d ( ε, R, Ω) > such that we havefor all t ≥ f ( t ) ⊂ B x ( a, d ) × R ( Ω × R . Furthermore, there exists ρ ∞ ∈ L ∞ c (Ω) with R Ω ρ ∞ ( x ) d x = 1 and such that W ( f ( t ) , ρ ∞ ⊗ δ v =0 ) −→ t → + ∞ , exponentially fast. We state two lemmas that give estimates for the size of the image of the initial supportB x ( a, ε ) × B v (0 , R ) under the flow t X( t ; 0 , x, v ). This is achieved thanks to an appropriatecontrol on the L L ∞ norm of u . Lemma 8.2.
Let u ∈ L ( R + ; H (Ω)) ∩ L ( R + ; W , ∞ (Ω)) . For all t ≥ , for all x ∈ Ω andfor all v ∈ B(0 , R ) , we have | V( t ; 0 , x, v ) − v | ≤ k u k L (0 ,t ;L ∞ (Ω)) + R, | X( t ; 0 , x, v ) − x | ≤ k u k L (0 ,t ;L ∞ (Ω)) + R. Moreover, we have V( t ; 0 , x, v ) ∈ B (cid:0) , k u k L (0 ,t ;L ∞ (Ω)) + 2 R (cid:1) , and if x ∈ B( a, ε ) , we have X( t ; 0 , x, v ) ∈ B (cid:0) a, k u k L (0 ,t ;L ∞ (Ω)) + R + ε (cid:1) . Proof.
Recall that we have the formula for t ≥ t ; 0 , x, v ) = e − t v + Z t e τ − t ( P u )( τ, X( τ ; 0 , x, v )) d τ, X( t ; 0 , x, v ) = x + v − V( t ; 0 , x, v ) + Z t ( P u )( τ, X( τ, , x, v )) d τ. Therefore, for t ≥
0, the triangular inequality first leads to | V( t ; 0 , x, v ) − v | ≤ | V( t ; 0 , x, v ) − e − t v | + | e − t v − v |≤ k P u k L (0 ,t ;L ∞ ( R )) + (1 − e − t ) | v |≤ k u k L (0 ,t ;L ∞ (Ω)) + R, if v ∈ B v (0 , R ). Then, in a similar way we get | X( t ; 0 , x, v ) − x | ≤ k P u k L (0 ,t ;L ∞ ( R )) + | V( t ; 0 , x, v ) − v |≤ k P u k L (0 ,t ;L ∞ ( R )) + k u k L (0 ,t ;L ∞ (Ω)) + R ≤ k u k L (0 ,t ;L ∞ (Ω)) + R. Finally, the triangular inequality gives the last statements.39 orollary 8.3.
Let u ∈ L ( R + ; H (Ω)) ∩ L ( R + ; W , ∞ (Ω)) . Let ε > and a ∈ Ω such that B x ( a, ε ) ⊂ Ω with d (B x ( a, ε ) , ∂ Ω) > and R > such that R < d (B x ( a, ε ) , ∂ Ω) . (8.2) Suppose that the velocity field u satisfies k u k L ( R + ;L ∞ (Ω)) ≤ δ, (8.3) where δ := 12 (cid:18) d (B x ( a, ε ) , ∂ Ω)2 − R (cid:19) . Then for all T ≥ and for all ( x, v ) ∈ B x ( a, ε ) × B v (0 , R ) , we have X( T ; 0 , x, v ) ∈ Ω , (8.4) and more precisely X( T ; 0 , x, v ) ∈ B (cid:16) a, ε + R + 2 δ (cid:17) ( Ω . (8.5) Proof.
Let T ≥
0. We first apply Lemma 8.2 until time T to see that for all σ ∈ [0 , T ]X( σ ; 0 , x, v ) ∈ B (cid:0) a, ε + R + 2 k u k L (0 ,T ;L ∞ (Ω)) (cid:1) , if ( x, v ) ∈ B x ( a, ε ) × B v (0 , R ).Then, thanks to the global assumption (8.3), we get for all σ ∈ [0 , T ] and for all ( x, v ) ∈ B x ( a, ε ) × B v (0 , R ) X( σ ; 0 , x, v ) ∈ B( a, L ) , where L := ε + R + 2 δ. Thus, the point is just to ensure that B( a, L ) ( Ω . Since d ( a, ∂ Ω) = ε + d (B x ( a, ε ) , ∂ Ω), the inequality
L < d ( a, ∂ Ω) will be satisfied if we provethat R + 2 δ < d (B x ( a, ε ) , ∂ Ω) . By the definition of δ , we have indeed R + 2 δ = d (B x ( a, ε ) , ∂ Ω)2 < d (B x ( a, ε ) , ∂ Ω) , and thus infer that B( a, L ) ( Ω , with d (cid:16) B( a, L ) , ∂ Ω (cid:17) > , leading to the conclusion (8.4). 40t means that, for a well chosen small support of f , the support of f ( t ) stays inside Ω × R and far from the boundary as long as k u k L T L ∞ x (Ω) is small enough for all positive times. Thisassumption will be actually essentially satisfied by using the previous results of the bootstrapargument of Section 6.We eventually turn to the proof of Proposition 8.1. Proof. • Recall that we work with an initial distribution f such that supp f ⊂ B x ( a, ε ) × B v (0 , R ) ⊂ Ω × R and under the assumption (8.1), namely2 R < d (B x ( a, ε ) , ∂ Ω) , which is exactly the previous condition (8.2).First, we define δ = δ ( ε, R, Ω) > u ∈ L (0 , , L ∞ (Ω))(see Proposition 4.8) together with the dominated convergence theorem to obtain t ∈ ]0 ,
1[ suchthat the condition k u k L (0 ,t ;L ∞ (Ω)) < δ , holds.Furthermore, from the assumption (1.16) on the initial data, we can perform the sameanalysis as in the bootstrap argument of Section 6 to get that the quantity k∇ u k L ( t, + ∞ ;L ∞ (Ω)) is as small as we want if we reduce E(0). Thanks to the Poincar´e inequality on H ∩ W , ∞ (Ω),we see that we are able to get the control k u k L ( t, + ∞ ;L ∞ (Ω)) ≤ C Ω k∇ u k L ( t, + ∞ ;L ∞ (Ω)) < δ , (this is possible because we work on an interval of time far from 0). Combining these two pieces,we obtain the fact that the global condition (8.3) is satisfied. Thanks to Corollary 8.3 togetherwith the inclusion supp f ( t ) ⊂ X( t ; 0 , supp f ) × V( t ; 0 , supp f ) , we get for all t ∈ R + supp f ( t ) ( Ω × R . In particular, with (8.5) of Corollary 8.3, we have for all t ∈ R + supp f ( t ) ⊂ B x (cid:16) a, ε + R + 2 δ (cid:17) × R . Remark 8.4.
In such a situation where the particles never leave the domain, we observe thatthe trace of f vanishes for all times there is conservation of the mass. Thus, it is now possibleto prove a convergence of the typeW , Ω × R ( f ( t ) , ρ ∞ ⊗ δ v =0 ) ≤ E(0) / C λ exp( − λt ) , in Corollary 1.9, where W , Ω × R stands for the 1-Wasserstein distance on Ω × R , and withW , Ω ( ρ f ( t ) , ρ ∞ ) → + ∞ −→ , (8.6)41here W , Ω stands for the 1-Wasserstein distance on Ω, as it was already observed in the caseof the torus (see [24]). Indeed, thanks to vanishing trace of f , we can now take smooth testfunctions on Ω, whose Lipschitz constant is less than 1, in the proof of Proposition 2.4 becausethere are no boundary terms.In particular, the previous convergence (8.6) and the fact that the support of each ρ f ( t ) isuniformly included in a ball imply that the limit ρ ∞ is also compactly supported in the sameball. Here, we deal with the situation where all particles escape from the domain Ω after a finite timeso that the kinetic distribution vanishes uniformly after this time. More precisely, we have thefollowing Proposition.
Proposition 8.5.
Let ( u , f ) be an admissible initial condition in the sense of Definition 1.1and take any weak solution ( u, f ) to the Vlasov-Navier-Stokes system with initial data ( u , f ) .There exists L = L (Ω) for which if ε, T, R > are choosen such that supp f ⊂ B x ( a, ε ) × (cid:16) R \ B v (0 , R ) (cid:17) ⊂ Ω × R , (8.7) L > ε, R > L + ε − e − T , (8.8) then the following holds. If the smallness condition (1.16) is satisfied, then for any weak solution ( u, f ) to the Vlasov-Navier-Stokes system with initial data ( u , f ) , we have ∀ t ≥ T, f ( t ) = 0 a.e. (8.9)We first show that the particle trajectory always enters in the complementary of any ballafter a finite time, provided that the initial support in velocity contains only ”high” velocitiesand that the L L ∞ norm of the fluid is small enough. Lemma 8.6.
Let
L, ε, R, T > such that L > ε, (8.10)
R > ε + 2 L − e − T . (8.11) If k u k L ( R + ;L ∞ (Ω)) < L , (8.12) then for all ( x, v ) ∈ B x ( a, ε ) × ( R \ B v (0 , R )) , there exists T x,v ∈ (0 , T ] such that X( T x,v ; 0 , x, v ) ∈ R \ B( a, L ) . Proof.
First, we write the formula (7.1) under the form X( s ; t, x, v ) + V( s ; t, x, v ) = x + v + Z st ( P u )( τ, X( τ ; t, x, v )) d τ, V( s ; t, x, v ) = e − s + t v + Z st e τ − s ( P u )( τ, X( τ ; t, x, v )) d τ. (8.13)42or all ( x, v ) ∈ B x ( a, ε ) × ( R \ B v (0 , R )) and t ∈ [0 , T ], we use (8.13) and the triangularinequality to write | X( t ; 0 , x, v ) − a | ≥ (cid:12)(cid:12)(cid:12)(cid:12) ( e − t − v + Z t e τ − t ( P u )( τ, X( τ, , x, v ))d τ (cid:12)(cid:12)(cid:12)(cid:12) − | x − a | − (cid:12)(cid:12)(cid:12)(cid:12)Z t ( P u )( τ, X( τ, , x, v ))d τ (cid:12)(cid:12)(cid:12)(cid:12) , so that | X( t ; 0 , x, v ) − a | ≥ (1 − e − t ) R − ε − Z t | ( P u )( τ, X( τ, , x, v )) | d τ ≥ (1 − e − t ) R − ε − k u k L ( R + ;L ∞ (Ω)) . We thus end up withsup τ ∈ [0 ,T ] | X( τ ; 0 , x, v ) − a | ≥ (1 − e − T ) R − ε − k u k L ( R + ;L ∞ (Ω)) . Since we have (1 − e − T ) R − ε > L thanks to assumption (8.11), we observe that the condition(8.12) implies ∀ ( x, v ) ∈ B x ( a, ε ) × ( R \ B v (0 , R )) , sup τ ∈ [0 ,T ] | X( τ ; 0 , x, v ) − a | > L. It means that that for all ( x, v ) ∈ B x ( a, ε ) × ( R \ B v (0 , R )), there exists at least one time T x,v ≥ | X( T x,v ; 0 , x, v ) − a | > L. We observe that T x,v > L > ε by assumption (8.10) , and this concludes the proof.We now give a proof of Proposition 8.5.
Proof.
Let us recall that we work in a small data regime (1.16) and under the assumption ofhigh initial velocity (8.7)-(8.8), that is
R > ε + 2 L − e − T , supp f ⊂ B x ( a, ε ) × ( R \ B v (0 , R )) ⊂ Ω × R , for some L to be determined later. In short, this condition will allow us to ensure that all theparticles have left the domain juste after time T (which is fixed). If we want this time T to besmall, it imposes high initial speeds ( L being fixed).Thanks to the representation formula for the density and by using the same change ofvariable as in the proof of Proposition 3.8, we have for any t ≥ T Z Ω × R f ( t, x, v ) d x d v = Z O t e t f (Z ,t ( x, v )) d x d v = Z Ω × R τ + (0 ,z ) >t f ( z ) d z ≤ Z supp f τ + (0 ,z ) >T f ( z ) d z, f ≥ a.e. Since f ≥ a.e , it is enough to show that the previous time T > f and Ω) satisfies ∀ z ∈ supp f , τ + (0 , z ) ≤ T, for the integral R Ω × R f ( t, x, v ) d x d v to vanish for all t ≥ T .Recall that because of the definition of the extension operator P and because Ω is bounded,we have supp P u ⊂ Ω ( B( a, L ) , for some L >
0. We then set L := 2 L > ε .First, we use the fact that u ∈ L ( R + , L ∞ (Ω)) (see Proposition 4.8) together with thedominated convergence theorem to obtain a time t ∈ ]0 , T [ such that k u k L (0 ,t ;L ∞ (Ω)) < L/ k L ( t, + ∞ ;L ∞ (Ω)) is as small as we want. Combining thisargument with the Poincar´e inequality on H ∩ W , ∞ (Ω), we get k u k L ( t, + ∞ ;L ∞ (Ω)) ≤ C Ω k∇ u k L ( t, + ∞ ;L ∞ (Ω)) < L . Therefore, the condition (8.12) is indeed satisfied.Take now ( x, v ) ∈ supp f . Now, let us define t = t ( x, v ) as the first exit time of thetrajectory from the previous ball, namely t ( x, v ) := sup n s ≥ | ∀ σ ∈ [0 , s ] , X( σ, , x, v ) ∈ B( a, L ) o . Then, we use Lemma 8.6 with our choice of L to obtain a finite time t = t ( x, v ) > t ≤ T, X( t ; 0 , x, v ) ∈ R \ B( a, L ) . It therefore implies that t < t ≤ T < ∞ and we emphasize the fact that T does not dependon ( x, v ). By definition, we have τ + (0 , x, v ) ≤ t ( x, v ) so that the control ∀ ( x, v ) ∈ supp f , τ + (0 , x, v ) ≤ T, is indeed satisfied. It therefore implies that f ≡ T , which iswhat we wanted to prove. Remark 8.7.
In such a scenario, we have ρ ∞ = 0 in the statement of Corollary 1.9. The aim of this subsection is to provide a proof of Proposition 1.12. We want to show that, fora fixed α ∈ [0 , u , f ) which gives rise to weak solutions whosekinetic part concentrates in velocity, with a spatial asymptotic profile of total mass α .44o do so, in view of the two previous sections, we consider L = L (Ω) given in Proposition8.5 and we choose ε, T, R , R > R < R , supp f ⊂ B x ( a, ε ) × (cid:16) B v (0 , R ) ⊔ (cid:0) R \ B v (0 , R ) (cid:1) (cid:17) , Z Ω × R f | v |
Appendix
A.1 Boundary value problem in Ω × R for the kinetic equation Theorem A.1.
Take f ∈ L ∩ L ∞ (Ω × R ) and a vector field u ∈ L ( R + ; W , (Ω)) . Considerthe following kinetic boundary value problem on Ω × R . ∂ t f + v · ∇ x f + div v (( u − v ) f ) = 0 ,f | t =0 = f ,f = 0 , on Σ − . Then we have, for all fixed
T > • Well-posedness: There exists a unique f ∈ L ∞ loc ( R + ; L ∩ L ∞ (Ω × R )) which is a weaksolution of the previous Cauchy problem. Furthermore, f ∈ C ( R + ; L p loc (Ω × R )) , for all p ∈ [1 , ∞ ) and the function f has a trace on ∂ Ω × R defined in the following sense:there exists a unique element γf ∈ L ∞ ([0 , T ] × ∂ Ω × R ) such that for any test function ψ ∈ C ∞ ([0 , T ] × Ω × R ) with compact support in velocity, and for all ≤ t ≤ t ≤ T Z t t Z Ω × R f ( t, x, v ) [ ∂ t ψ + v · ∇ x ψ + ( u − v ) · ∇ v ψ ] ( t, x, v ) d v d x d t = Z Ω × R f ( t , x, v ) ψ ( t , x, v ) d v d x − Z Ω × R f ( t , x, v ) ψ ( t , x, v ) d v d x + Z t t Z ∂ Ω × R [( γf ) ψ ( t, x, v )] v · n ( x ) d v d σ ( x ) d t. • Renormalization: For every β ∈ C ( R ) , for all test function ψ ∈ C ∞ ([0 , T ] × Ω × R ) with compact support in velocity, and for all ≤ t ≤ t ≤ T , we have Z t t Z Ω × R β ( f ( t, x, v )) [ ∂ t ψ + v · ∇ x ψ + ( u − v ) · ∇ v ψ ] ( t, x, v ) d v d x d t = Z Ω × R β ( f ( t , x, v )) ψ ( t , x, v ) d v d x − Z Ω × R β ( f ( t , x, v )) ψ ( t , x, v ) d v d x + Z t t Z ∂ Ω × R [ β ( γf ) ψ ( t, x, v )] v · n ( x ) d v d σ ( x ) d t − Z t t Z Ω × R ψ (cid:2) f β ′ ( f ) − β ( f ) (cid:3) ( t, x, v ) d v d x d t. • Stability: If u n → u in L ( R + ; L (Ω)) and f ,n → f in L (Ω × R ) , the corresponding sequence of solutions ( f n ) satisfies for all p < ∞ , f n −→ f in L ∞ loc ( R + ; L p (Ω × R )) . Such a result can be found in [6, Theorem 3.2 - Proposition 3.2] for the well-posedness andrenormalization properties and in [8, Theorem VI.1.9] for the stability property.46 .2 Proof of Proposition 3.2
This Section aims at giving a proof for the representation formula (3.8), which holds for theweak solution to the Vlasov equation. We use the notations and definitions of Section 3.For ( x, v ) ∈ Ω × R and for any t ≥
0, we first define τ + ( t, x, v ) := sup { s ≥ t | ∀ σ ∈ [ t, s ] , X( σ ; t, x, v ) ∈ Ω } . If t ≥ O t := (cid:8) ( x, v ) ∈ Ω × R | τ − ( t, x, v ) < (cid:9) , where τ − ( t, x, v ) has been defined in (3.6), If t ≥ O t = \ σ ∈ [0 ,t ] Z t,σ (Ω × R ) , Z ,t ( O t ) = (cid:8) ( x, v ) ∈ Ω × R | τ + (0 , x, v ) > t (cid:9) = \ σ ∈ [0 ,t ] Z ,σ (Ω × R ) . By continuity, we have X τ + (0 ,z ) , ( z ) ∈ ∂ Ω if z ∈ Ω × R and τ + (0 , z ) < + ∞ . More precisely, wehave the following result. Lemma A.2.
For z = ( x, v ) ∈ Ω × R , if τ + (0 , z ) < + ∞ then we have Z τ + (0 ,z ) , ( z ) = (X τ + (0 ,z ) , ( z ) , V τ + (0 ,z ) , ( z )) ∈ Σ + ∪ Σ . (A.1) Proof.
Let us suppose by contradiction that V τ + (0 ,z ) , ( z ) · n (X τ + (0 ,z ) , ( z )) <
0. Note thatsince z ∈ Ω × R , we have τ + (0 , z ) >
0. Since Ω is smooth, there exists r > ∈ C (B(X τ + (0 ,z ) , ( z ) , r )) such that for all y ∈ R , y ∈ B (cid:0) X τ + (0 ,z ) , ( z ) , r (cid:1) ∩ Ω ⇔ Ψ( y ) > ,y ∈ B (cid:0) X τ + (0 ,z ) , ( z ) , r (cid:1) ∩ ∂ Ω ⇔ Ψ( y ) = 0 . For all τ > τ + := τ + (0 , z ), we haveΨ (X τ, ( z )) = Ψ (cid:0) X τ + , ( z ) (cid:1) + ( τ − τ + ) ˙X τ + , ( z ) · ∇ Ψ (cid:0) X τ + , ( z ) (cid:1) + o ( τ − τ + )= ( τ + − τ )V τ + , ( z ) · n (cid:0) X τ + , ( z ) (cid:1) (cid:12)(cid:12) ∇ Ψ (cid:0) X τ + , ( z ) (cid:1)(cid:12)(cid:12) + o ( τ − τ + ) . Thus, for τ > τ + , Ψ (X τ, ( z )) <
0. This means there exists ε > τ ∈ ( τ + (0 , z ) − ε, τ + (0 , z )) ⊂ R + , X τ, ( z ) / ∈ Ω, which is in contradiction with the definition of τ + (0 , z ).We now turn to the proof of Proposition 3.2. Since the Vlasov equation has a unique solutionwhen u is fixed (see Appendix A.1), we have to check that R + × Ω × R −→ R ( t, x, v ) e t O t ( x, v ) f (Z ,t ( x, v )) , is a weak solution to this equation associated to the velocity field u and starting at f . Letus fix T >
0. We thus take ϕ ∈ C ∞ c ([0 , T ] × Ω × R ) such that ϕ ( T ) = 0 and vanishing on[0 , T ] × (Σ + ∪ Σ ) and we want to show that Z T Z Ω × R e s O s ( x, v ) f (Z ,s ( x, v )) h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, x, v ) d v d x d s = − Z Ω × R f ( x, v ) ϕ (0 , x, v ) d v d x. ,s ( O s ) = (cid:8) ( x, v ) ∈ Ω × R | τ + (0 , x, v ) > s (cid:9) , to write Z T Z Ω × R e s O s ( x, v ) f (Z ,s ( x, v )) h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, x, v ) d v d x d s = Z T Z O s e s f (Z ,s ( x, v )) h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, x, v ) d v d x d s = Z T Z Ω × R τ + (0 ,z ) >s e s f ( z ) h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, Z s, ( z ))J s ( z ) d z d s, where we have used the change of variable z = Z ,s ( x, v ) and where J s ( z ) stands for the Jacobianof the map z Z s, ( z ), whose value is J s ( z ) = e − s for all s ≥
0. Furthermore, by usingdd s h ϕ ( s, Z s, ( z )) i = h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, Z s, ( z )) , we get Z T Z Ω × R τ + (0 ,z ) >s e s f ( z ) h ∂ t ϕ + v · ∇ x ϕ + ( u − v ) · ∇ v ϕ i ( s, Z s, ( z ))J s ( z ) d z d s = Z T Z Ω × R τ + (0 ,z ) >s f ( z ) dd s h ϕ ( s, Z s, ( z )) i d z d s = Z T Z Ω × R τ + (0 ,z ) >s dd s h f ( z ) ϕ ( s, Z s, ( z )) i d z d s = Z Ω × R (Z min( T,τ + (0 ,z ))0 dd s h f ( z ) ϕ ( s, Z s, ( z )) i d s ) d z, where we have used Fubini’s Theorem. We then write Z Ω × R (Z min( T,τ + (0 ,z ))0 dd s h f ( z ) ϕ ( s, Z s, ( z )) i d s ) d z = (I) + (II) , where (I) := Z Ω × R τ + (0 ,z ) >T (Z min( t,τ + (0 ,z ))0 dd s h f ( z ) ϕ ( s, Z s, ( z )) i d s ) d z, (II) := Z Ω × R τ + (0 ,z ) ≤ T (Z min( t,τ + (0 ,z ))0 dd s h f ( z ) ϕ ( s, Z s, ( z )) i d s ) d z. First, we have(I) = Z Ω × R τ + (0 ,z ) >T f ( z ) ϕ ( T, Z T, ( z )) d z − Z Ω × R τ + (0 ,z ) >T f ( z ) ϕ (0 , z ) d z. Since ϕ ( T ) = 0, the first integral vanishes and we obtain(I) = Z Ω × R e T O T ( x, v ) f (Z ,T ( x, v )) ϕ ( T, x, v ) d v d x − Z Ω × R τ + (0 ,z ) >T f ( z ) ϕ (0 , z ) d z = − Z Ω × R τ + (0 ,z ) >T f ( z ) ϕ (0 , z ) d z, Z Ω × R τ + (0 ,z ) ≤ T f ( z ) ϕ ( τ + (0 , z ) , Z τ + (0 ,z ) , ( z )) d z − Z Ω × R τ + (0 ,z ) ≤ T f ( z ) ϕ (0 , z ) d z. Thanks to (A.1) and to the fact that ϕ vanishes on Σ + ∪ Σ , we see that the first integral isactually 0.Eventually, gathering all the previous pieces, we end up with(I) + (II) = − Z Ω × R f ( x, v ) ϕ (0 , x, v ) d v d x. The proof of Proposition (3.2) is finally complete.
A.3 The Wasserstein distance
In this section, X stands for a separable and complete subset of R d or R d × R d (in the previoussections, we used X = Ω or X = Ω × R d where Ω is an open subset of R d ).We recall the definition of the 1-Wasserstein distance on X and the useful and classicalMonge-Kantorovich formula (see [17, Section 11.8] for instance). Definition A.3.
For all m >
0, we define M ,m ( X ) the set of positive measures µ on X suchthat Z X | x | d µ ( x ) < ∞ , µ ( X ) = m. Definition A.4.
For all m >
0, if µ et ν are two measures belonging to M ,m ( X ), we definethe Wasserstein distance W ( µ, ν ) as the quantityW ( µ, ν ) := inf γ ∈ Π( µ,ν ) Z X × X | x − x ′ | d γ ( x, x ′ ) , where Π( µ, ν ) stands for the set of positive measures on X × X whose first marginal is µ andsecond marginal ν . Proposition A.5.
Fix m > . Given ( µ n ) n ∈ M ,m ( X ) N and µ ∈ M ,m ( X ) , the two followingstatements are equivalent(i) For all f ∈ C b ( X ) , Z X ( f ( z ) + | z | ) d µ n ( z ) n → + ∞ −→ Z X ( f ( z ) + | z | ) d µ ( z ) . (ii) (W ( µ n , µ )) n n → + ∞ −→ . Theorem A.6 (Duality formula of Monge-Kantorovich). If µ et ν are two measuresbelonging to M ,m ( X ) , we have the following formula W ( µ, ν ) = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z X ψ ( x )d µ ( x ) − Z X ψ ( x )d ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ψ ∈ Lip( X ) , k∇ ψ k ∞ ≤ (cid:27) . .4 Extension of Lipschitz functions vanishing at the boundary Theorem A.7.
Let Ω be a smooth bounded and open subset of R d . Take u ∈ W , ∞ (Ω) suchthat u | ∂ Ω = 0 . If we set ∀ x ∈ R d , ( P u )( x ) := ( u ( x ) , if x ∈ Ω , if x ∈ R \ Ω , then the function P u ∈ W , ∞ ( R d ) and we have k P u k L ∞ ( R d ) = k u k L ∞ (Ω) , k∇ ( P u ) k L ∞ ( R ) ≤ k∇ u k L ∞ (Ω) . Proof.
Let us show that the extension
P u is Lipschitz on R d . To do so, take x, y ∈ R d . • If x ∈ Ω and y ∈ Ω, then we have | ( P u )( x ) − ( P u )( y ) | = | u ( x ) − u ( y ) | ≤ k∇ u k L ∞ (Ω) | x − y | . • If x / ∈ Ω and y / ∈ Ω, then | ( P u )( x ) − ( P u )( y ) | = 0 ≤ k∇ u k L ∞ (Ω) | x − y | . • If x ∈ Ω and y / ∈ Ω, then we use the connectedness of the segment [ x, y ] to find a point z ∈ [ x, y ] ∩ ∂ Ω. We thus write | ( P u )( x ) − ( P u )( y ) | = | u ( x ) | = | u ( x ) − u ( z ) | ≤ k∇ u k L ∞ (Ω) | x − z | ≤ k∇ u k L ∞ (Ω) | x − y | . Here, we have used the fact that the Lipschitz semi-norm of u on Ω is smaller than k∇ u k L ∞ (Ω) (see e.g: [8, Proposition III.2.9]). All in all, we end up with | ( P u )( x ) − ( P u )( y ) | ≤ k∇ u k L ∞ (Ω) | x − y | , so that P u ∈ W , ∞ ( R ) with the desired inequalities, because of the identification W , ∞ ( R ) =Lip ∞ ( R ). A.5 A variant of Gronwall lemma
We state a Gronwall lemma under integral form in which exponential decay is obtained.
Lemma A.8.
Let g : R + → R + be an integrable and decaying function such that g (0) is welldefined and which satisfies, for some λ > , for almost every t ≥ λ Z + ∞ t g ( τ ) d τ ≤ g ( t ) . Then, for almost every t ≥ , we have g ( t ) . λ g (0) e − λt , where . λ refers to a constant only depending on λ . A proof can be found in [24, Appendix]. 50 .6 Agmon inequality on a bounded domain of R Proposition A.9.
Let Ω be a smooth bounded domain of R . For all u ∈ H (Ω) , we have theinequality k u k L ∞ (Ω) . k u k / (Ω) k u k / (Ω) , where . only depends on Ω . We refer to [13, Lemma 4.10] for a proof.
A.7 Gagliardo-Nirenberg-Sobolev inequality on a bounded do-main
Theorem A.10.
Let Ω be a smooth bounded domain of R d . Let ≤ p, q, r ≤ ∞ and m ∈ N .Suppose j ∈ N and α ∈ [0 , satisfy the relations p = jd + (cid:18) r − md (cid:19) α + 1 − αq ,jm ≤ α ≤ , with the exception α < if m − j − d/r ∈ N .Then for all g ∈ L q (Ω) , if D m g ∈ L r (Ω) , we have D j g ∈ L p (Ω) with the estimate k D j g k L p (Ω) . k D m g k α L r (Ω) k g k − α L q (Ω) + k g k L s (Ω) , where ≤ s ≤ max { q, r } and where . only depends on Ω .Moreover, if g has a vanishing trace at ∂ Ω , we can drop the last term k g k L s (Ω) in the r.h.sof the inequality. This result can be found in [11, Thm 1.5.2].
A.8 Maximal L p L q regularity for the Stokes system on a boundeddomain Let Ω be a smooth bounded domain of R d and 1 < q < ∞ . Each vector field u ∈ L q (Ω) isuniquely decomposed as u = e u + ∇ p, e u ∈ L q div (Ω) , p ∈ L q (Ω) , ∇ p ∈ L q (Ω) . where L q div (Ω) stands for the closure in L q (Ω) of D div (Ω). In this so called Helmoltz decompo-sition, we recall that the projection P q : u e u is continuous from L q (Ω) to L q div (Ω).For 1 < q < ∞ , we consider the following Stokes operator: A q := − P q ∆ u, D ( A q ) := L q div (Ω) ∩ W ,q (Ω) ∩ W ,q (Ω) . We also set D − s ,sq (Ω) := (cid:0) D ( A q ) , L q div (Ω) (cid:1) /s,s , where ( , ) /s,s refers to the real interpolation space of exponents (1 /s, s ).51 heorem A.11. Consider < T ≤ ∞ and < q, s < ∞ . Then, for every u ∈ D − s ,sq (Ω) which is divergence free and f ∈ L s (0 , T ; L q div (Ω)) , there exists a unique solution u of the Stokessystem ∂ t u + A q u = f,u (0 , x ) = u ( x ) , satisfying u ∈ L s (0 , T ′ ; D ( A q )) for all finite T ′ ≤ T,∂ t u ∈ L s (0 , T ; L q (Ω)) , and k ∂ t u k L s (0 ,T ;L q (Ω)) + k D u k L s (0 ,T ;L q (Ω)) ≤ C (cid:18) k u k D − s ,sq (Ω) + k f k L s (0 ,T ;L q (Ω)) (cid:19) , where C = C ( q, s, Ω) .Furthermore, if u ∈ H (Ω) and if s ∈ (1 , , the statement holds and we can replace k u k D − /s,sq (Ω) by k u k H (Ω) in the right hand side of the previous inequality. A proof of this result and further references on the theory can be found in [19]. The laststatement about the fact that H (Ω) = D ( A ) ֒ → D − /s,sq (Ω) for s ∈ (1 ,
2) comes from [19,Remark 2.5].
A.9 Parabolic regularization for the Navier-Stokes system witha source term on a bounded domain
Theorem A.12.
Let Ω be a regular bounded domain of R . There exist two universal constants C , C > such that the following holds. Consider u ∈ H (Ω) and F ∈ L ( R + ; L (Ω)) and T > such that k∇ u k (Ω) + C Z T k F ( s ) k (Ω) d s ≤ √ C C . (A.2) Then, there exists on [0 , T ] a unique Leray solution to the Navier-Stokes equations with initialdata u and source F . This solution u belongs to L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) and satisfiesfor a.e. ≤ t ≤ T k∇ u ( t ) k (Ω) + 12 Z t k Au ( s ) k (Ω) d s ≤ k∇ u k (Ω) + C Z T k F ( s ) k (Ω) d s. (A.3) Proof.
First, if such a Leray solution to the Navier-Stokes equations exists, we have directly u ∈ L (0 , T ; H (Ω)), which corresponds to a classical tridimensional case of weak-strong uniquess(see for instance [10, Theorem 3.3]). Hence, it remains to prove that such a solution does existon [0 , T ]. We proceed in the following way.We rely on an approximation procedure by regularising the data F and u and by consideringa standard Galerkine approximation ( u N ) N of the corresponding Navier-Stokes system. Theclassical idea is to obtain the desired parabolic estimations on the sequence ( u N ) N on [0 , T ].Combining these new estimates with the classical energy estimates for the Leray solution ofthe Navier-Stokes system on [0 , T ], we can use a compactness argument to produce a solution52ith the L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) regularity and which satisfies the estimate (A.3)on [0 , T ].Thus, in the following, it is sufficient to work with a smooth solution u and with smoothdata. We first apply the Leray projection P to the Navoer-Stokes equations and multiply thisby Au to obtain dd t k∇ u k (Ω) + k Au k (Ω) ≤ C k F k (Ω) + C k∇ u k (Ω) , (A.4)where C >
0. Moreover, there exists an another universal constant C > k∇ u k (Ω) ≤ C k Au k (Ω) . So we rewrite the inequality (A.4) asdd t k∇ u k (Ω) + 12 k Au k (Ω) ≤ C k F k (Ω) + C k∇ u k (Ω) − C k∇ u k (Ω) , on [0 , T ]. Now, if we set x ( t ) := k∇ u ( t ) k (Ω) , y ( t ) := k Au ( t ) k (Ω) , z ( t ) := k F ( t ) k (Ω) , (A.5)and by integrating the previous inequality between 0 and t , we infer that for almost every0 ≤ t ≤ T , x ( t ) + 12 Z t y ( s ) d s ≤ x (0) + Z t C z ( s ) d s + Z t C x ( s ) (cid:18) x ( s ) − C C (cid:19) d s. (A.6)To obtain (A.3), it is sufficient to prove that x ( t ) ≤ / √ C C for t ∈ [0 , T ). To do so, weremark that the assumption (A.2) precisely corresponds to x (0) + Z t C z ( s ) d s ≤ √ C C , and in particular, x (0) ≤ / √ C C . So, by continuity of t x ( t ), there exists a maximaltime T ∈ ]0 , T ] such that x ( t ) ≤ / √ C C on [0 , T [. If T = T , there is nothing to do. So, weargue by contradiction by assuming that T < T . We thus have for all 0 ≤ s ≤ T x ( s ) − C C ≤ , and the inequality (A.6) with t = T turns into x ( T ) + 12 Z T y ( s ) d s ≤ x (0) + Z T C z ( s ) d s, ≤ x (0) + Z T C z ( s ) d s, which implies in particular x ( T ) ≤ √ C C < √ C C . Again by continuity of t x ( t ), there exists ε > t ∈ [ T , T + ε ], x ( t ) ≤ / √ C C , which is a contradiction with the definition of T . It thus implies that for all t ∈ [0 , T ), x ( t ) ≤ / √ C C and we finally end up with the following inequality on [0 , T ] x ( t ) + 12 Z t y ( s ) d s ≤ x (0) + Z t C z ( s ) d s ≤ √ C C . By a view of the definition (A.5), this finally proves the inequality (A.3).
Acknowledgements.
Partial support by the grant ANR-19-CE40-0004 is acknowledged.53 eferences [1] O. Anoshchenko and A. Boutet de Monvel-Berthier. The existence of the global generalizedsolution of the system of equations describing suspension motion.
Mathematical methodsin the applied sciences , 20(6):495–519, 1997.[2] C. Bardos and P. Degond. Global existence for the vlasov-poisson equation in 3 spacevariables with small initial data. In
Annales de l’Institut Henri Poincare (C) Non LinearAnalysis , volume 2, pages 101–118. Elsevier, 1985.[3] E. Bernard, L. Desvillettes, F. Golse, and V. Ricci. A derivation of the Vlasov-Navier-Stokes model for aerosol flows from kinetic theory.
Commun. Math. Sci. , 15(6):1703–1741,2017.[4] E. Bernard, L. Desvillettes, F. Golse, and V. Ricci. A derivation of the Vlasov-Stokessystem for aerosol flows from the kinetic theory of binary gas mixtures.
Kinet. Relat.Models , 11(1):43–69, 2018.[5] L. Boudin, L. Desvillettes, C. Grandmont, and A. Moussa. Global existence of solutionsfor the coupled Vlasov and Navier-Stokes equations.
Differential Integral Equations , 22(11-12):1247–1271, 2009.[6] L. Boudin, C. Grandmont, and A. Moussa. Global existence of solutions to the incompress-ible Navier-Stokes-Vlasov equations in a time-dependent domain.
J. Differ. Equations ,262:1317–1340, 2017.[7] L. Boudin, D. Michel, and A. Moussa. Global existence of weak solutions to the incom-pressible Vlasov-Navier-Stokes system coupled to convection-diffusion equations.
Math.Models Methods Appl. Sci. , 2020.[8] F. Boyer and P. Fabrie.
Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models , volume 183. Springer Science & Business Media,2012.[9] K. Carrapatoso and M. Hillairet. On the derivation of a Stokes–Brinkman problem fromStokes equations around a random array of moving spheres.
Communications in Mathe-matical Physics , 373(1):265–325, 2020.[10] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier.
Mathematical geophysics: Anintroduction to rotating fluids and the Navier-Stokes equations , volume 32. Oxford Univer-sity Press, 2006.[11] P. Cherrier and A. Milani.
Linear and quasi-linear evolution equations in Hilbert spaces .American Mathematical Society Providence, 2012.[12] Y.-P. Choi and B. Kwon. Global well-posedness and large-time behavior for the inhomo-geneous Vlasov–Navier–Stokes equations.
Nonlinearity , 28(9):3309, 2015.[13] P. Constantin and C. Foias.
Navier-Stokes equations . University of Chicago Press, 1988.[14] L. Desvillettes. Some aspects of the modeling at different scales of multiphase flows.
Com-puter methods in applied mechanics and engineering , 199(21-22):1265–1267, 2010.[15] L. Desvillettes, F. Golse, and V. Ricci. The mean-field limit for solid particles in a Navier-Stokes flow.
Journal of Statistical Physics , 131(5):941–967, 2008.5416] R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory andSobolev spaces.
Inventiones mathematicae , 98(3):511–547, 1989.[17] R. M. Dudley.
Real analysis and probability . CRC Press, 2018.[18] C. Foias, O. Manley, R. Rosa, and R. Temam.
Navier-Stokes equations and turbulence ,volume 83. Cambridge University Press, 2001.[19] Y. Giga and H. Sohr. Abstract Lp estimates for the Cauchy problem with applications tothe Navier-Stokes equations in exterior domains.
Journal of functional analysis , 102(1):72–94, 1991.[20] O. Glass, D. Han-Kwan, and A. Moussa. The Vlasov-Navier-Stokes system in a 2d pipe:Existence and stability of regular equilibria.
Archive for Rational Mechanics and Analysis ,230(2):593–639, 2018.[21] T. Goudon, L. He, A. Moussa, and P. Zhang. The Navier-Stokes-Vlasov-Fokker-Plancksystem near equilibrium.
SIAM J. Math. Anal. , 42(5):2177–2202, 2010.[22] K. Hamdache. Global existence and large time behaviour of solutions for the Vlasov-Stokesequations.
Japan journal of industrial and applied mathematics , 15(1):51, 1998.[23] D. Han-Kwan. Large time behavior of small data solutions to the Vlasov–Navier–Stokessystem on the whole space. arXiv preprint arXiv:2006.09848 , 2020.[24] D. Han-Kwan, A. Moussa, and I. Moyano. Large time behavior of the Vlasov-Navier-Stokessystem on the torus.
Archive for Rational Mechanics and Analysis , 236(3):1273–1323, 2020.[25] M. Hillairet. On the homogenization of the Stokes problem in a perforated domain.
Archivefor Rational Mechanics and Analysis , 230(3):1179–1228, 2018.[26] M. Hillairet, A. Moussa, and F. Sueur. On the effect of polydispersity and rotation on theBrinkman force induced by a cloud of particles on a viscous incompressible flow.
Kinet.Relat. Models , 12, 2019.[27] R. M. H¨ofer. Sedimentation of inertialess particles in Stokes flows.
Communications inMathematical Physics , 360(1):55–101, 2018.[28] P. Jabin. Large time concentrations for solutions to kinetic equations with energy dissipa-tion.
Communications in Partial Differential Equations , 25(3-4):541–557, 2000.[29] A. Mecherbet. Sedimentation of particles in Stokes flow.
Kinet. Relat. Models , 12, 2019.[30] S. Mischler. On the trace problem for solutions of the Vlasov equation: The trace problemfor solutions.
Communications in Partial Differential Equations , 25(7-8):1415–1443, 2000.[31] P. J. O’Rourke. Collective drop effects on vaporizing liquid sprays. Technical report, LosAlamos National Lab., NM (USA), 1981.[32] J. C. Robinson, J. L. Rodrigo, and W. Sadowski.
The three-dimensional Navier–Stokesequations: Classical theory , volume 157. Cambridge university press, 2016.[33] F. A. Williams.