Nonlinear Analysis. On the dynamics of a fluid particle interaction model: The bubbling regime

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1 Nonlinear Analysis 74 (211) Contents lists available at ScienceDirect Nonlinear Analysis journal omepage: On te dynamics of a fluid particle interaction model: Te bubbling regime J.A. Carrillo a, T. Karper b,, K. Trivisa c a ICREA Departament de Matemàtiques, Universitat Autònoma de Barcelona, 8193 Bellaterra, Spain b Department of Matematical Sciences, Norwegian University of Science and Tecnology, N-7491 Trondeim, Norway c Department of Matematics, University of Maryland, College Park, MD , USA a r t i c l e i n f o a b s t r a c t Article istory: Received 4 November 21 Accepted 29 December 21 Keywords: Global-in-time existence Large data Large-time beaviour Fluid particle interaction model Compressible and viscous fluid Smolucowski equation Tis article deals wit te issues of global-in-time existence and asymptotic analysis of a fluid particle interaction model in te so-called bubbling regime. Te mixture occupies te pysical space R 3 wic may be unbounded. Te system under investigation describes te evolution of particles dispersed in a viscous compressible fluid and is expressed troug te conservation of fluid mass, te balance of momentum and te balance of particle density often referred as te Smolucowski equation. Te coupling between te dispersed and dense pases is obtained troug te drag forces tat te fluid and te particles exert mutually by te action reaction principle. We sow tat solutions exist globally in time under reasonable pysical assumptions on te initial data, te pysical domain, and te external potential. Furtermore, we prove te large-time stabilization of te system towards a unique stationary state fully determined by te masses of te initial density of particles and fluid and te external potential. 211 Elsevier Ltd. All rigts reserved. 1. Introduction Fluid particle interactions arise in many practical applications in biotecnology, medicine [1], reactive gases formed by fuel droplets in combustion [2 5], recycling and mineral processes [6,7], and atmosperic pollution [8]. Aerosols and sprays can be modelled by fluid particle type interactions in wic bubbles of suspended substances are seen as solid particles. Two-pase flow ydrodynamic models ave also been proposed [9]. Here, we focus on a particular system derived by formal asymptotics from a mesoscopic description. Tis is based on a kinetic equation for te particle distribution of Fokker Planck type coupled to fluid equations. Different macroscopic equations can be obtained as scaling limits; see [1] for a complete description of te modelling issues. In tese models, te fluid is eiter incompressible [11,12] or compressible [1]. Te coupling between te kinetic and te fluid equations is obtained troug te friction forces tat te fluid and te particles exert mutually. Te friction force is assumed to follow te Stokes law and tus is proportional to te relative velocity vector, i.e., is proportional to te fluctuations of te microscopic velocity ξ R 3 around te fluid velocity field u. More precisely, te cloud of particles is described by its distribution function f ε (t, x, ξ) on pase space, wic is te solution to te dimensionless Vlasov Fokker Planck equation t f ε + 1 ξ x f ε x Φ ξ f ε = 1 ξ ε ε div ξ εuε f + ξ f ε. (1.1) Corresponding autor. addresses: carrillo@mat.uab.es (J.A. Carrillo), karper@mat.ntnu.no (T. Karper), trivisa@mat.umd.edu (K. Trivisa) X/$ see front matter 211 Elsevier Ltd. All rigts reserved. doi:1.116/j.na

2 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Here, ϵ > is a dimensionless parameter and we ave a drag force independent of te fluid density ϱ ε, but proportional to te relative velocity of te fluid and te particles given by ξ u ε (t, x). Te rigt-and side of te moment equation in te Navier Stokes system takes into account te action of te cloud of particles on te fluid troug te forcing term ξ F ε = ε u ε(t, x) f (t, x, ξ) dξ. R 3 Te density of te particles η ε (t, x) is related to te probability distribution function f ε (t, x, ξ) troug te relation η ε (t, x) = f ε (t, x, ξ) dξ. R 3 Te well-posedness of tis kinetic yperbolic coupled fluid particle system as been addressed in [13] in te case of compressible models for te fluid equations. Tere are two different scaling limits for tis model, te so-called bubbling and flowing regimes. Tey correspond to te diffusive approximation of te kinetic equation, te bubbling regime, written in (1.1), and te strong drag force and strong Brownian motion for te flowing regime. Tis last regime as been studied in [14] were it is obtained rigorously as te limit from te mesoscopic description for local-in-time solutions and initial data bounded away from zero for te densities. We also refer te reader to [15] for asymptotic preserving numerical scemes in relation to tese scaling limits. In all te above mentioned studies, te viscosity of te fluid was neglected altoug it is te source of te drag forces. Te viscosity is present in te dimensionless systems altoug negligible, as noted in [1, Remark 3]. In tis work, we will deal wit te resulting formal macroscopic fluid particle system obtained troug te scaling limit in (1.1) as ε by te standard Hilbert-expansion procedure. In tis scaling limit, particles are supposed to ave a negligible density wit respect to te fluid, and tus, due to buoyancy effects, tey will typically move upwards in a system under gravity, from were we get te name of bubbling. Tis situation is typically complemented wit no-flux boundary conditions in a bounded domain. More generally, we can ask ourselves under wat conditions on te external potential in unbounded domains we can assert convergence towards stationary integrable states. We refer te reader to [1,15] for a detailed account of te pysical meaning and validity of te scaling limits. Summarizing, te state of suc flows in tis macroscopic description is, in general, caracterized by te variables: te total mass density ϱ(t, x), te velocity field u(t, x), as well as te density of particles in te mixture η(t, x), depending on te time t (, T) and te Eulerian spatial coordinate x R 3. In tis section, we present te primitive conservation equations governing fluid particle flows in te bubbling regime. Tese equations express te conservation of mass, te balance of momentum, and te balance of particle densities often referred as te Smolucowski equation: t ϱ + div x (ϱu) =, t (ϱu) + div x (ϱu u) + x (p(ϱ) + η) µu λ x div x u = (η + βϱ) x Φ, (1.3) t η + div x (η(u x Φ)) η =. Here, p denotes te pressure p(ϱ) = aϱ γ, a >, γ > 1, β, and Φ denotes te external potential (typically incorporating gravity and boyancy). In tis paper, we require te potential to satisfy suitable confinement conditions (HC) (see Section 2), wic does not limit te pysical relevance of our results. Te viscosity parameters µ > and λ + 2 µ are non-negative constants and 3 β > if is unbounded. Anoter macroscopic effect is tat te total pressure function in te momentum equation depends on bot te particle and te fluid densities p(ϱ) + η. We impose te no-slip boundary condition for te velocity vector leading to a no-flux condition for te fluid density troug te boundaries and te no-flux condition for te particle density u = x η ν + η x Φ ν = on (, T), (1.5) wit ν denoting te outer normal vector for te boundary. Our problem is supplemented wit te initial data {ϱ, m, η } suc tat ϱ(, x) = ϱ L γ () L 1 + (), (ϱu)(, x) = m L 6 5 () L 1 (), η(, x) = η L 2 () L 1 + (). Motivated by te stability arguments in [1], te numerical investigation presented in [15], as well as a number of studies on numerical experiments and scale analysis on te proposed model (see [16]), we investigate te issues of global existence and asymptotic analysis for fluid particle interaction flows, providing a rigorous matematical teory based on te principle of balance laws. Te total energy of te mixture is given by E(η, ϱ, u)(t) := [ 1 2 ϱ(t) u(t) 2 + a γ 1 ϱγ (t) + (η log η)(t) + (βϱ + η)(t)φ (1.2) (1.4) (1.6) ] dx. (1.7)

3 278 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) At te formal level, te total energy can be viewed as a Lyapunov function satisfying te energy inequality de dt + µ x u 2 + λ div x u x η + η x Φ 2 dx. (1.8) Terefore, it is reasonable to anticipate tat, at least for some sequences t n, η(t n ) η s, ϱ(t n ) ϱ s, ϱu(t n ), were η s, ϱ s satisfy te stationary problem x (p(ϱ s ) + η s ) = (η s + βϱ s ) x Φ on. Te energy estimate written in te form E(t) + T now implies tat ( x u 2 L 2 + div x u 2 L 2 )dt + T 2 x η + η x Φ 2 dxdt E(), 2 x ηs + η s x Φ 2 =. (1.9) Te aim of tis paper is to sow tat, in fact, any weak solution converges to a fixed stationary state as time goes to infinity, or more precisely, ϱ(t) ϱ s strongly in L γ (), ess sup ϱ(τ) u(τ) 2 dx, τ>t η(t) η s strongly in L p (), as t under te confinement ypotesis on te domain and te external potential Φ given in (HC) (cf. Definition 2.1). Indeed, it can be sown tat te sequences (ϱ n, η n ) of te time sifts defined as ϱ n (t, x) := ϱ(t + τ n, x), τ n, η n (t, x) := η(t + τ n, x), τ n, contain subsequences, denoted by te same index w.l.o.g., suc tat and ϱ n ϱ s strongly in L 1 loc ((, 1) ), η n η s strongly in L p 1 ((, T); L p 2 )() for some p 1, p 2 > 1, were (ϱ s, η s ) solve te stationary problem x p(ϱ s ) = βϱ s x Φ, x η s = η s x Φ. It is wort noting tat te confinement ypotesis is bot necessary and sufficient for te stationary problem (1.1) to admit a unique solution (ϱ s, η s ) (Section 4). Te main ingredients of our approac can be formulated as follows: A suitable weak formulation of te underlying pysical principles is introduced. Pysically grounded ypoteses are imposed on te system as follows. Te mixture occupies te pysical space R 3. Te boundary conditions are cosen in suc a way tat te dissipation of energy is guaranteed, wereas te pressure of te mixture takes into account bot te density of te fluid and te density of particles. A priori estimates are establised, based solely on te boundedness of te initial energy and entropy of te system. A suitable approximating sceme is introduced for te construction of te solution based on a two-level approximating procedure: te first level involves an artificial pressure approximation, wereas te second-level approximation employs a time discretization sceme. Te sequence of approximate solutions is constructed wit te aid of a fixed point argument coupling te time discretized compressible isentropic Navier Stokes equations to a discretization in time of te equation for η. Pysically grounded ypoteses are imposed on te domain and te external potential Φ (confinement ypoteses (HC)). Te analysis in te present article treats bot te case of a bounded pysical domain and te case of an unbounded domain. We remark tat te confinement ypotesis (HC) on (, Φ) plays a crucial role in providing control of te negative contribution of te pysical entropy η log η in te free energy bounds for unbounded domains. (1.1)

4 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Hig integrability properties for te density need to be establised for te limit passage in te family of approximate solutions and in particular in taking te vanising artificial pressure limit. We remark tat in te present context, te potential Φ is not integrable on unbounded domains. To deal wit tis new difficulty we employ te Fourier multipliers in te spirit of [17,18], wile taking into consideration te new features of our problem. We remark tat bot te total fluid mass and te total particle mass are constants of motion. In particular, we are able to conserve te total masses also in te large-time limit allowing us to uniquely determine te long-time asymptotics (cf [19]). Te paper is organized as follows. In Section 2 we collect all te necessary ypoteses imposed on te external potential (confinement ypoteses (HC)), and we present te notion of free energy solutions and te main results of tis article. Section 3 is devoted to te proof of te global existence of weak solutions (Teorem 2.1). First, a suitable approximation sceme based on an artificial pressure approximation and on a time discretization sceme is introduced. Te remaining section is devoted to te limit passage in te family of approximate solutions. Te most delicate part of te analysis concerned wit te vanising artificial pressure limit is presented in Section 3.5. Te large-time asymptotic analysis is described in Section Free energy solutions and main results In tis work, we analyse te existence and large-time asymptotics of certain kinds of weak solutions to te two-pase flow problems (1.2) (1.4) coupled wit no-flux boundary conditions (1.5) under two different geometrical constraints of interest in te applications: for bounded domains and for unbounded domains under confinement conditions due to te external potential. We will collect all assumptions concerning te geometry and te external potential Φ under te generic name of confinement conditions. Let us remark tat te external potential Φ is always defined up to a constant. Terefore, for external potentials Φ bounded from below, we can always assume witout loss of generality, by adding a suitable constant, tat inf Φ(x) =. x (2.1) Definition 2.1. Given a domain C 2,ν, ν >, R 3, and given an external potential Φ : R + bounded from below, satisfying (2.1), we will say tat (, Φ) verifies te confinement ypoteses (HC) for te two-pase flow system (1.2) (1.4) coupled wit no-flux boundary conditions (1.5) wenever: (HC Bounded) If is bounded, Φ is bounded and Lipscitz continuous in and te sub-level sets [Φ < k] are connected in for any k >. (HC Unbounded) If is unbounded, we assume tat Φ W 1, loc (), β >, te sub-level sets [Φ < k] are connected in for any k >, and e Φ/2 L 1 (), Φ(x) c 1 x Φ(x) c 2 Φ(x), x > R, (2.2) for some large R >. Remark 2.1. Te confinement assumption (HC) as pysical relevance in our setting as it is verified for several domains wit Φ being te gravitational potential. For instance, 1. wen = {x R 3 (x 1, x 2 ) [a, b] 2, x 3 [, H]} and Φ(x) = gx 3, were β = 1 ϱ F ϱ P, 2. wen = {x R 3 (x 1, x 2 ) [a, b] 2, x 3 > } and Φ(x) = gx 3, were β = 1 ϱ F ϱ P and ϱ F < ϱ P, 3. wen = R 3 \ B(, R) and Φ(x) = g x, were B(, R) is te ball centred at te origin wit radius R and β >. Here, ϱ F and ϱ P are te typical mass densities of fluid and particles, respectively. Remark tat 1. corresponds to te standard bubbling case (see [1]) in wic particles move upwards due to buoyancy. Let us now specify te kinds of weak solutions for te two-pase flow system (1.2) (1.4) tat we will be dealing wit. Definition 2.2. Let us assume tat (, Φ) satisfy te confinement ypoteses (HC); we say tat {ϱ, u, η} is a free energy solution of problem (1.2) (1.4) supplemented wit boundary data for wic (1.5) olds and initial data {ϱ, m, η } satisfying (1.6) provided tat te following old: ϱ represents a renormalized solution of Eq. (1.2) on (, ) : for any test function ϕ D([, T) ), any T >, and any b suc tat b L C[, ), B(ϱ) = B(1) + ϱ 1 b(z) dz, z 2

5 2782 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) te following integral identity olds: B(ϱ) t ϕ + B(ϱ)u x ϕ b(ϱ)div x uϕ Te balance of momentum olds in a distributional sense, namely ϱu t v + ϱu u : x v + (p(ϱ) + η) div x v dx dt = dx dt = B(ϱ )ϕ(, ) dx. (2.3) µ x u x v + λdiv x udiv x v (η + βϱ) x Φ v dx dt m v(, ) dx, (2.4) for any test function v D([, T); D(; R 3 )) and any T > satisfying ϕ =. All quantities appearing in (2.4) are supposed to be at least integrable. In particular, te velocity field u belongs to te space L 2 (, T; W 1,2 (; R 3 )); terefore it is legitimate to require u to satisfy te boundary conditions (1.5) in te sense of traces. η is a weak solution of (1.4). Tat is, te integral identity η t ϕ + ηu x ϕ η x Φ x ϕ x η x ϕ dxdt = η ϕ(, ) dx (2.5) is satisfied for test functions ϕ D([, T) ) and any T >. All quantities appearing in (2.5) must be at least integrable on (, T). In particular, η belongs L 2 ([, T]; L 3 ()) L 1 (, T; W 1, 3 2 ()). Given te total free energy of te system 1 E(ϱ, u, η)(t) := 2 ϱ u 2 + a γ 1 ϱγ + η log η + (βϱ + η)φ dx, ten E(ϱ, u, η)(t) is finite and bounded by te initial energy of te system, i.e., E(ϱ, u, η)(t) E(ϱ, u, η ) a.e. t >. Moreover, te following free energy dissipation inequality olds: µ x u 2 + λ div x u x η + η x Φ 2 dx dt E(ϱ, u, η ). (2.6) Now, we ave all te ingredients to state te main results of tis work. Teorem 2.1 (Global Existence). Let us assume tat (, Φ) satisfy te confinement ypoteses (HC). Ten, te problem (1.2) (1.4) supplemented wit boundary conditions (1.5) and initial data satisfying (1.6) admits a weak solution {ϱ, u, η} on (, ) in te sense of Definition 2.2. In addition, (i) te total fluid mass and particle mass given by M ϱ (t) = ϱ(t, )dx and M η (t) = η(t, )dx, respectively, are constants of motion. (ii) te density satisfies te iger integrability result ϱ L γ +Θ ((, T) ), for any T >, were Θ = min 2 3 γ 1, 1 4. We can now completely caracterize te large-time beaviour of free energy solutions to (1.2) (1.7). Teorem 2.2 (Large-Time Asymptotics). Let us assume tat (, Φ) satisfy te confinement ypoteses (HC). Ten, for any free energy solution (ϱ, u, η) of te problem (1.2) (1.4), in te sense of Definition 2.2, tere exist universal stationary states ϱ s (x), η s (x), suc tat ϱ(t) ϱ s strongly in L γ (), ess sup ϱ(τ) u(τ) 2 dx, τ>t η(t) η s strongly in L p 2 () for, p 2 > 1, as t, were (η s, ϱ s ) are caracterized as te unique free energy solution of te stationary state problem: x p(ϱ s ) = βϱ s x Φ, ϱ x η s = η s x Φ, s dx = ϱ dx, η s dx = η dx (2.7)

6 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) given by te formulas 1 γ 1 + γ 1 ϱ s = βφ + Cϱ, aγ ηs = C η exp( Φ), were C η and C ϱ are uniquely given by te initial masses due to (2.7). 3. Global-in-time existence Tis section is devoted to te proof of te existence result (Teorem 2.1) wic will be acieved by patcing local-in-time solutions wit te aid of suitable global estimates. Lemma 3.1. Te conclusion of Teorem 2.1 olds true on any time space cylinder [, T), were T > is any given finite time. Taking Lemma 3.1 for granted, a weak solution verifying Teorem 2.1 can be constructed as follows. Proof of Teorem 2.1. Fix any T >. From Lemma 3.1, we ave te existence of a weak solution (ϱ 1, u 1, η 1 ) on [, 2T ). To proceed, we will need te function 1, t [, T ϵ], ζ ϵ (t) = dist(t, T ), t (T ϵ, T ),, oterwise. By setting ϕζ ϵ as a test function in te continuity formulation (2.3), we obtain T T B(ϱ 1 )ϕϵ 1 dxdt + ζ ϵ B(ϱ 1 ) t ϕ + ζ ϵ B(ϱ 1 )u 1 x ϕ ζ ϵ b(ϱ 1 )div x u 1 ϕ dx dt T ϵ = B(ϱ )ϕ(, )ζ ϵ () dx, ϕ C c ([, T] ). Sending ϵ in tis formulation yields T B(ϱ 1 ) t ϕ + B(ϱ 1 )u 1 x ϕ b(ϱ 1 )div x u 1 ϕ dx dt = B(ϱ(T, ))ϕ(t, ) dxdt B(ϱ )ϕ(, ) dx, ϱ C c ([, T] ). Hence, (ϱ 1, u 1 ) is a renormalized solution of te continuity equation on te closed interval [, T ] wit additional boundary terms at t = T. By applying similar arguments to te momentum equation (2.4) and particle density equation (2.5), we conclude tat (ϱ 1, u 1, η 1 ) is a weak solution on [, T ] (wit additional boundary terms at t = T ). By te various uniform-in-time bounds (cf. (2.6)), (ϱ 1, u 1, η 1 ) at time T as sufficient integrability to serve as initial data for a new solution (ϱ 2, u 2, η 2 ) defined on [T, 2T ]. Te energy of te new solution is less tan te initial energy E(ϱ, u, η ) and te triple (ϱ, u, η) given by (ϱ, u, η)(t) = (ϱ i, u i, η i ), t ((i 1)T, it ], is a weak solution on [, 2T ]. A solution for all times is readily obtained by iterating tis process. In te remaining parts of tis section we prove Lemma 3.1. Te overall strategy can be outlined as follows. First, we prove tat tere exists a sequence of approximate solutions {ϱ δ,, u δ,, η δ, } > (see te ensuing subsection for details). Ten, by first sending and subsequently δ in tis sequence, we prove tat in te limit we obtain a weak solution to te system (1.2) (1.4) in te sense of Definition Te approximation sceme Te approximation sceme is realized in two layers. In te first layer we add a regularization term to te pressure. Definition 3.1 (Artificial Pressure Approximation). For a given δ >, we say tat te triple {ϱ δ, u δ, η δ } is a weak solution to te artificial pressure approximation sceme provided tat {ϱ δ, u δ, η δ } is a weak solution in te sense of Definition 2.2 but wit te pressure p δ (ϱ δ ) given by p δ (ϱ δ ) = p(ϱ δ ) + δϱ 6 δ, and initial data (ϱ δ,, m,δ, η δ, ) = (ϱ, m, η ) κ δ, wit κ δ denoting te standard smooting kernel.

7 2784 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Te second layer of approximation is a discretization of te equations in time. To define tis layer of approximation, we sall make use of te space W() = L 1 () L 6 () W 1,2 () W 1, 3 2 () L 3 () L 1 (). Definition 3.2 (Time Discretization Sceme). Let δ > be fixed. Given a time step >, we discretize te time interval [, T] in terms of te points t k = k, k =,..., M, were we assume tat M = T. Now, we sequentially determine functions suc tat: {ϱ k, δ, uk, δ, ηk δ, } W(), k = 1,..., M, Te time discretized continuity equation, d t [ϱk δ, ] + div x(ϱ k δ, uk δ, ) =, (3.1) olds in te sense of distributions on. Te time discretized momentum equation wit artificial pressure, d t [ϱk δ, uk ] + δ, div x(ϱ k δ, uk δ, uk ) δ, µuk λ δ, xdiv x u k + δ, x pδ (ϱ k ) + δ, δ, ηk = (βϱ k δ, + ηk δ, ) xφ, olds in te sense of distributions on. Te time discretized particle density equation, (3.2) d t [ηk ] + δ, div x η k δ, (u k δ, xφ) η k δ, =, (3.3) olds in te sense of distributions on. In te above equations, d t [φk ] = φk φ k 1 denotes implicit time stepping. For eac fixed >, te time discretized solution {ϱ k, δ, uk, δ, ηk δ, }M k=1 is extended to te wole of (, T) by setting (ϱ δ,, u δ,, η δ, )(t) = (ϱ k δ,, uk δ,, ηk δ, ), t (tk 1, t k ], k = 1,..., M. (3.4) In addition, we set te initial data ϱδ, (), (ϱ δ, u δ, )(), η δ, () = (ϱ δ,, m δ,, η δ, ). Te next two subsections treat te existence of a well-defined approximation sceme in bot relevant cases, bounded or not, under te confinement conditions (HC) Well-defined approximations: bounded To prove te existence of a solution to te time discretized approximate equations (3.1) (3.3) we will utilize a fixed point argument. For tis purpose we define te operator T [ ] : W() W(), as te solution (ϱ, u, η) = T [ψ, z, ζ ] to te system of equations and ϱ ϱ k 1 + div x (ϱu) =, ϱu ϱ k 1 u k 1 + div x (ϱu u) µu λ x div x u = x (p δ (ϱ) + ζ ) (βϱ + ζ ) x Φ, (3.6) η η k 1 + div x (η(z x Φ)) η =, (3.7) in te sense of distributions on, were (ϱ k 1, u k 1, η k 1 ) W() is given. Observe tat for a fixed >, δ >, and k = 1,..., M, any fixed point of te operator T [ ] will be a distributional solution to te time discretized approximate equations (3.1) (3.3). i.e. te fixed point (ϱ k, δ, uk, δ, ηk ) = δ, T ϱ k δ,, u k, δ, δ, ηk is precisely te desired solution at eac time step. (3.5)

8 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) T is well-defined Te existence of functions (ϱ, u, η) W() satisfying (ϱ, u, η) = T [ψ, z, ζ ] follows from Lemmas 3.2 and 3.4. Lemma 3.2 ([18], Teorem 6.1). Let > be fixed and arbitrary, and assume tat 1 ϱk 1 L γ () L 6 (), 1 mk 1 L 6 5 () L p () for some p > 1, ζ W 1, 3 2 (), and x Φ L p (), for all p. Tere exists a weak solution (ϱ, u) L γ () L 6 () W 1,2 () of (3.5) (3.6) satisfying p δ (ϱ) L 2 (), (3.8) and x curl x u, x div x u 1 λ + µ p δ(ϱ), L q (K), (3.9) were q = 15 > 6 and K is any compact subset Notation. Trougout te paper we use overbars to denote weak limits; just to illustrate tis notational use, we will often use convexity arguments based on te following classical lemma [17]: Lemma 3.3. Let O be a bounded open subset of R M wit M 1. Suppose g: R (, ] is a lower semicontinuous convex function and {v n } n 1 is a sequence of functions on O for wic v n v in L 1 (O), g(v n ) L 1 (O) for eac n, g(v n ) g(v) in L 1 (O). Ten g(v) g(v) a.e. on O, g(v) L 1 (O), and g(v) O dy lim inf n g(v O n) dy. If, in addition, g is strictly convex on an open interval (a, b) R and g(v) = g(v) a.e. on O, ten, passing to a subsequence if necessary, v n (y) v(y) for a.e. y {y O v(y) (a, b)}. Lemma 3.4. Assume tat is bounded. Let η k 1 L 1 () W 1, 3 2 () {η k 1 }, and z W 1,2 () be given functions. Ten, for eac fixed >, tere exists a non-negative function η W 1, 3 2 () L 1 () satisfying (3.7) in te sense of distributions on. Moreover, 1 [η log η + ηφ] + 2 x η + η x Φ 2 dx C, (3.1) were te constant C > depends on z W 1,2 (), η k 1 L 3 (), and Φ W 1, (). Proof. For eac ϵ >, we let z ϵ = z κ ϵ, were is te convolution product and κ ϵ is te standard smooting kernel. From [2, Proposition 4.29], we can assert te existence of a unique weak solution η ϵ W 2,2 (), η ϵ to te linear elliptic equation η ϵ η k 1 + div x (η ϵ (z ϵ x Φ)) η ϵ = in, (3.11) satisfying te boundary condition η ϵ x Φ ν + x η ϵ ν =. By integrating over (using te boundary condition), we observe tat η ϵ dx = η k 1 dx C, and ence η ϵ L 1 () independently of ϵ. Let te sequence {B l } l=1 be given by log y, y > l B l (y) = 1, log l 1, y l 1. Moreover, let l = {x : η ϵ (x) > l 1 } and c l = \ l. Multiply (3.11) wit B l (η ϵ ) and integrate by parts to obtain 1 η ϵ log η ϵ η k 1 log η ϵ dx 1 η ϵ log l η k 1 log l dx l c l + x Φ x η ϵ + 1 l η xη ϵ 2 ϵ z ϵ x η ϵ dx =. (3.12)

9 2786 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Applying te identity 1 η ϵ x η ϵ 2 = 4 x ηϵ 2 and reordering terms in (3.12), 1 η ϵ log η ϵ + 4 x ηϵ 2 dx = 1 η ϵ log l η k 1 log l dx + 1 η k 1 log η ϵ dx l c l l 2 η ϵ x Φ x ηϵ 2 η ϵ z ϵ x η ϵ dx l 1 log l 1 l dx + 1 η k 1 η ϵ dx l c l + 2 x ηϵ 2 L 2 ( l ) ηϵ x Φ L 2 ( l ) + η ϵ z ϵ L 2 ( l ) 1 log l 1 l c l + l 1 η k 1 η ϵ dx + 2 η ϵ 2 L 2 ( l ) + η ϵ z ϵ 2 L 2 ( l ) + η ϵ x Φ L 2 ( l ), were te last inequality is te Caucy inequality. Te last term is bounded since η ϵ, η k 1 L 1 () and Φ W 1, (), independently of ϵ and l. Using te Sobolev embedding W 1,2 () L 6 () togeter wit η k 1 L 3 (), we acieve 1 η ϵ log η ϵ + 2 x ηϵ 2 dx C + η ϵ 1 η k 1 + z ϵ 2 dx C(1 + η ϵ 3 ). (3.13) l L 2 ( l l ) Applying te Hölder inequality, Sobolev embedding, and Young s inequality (wit epsilon), respectively, we bound te last term as follows: η ϵ 3 2 ( l ) ηϵ 1 3 η ϵ 4 3 C L 1 () L 4 ( l ) x ηϵ 4 3 L 2 ( l ) C + β x ηϵ 2 L 2 ( l ). L Inserting tis expression in (3.13) and fixing β small, we gater l 1 η ϵ log η ϵ + x ηϵ 2 dx C, (3.14) were te constant C is independent of bot ϵ and l. Since is bounded, it follows tat η ϵ log η ϵ L 1 (), η ϵ W 1, 3 2 (), independently of ϵ. Since W 1, 3 2 () is compactly embedded in L p () for all p < 3, we can conclude tat, passing to a subsequence if necessary, η ϵ η in W 1, 3 2 (), η ϵ η in L p (), p < 3, as ϵ. Consequently, we can take te limit ϵ in (3.11) to conclude tat η satisfies (3.7) in te sense of distributions on. Next, since η ϵ η in L p (), p < 3, we can also conclude tat ηϵ η in L q (), q < 6, x ηϵ x η in L 2 (), as ϵ. Using tis, we can send ϵ in (3.14) to obtain 1 η log η + x η 2 dx C. By weak lower semicontinuity, x η 2 L 2 () x η 2 L 2 (). Tus, 1 η log η + x η 2 C. Ten, using te identity 2 x Φ x η ϵ + η ϵ x Φ x ηϵ 2 = 2 x ηϵ + η ϵ x Φ 2,

10 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) we are led to te conclusion tat 1 [η log η + ηφ] + 2 x ηϵ + η ϵ x Φ 2 dx C + 1 ηφ + 4 x η η x Φ + η x Φ 2 dx C 1 + η L 1 () Φ L () + x Φ 2 L () + xφ L () x η L () 2 C, wic is (3.1) and te proof is complete. Remark 3.1. Since x η L2 (), te set of particle vacuum regions ave measure zero; {x ; η = } = T admits a fixed point We now prove tat T [ ] admits a fixed point and consequently tat te time discretization sceme in Definition 3.2 is well-defined. Te key observation made is tat te L 2 bound on te pressure enables us to obtain an energy equality. Tis equality in turn yields compactness of te operator T [ ]. Lemma 3.5. Assume te case of bounded domain. Let {ϱ k 1, m k 1, η k 1 } L 2 () L 6 5 () L p () L γ (), p > 1, be given functions. Ten, for eac fixed > tere exists a fixed point {ϱ, u, η} W() for te operator T [ ]; i.e. (ϱ, u, η) = T [ϱ, u, η]. As a consequence, te time discretization sceme given by Definition 3.2 is well-defined for bounded domains. Proof. We will prove te existence of a fixed point by verifying te postulates of te Scauder corollary to te Scaefer fixed point teorem [21]; if te operator T [ ] is continuous, compact, and te set {s [, 1], x W() : x = T [sx]} is uniformly bounded, ten te operator T [ ] admits a fixed point. First, we observe tat te operator T [ ] is clearly bounded and continuous. Next, we prove tat te operator T [ ] is compact. For tis purpose, let {, z n, η n } n=1 be a sequence suc tat (z n, η n ) b W 1,2 () W 1, 3 2 () for all n = 1,...,, and construct a sequence {ϱ n, u n, η n } n=1 by setting (ϱ n, u n, η n ) = T [, z n, ζ n ], n = 1,...,. Ten, from te previous lemmas we know tat (ϱ n, u n, η n ) W() independently of n. Hence, we ave te existence of functions (ϱ, u, η) W() suc tat, by passing along a subsequence if necessary, (ϱ n, u n, η n ) (ϱ, u, η), in W(). Moreover, by compact Sobolev embedding, we clearly ave te existence of (, z, η) W() suc tat ζ n ζ a.e. in and z n z a.e. in. Now, we claim tat in fact (ϱ n, u n, η n ) (ϱ, u, η) in W(), were (ϱ, u, η) = T [, z, ζ ], and consequently tat te operator T [ ] is compact. To prove tis claim, we first note tat compactness of η n in W 1, 3 2 () is immediate from te linearity of (3.3). Tat is, since η n W 1, 3 2 () we ave by Sobolev embedding tat η n η a.e. in and tus by setting log η log η n (cf. Remark 3.1) as a test function in (3.7) one discovers lim x ηn 2 x η 2 dx =, n wic immediately implies compactness in W 1, 3 2 (). Moreover, in te limit we ave tat η satisfies η η k 1 + div x (η(z x Φ)) η =, (3.15) in te sense of distributions on. We continue proving compactness of te operator T [ ] by first proving strong convergence of te density ϱ n ϱ a.e. in. Compactness of te velocity u n in W 1,2 () will ten follow from an energy equality. In order to prove strong convergence of te density we will need weak sequential continuity of te effective viscous flux. Tat is, lim n [(λ + µ)div x u n p δ (ϱ n )] ψϱ n dx = (λ + µ)div x u p δ (ϱ) ψϱ dx, ψ C c (). (3.16) Here, (3.16) is an immediate consequence of (3.9) and compact embedding of Sobolev spaces.

11 2788 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Before we proceed to prove strong convergence of ϱ n, we first note tat tere is no problem wit taking te limit in (3.1) to obtain in te limit 1 ϱ + div x(uϱ) = 1 ϱ k 1, in te sense of distributions on. Hence, since in particular ϱ L 6 () and u W 1,2 () we can conclude tat, for any B C[, ) C 1 (, ), 1 (3.17) ϱb (ϱ) + div x (B(ϱ)u) + ((ϱb (ϱ) B(ϱ))div x u) = 1 ϱ k 1B (ϱ), (3.18) in te sense of distributions on. Ten, by setting B(z) = z log z we obtain te identity ϱ n log ϱ n ϱ log ϱ + ϱ n ϱ dx = ϱdiv x u ϱ n div x u n + ϱ k 1 (log ϱ n log ϱ) dx. Tus, by taking te limit n, we see tat ϱ log ϱ ϱ log ϱ dx = lim lim ψ l ϱ div x u ψ l ϱ n div x u n dx + ϱ k 1 (log ϱ log ϱ) dx, (3.19) l n were ψ l C c () {ψ l } is suc tat ψ l = 1 on te set x ; dist(x, ) > 1 l. By an application of (3.16) we see tat for eac l = 1..., lim ψ l ϱ div x u ψ l ϱ n div x u n dx = lim ψ l (p δ (ϱ n )ϱ p δ (ϱ n )ϱ n ) dx, n n were te last inequality follows from te convexity of p δ (ϱ n ). Similarly, te concavity of log ϱ n yields ϱ k 1 (log ϱ log ϱ) dx. However, ten (3.19) togeter wit te convexity of z log z gives ϱ log ϱ ϱ log ϱdx wic immediately yields ϱ n ϱ a.e. in due to Lemma 3.3. We are now in a position to prove tat u n u in W 1,2 (). For tis purpose, we first set u n as a test function in (3.2) to obtain te identity ϱ n u n 2 + ϱ k 1 u n 2 lim u nm k 1 + µ x u n 2 + λ div x u n 2 dx n 2 = lim (p δ (ϱ n ) + ζ n ) div x u n (ζ n + βϱ n ) x Φ u n dx n = (p δ (ϱ) + ζ ) div x u (ζ + βϱ) x Φ u dx. (3.2) Note tat in addition to te strong convergence of η n and ϱ n we really need (3.8) to conclude tis. Now, since ϱ n ϱ, ζ n ζ, and u n u a.e. in tere is no problem wit passing to te limit in (3.2) to obtain ϱu m k 1 +div x (ϱu u) µu λ x div x u = x (p δ (ϱ) + ζ ) (βϱ + ζ ) x Φ, in te sense of distributions on. Hence, in view of (3.15) and (3.17), we can conclude tat (ϱ, u, η) = T [, z, ζ ]. Moreover, by using u as a test function for tis equation, we obtain te identity [ ϱu2 + ϱ k 1 u 2 um ] k 1 + µ x u 2 + λ div x u 2 dx = (p δ (ϱ) + ζ )div x u (ζ + βϱ) x Φ u dx 2 ϱu = 2 + ϱ k 1 u 2 um k 1 + µ x u 2 + λ div x u 2 dx, 2 were te last equality is (3.2). Tis can only appen if x u n x u. Tus, we can conclude tat T [ ] is a compact operator.

12 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Finally, let s [, 1] be arbitrary and assume tat tere exists a triple (ϱ, u, η) W() suc tat (ϱ, u, η) = T [sϱ, su, sη]. Ten, by setting u as a test function in (3.2) we get te identity ϱu 2 + ϱ k 1 u 2 um k 1 + µ x u 2 + λ div x u 2 dx = (p δ (ϱ) + sη)div x u (sη + βϱ) x Φ u dx. (3.21) 2 Using B(z) = Π δ (z) := 1 γ p(ϱ) + δ 1 5 ϱ6 as te renormalization function in (3.18) we also ave te identity p δ (ϱ)div x u dx = 1 Π δ (ϱ)(ϱ ϱ k 1) dx. (3.22) Similarly, using βφ as a test function in (3.5) gives β ϱ ϱ k 1 Φ βϱ u x Φdx =. Using Φ + log η as a test function in (3.7) (cf. Lemma 3.4) gives η η k 1 = (Φ + log η) sηu x Φ + sηdiv x u + 2 x η + η x Φ 2 dx. (3.23) By combining (3.22) (3.23), we obtain te identity p δ (ϱ)div x u + sηdiv x u (sη + βϱ) x Φ u dx = Π δ (ϱ) (ϱ ϱ k 1) η η k 1 (Φ + log η) β ϱ ϱ k 1 Φ 2 x η + η x Φ 2 dx. Ten, inserting tis into (3.21) and reordering terms gives ϱu 2 + ϱ k 1 u 2 um k 1 + ϱ ϱ k 1 Π δ (ϱ) + βφ + η η k 1 (Φ + log η)dx + µ x u 2 + λ div x u 2 dx 2 = 2 x η + η x Φ 2 dx. (3.24) Consequently, we can conclude tat η W 1, 3 2 () + ϱ L γ () + u W 1,2 () C, were te constant C depends on te data η k 1, ϱ k 1, m k 1, and Φ, togeter wit. However, C does not depend on te parameter s. We can now conclude te proof since we ave proved tat te operator T [ ] is bounded, continuous, compact, and tat te set {s [, 1], x W() : x = T [sx]} is uniformly bounded Well-defined approximations: unbounded At tis point we ave proved tat te approximation sceme is well-defined on bounded domains. Given an unbounded domain and an external potential Φ satisfying te assumptions (HC), we can always find an increasing sequence of domains r, wit r >, suc tat te r are bounded and ( r, Φ) satisfies (HC) approximating in te sense r> r =. Using te previous subsection, for any r >, tere is a solution on r. In tis subsection, we prove tat we can send r to obtain a solution in. Te following lemmas will be of use in te sequel Consequences of confinement: unbounded We sow in tis part ow to control te negative contribution of te pysical entropy η log η in te free energy bounds for unbounded domains. Here, te confinement conditions (HC) on (, Φ) are crucial. Most of tese lemmas can be seen in [22] but we include tem ere for te sake of completeness. We start wit a classical lemma from kinetic teory. Lemma 3.6. Assume tat (, Φ) satisfy te ypoteses (HC). For any density η L 1 + (), η(x) log η(x) dx 1 Φ(x)η(x) dx + 1 e Φ(x)/2 dx. 2 e Proof. Let η := η χ {η 1} and M = η(x) dx η(x) dx = M. Ten η(x) log η(x) + 12 Φ(x) dx = [U(x) log U(x)]µ dx M log Z

13 279 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) were U := η/µ, µ(x) = e Φ(x)/2 /Z wit Z = e Φ(x)/2 dx. Te Jensen inequality yields [U(x) log U(x)]µ dx U(x)µ dx log U(x)µ dx = M log M and 1 η(x) log η(x) dx = η(x) log η(x) dx M log M M log Z 2 Z e 1 Φ(x) η(x) dx, 2 from wic te desired claim follows. We can immediately use tis previous lemma to conclude te following consequence. Φ(x) η(x) dx Corollary 3.1. Assume tat (, Φ) satisfy te ypoteses (HC). For any density η L 1 + (), if η(x) log η(x) dx + Φ(x)η(x) dx C, ten η log η L 1 () and tere exists D > depending on C and Φ suc tat η(x) log + η(x) dx D and Φ(x)η(x) dx D. Finally, te above estimates can be used to control te mass of te densities η outside a large ball to avoid loss of mass at infinity. Lemma 3.7. Given any domain suc tat e Φ L 1 + () and any density η L1 + (), ten η(x) dx η(x) log η(x) dx + Φ(x)η(x) dx η(x) dx log. As a consequence, if e Φ L 1 + () and η(x) log η(x) dx + Φ(x)η(x) dx C, e Φ(x) dx ten for any ϵ > tere exists R > depending on C and Φ only suc tat η(x) dx < ϵ. (R 3 /B(,R)) Proof. A direct use of Jensen s inequality sows te first inequality by using te convexity of x x log x. To sow te second claim we start by applying te first inequality to te domain c := R (R3 /B(, R)) from wic we obtain η(x) η(x) c dx R dx log e Φ(x) D (3.25) dx c R c R for some D >, were Lemma 3.6 and Corollary 3.1 were used. Now, we argue by contradiction. If te second claim were not true, we would ave ϵ > R > R > R suc tat η(x) dx ϵ. Since e Φ L 1 + (), we can always assume tat R is large suc tat e Φ(x) dx e Φ(x) dx < ϵ η(x) dx c R c R and tus due to (3.25), η(x) dx c R c R c R c R e Φ(x) dx e D/ϵ e Φ(x) dx e D/ϵ. c R Tis leads to a contradiction since te rigt-and side can be made arbitrarily small by taking R large enoug.

14 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Te approximation sceme is well-defined Notation. In wat follows, we will often obtain a priori estimates for a sequence {v n } n 1 tat we write as v n b X for some functional space X. Wat tis really means is tat we ave a bound on v n X tat is independent of n. Lemma 3.8. Set (, Φ) satisfying (HC) wit unbounded and let (ϱ k 1, u k 1, η k 1 ) W() be given data. Te time discretization sceme (3.1) (3.3) admits a distributional solution in te sense of Definition 3.2. Proof. Let { r } r> be an increasing sequence of domains suc tat r> r = and suc tat, for eac fixed r, r is bounded and ( r, Φ) satisfies (HC). From te results of te previous subsection, we ave te existence of a triplet (ϱ r, u r, η r ) satisfying te time discretized equations (3.1) (3.3) in te sense of distributions on r. Consequently, we can define a family of suc solutions: {ϱ r, u r, η r } R<r<, (ϱ r, u r, η r ) W( r ), for eac fixed r (R, ), were R is fixed according to te requirements on te potential (see (2.2)). For tis construction, (3.24) yields 2 ϱ r u + r ϱk 1 u 2 r + ϱ r Π δ (ϱ r ) + βφ + η r r 2 (Φ + log η r)dx + µ x u r 2 + λ div x u r x ηr + η r x Φ 2 dx r u r m k η k 1 log η r dx r ϵ 1 [ η r 2 L 2 ( r ) + u 2 L 6 ( r ) ] + 1 4ϵ In addition, integrating (3.3) over r, η r L 1 ( r ) = η k 1 L 1 ( r ) C, [ η k 1 2 L 2 () + m k 1 6 L 5 () ]. (3.26) wit constant C independent of r. An interpolation inequality and te Sobolev Poincaré inequality allow us to conclude tat η r L 2 ( r ) η r 1 4 η L 1 ( r ) r 3 4 C L 3 ( r ) x ηr 3 4, L 2 ( r ) (3.27) were te constant C is independent of r. By applying (3.27) and te Sobolev Poincaré inequality to (3.26), we conclude tat r 2 ϱ r u + r ϱk 1 u 2 r 2 + ϱ r Π δ (ϱ r ) + βφ + η r (Φ + log η r)dx + (µ ϵ) x u r 2 + λ div x u r x ηr + η r x Φ 2 ϵ x ηr 2 dx C, (3.28) r were C is independent of r. Next, we observe tat (3.28) yields 4 x ηr x η r x Φ + η r x Φ 2 dx = 2 x ηr + η r x Φ 2 dx C. r r Reordering terms and integrating by parts, 4 x ηr 2 + η r x Φ 2 dxdt C + η r Φ dx. r r Ten, (2.2) and (3.28) gives η r Φ dx + η r Φ dx Φ L ( R ) η r dx + C η r Φ dx C, (3.29) R r \ R R r \ R wit C independent of r. Setting (3.29) into (3.28), fixing ϵ small, and applying Corollary 3.1 gives r 2 ϱ r u + r ϱk 1 u 2 r 2 + ϱ r Π δ (ϱ r ) + βφ + η r (Φ + log + η r )dx + µ x u r 2 + λ div x u r 2 + x ηr 2 + η r x Φ 2 dx C. r

15 2792 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Since η r 2 L 2 ( r ) = r η r dx C, te previous estimate allow us to conclude tat η r b L 3 ( r ) W 1, 3 2 (r ), u r b L 6 ( r ) W 1,2 ( r ), ϱ r b L 6 ( r ) L γ ( r ). Next, let (φ, v, ψ) [C c supp(φ, v, ψ) r. ()] 3 be arbitrary and fix a number r (R, ) large suc tat Ten, from te previous bounds, we ave te existence of functions (ϱ, u, η) W() suc tat, along a subsequence, (3.3) ϱ r ϱ, in L 6 ( r ), u r u, in W 1,2 ( r ), η r η, in W 1, 3 2 ( r ). Moreover, by compact Sobolev embedding, (3.31) η r η, in L 3 loc ( r), u r u, in L p loc ( r), p < 6. (3.32) Using (3.31) and (3.32), tere is no problem wit sending r in r d t [ϱ r]φ ϱ r u r x φ dx =, r d t [η r]ψ + (u r x Φ) x ψ x η r x ψ dxdt =, to conclude tat (ϱ, u, η) solves (3.1) (3.3) in te sense of distributions on (recall tat (φ, ψ) was cosen arbitrary). Similarly, we can pass to te limit in te momentum equation (3.2) to obtain d t [ϱu]v ϱu u : xv + µ x u x v + λdiv x udiv x v dx r + (βϱ + η) x Φv dx = lim p δ (ϱ r )div x v dx. r r r Hence, it remains to prove tat p δ (ϱ r ) p δ (ϱ) as r. Tis problem is also te core problem encountered in te existence analysis of Lions [18]. Due to te regularity properties of η r, te presence of te η variable does not impose any additional difficulties as compared wit [18]. However, some minor modifications are needed to incorporate te unbounded potential Φ. Te needed modifications for te present stationary case are almost identical to tose of te non-stationary case. Tus, we do not give te arguments ere but refer te reader to te more general arguments given in Section Tere exists an artificial pressure solution ( ) In te previous subsection we proved tat, for eac > and δ >, we can construct functions (ϱ δ,, u δ,, η δ, ) according to Definition 3.2 and (3.4). In tis section, we prove tat te corresponding sequence {(ϱ δ,, u δ,, η δ, )} >, wit δ > fixed, converges as te time step to an artificial pressure solution in te sense of Definition 3.1. Lemma 3.9 ([23, Corollary 4, p. 85]). Let X B Y be Banac spaces wit X B compactly. Ten, for 1 p <, {v : v L p (, T; X), v t L 1 (, T; Y)} is compactly embedded in L p (, T; B). Te following lemma is a variation of a result due to Lions (see [24]). Lemma 3.1. For a given T >, divide te time interval (, T) into M points suc tat (, T) = M k=1 (t k 1, t k ], were t k = k and we assume tat M = T. Let {f }, > {g } > be two sequences suc tat: {f } >, {g } > converge weakly to f, g respectively in L p 1 (, T; L q 1()), L p 2(, T; L q 2()) were 1 p1, q 1, 1 p p 2 = 1 q q 2 = 1. Te mapping t g (t, x) is constant on eac interval (t k 1, t k ], k = 1,..., M. Te discrete time derivative satisfies g (t, x) g (t, x) b L 1 (, T; W 1,1 ()). {f } > satisfies f n f n ( + ξ, t) L p 2 (,T;L q 2 ()), as ξ, uniformly in n. Ten, g f gf in te sense of distributions on (, T).

16 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Energy estimates Let δ > be fixed and let {ϱ δ,, u δ,, η δ, } > be a sequence of time discretized solutions constructed according to Definition 3.2 and (3.4). Since (3.24) olds for every k, we can sum tis equality over all k = 1,..., m, for any m [1, M], to obtain te energy equality t m t m E(ϱ δ,, u δ,, η δ, )(t m ) + µ x u δ, 2 + λ div x u δ, 2 dxdt + 2 x ηδ, + η δ, x Φ 2 dxdt + 1 m (γ 1)(ϱ k δ, γ 1 )γ + (ϱ k 1 δ, )γ γ (ϱ k 1 δ, )γ 1 ϱ k δ, dx + δ 5 m k=1 k=1 = E(ϱ δ,, u δ,, η δ, ), 5(ϱ k δ, )6 + (ϱ k 1 δ, )6 6(ϱ k 1 δ, )5 ϱ k δ, dx + m k=1 η k 1 log ηk 1 were t m = m (, T) and te energy E(,, ) is given by (2.6). By convexity, (γ 1)a γ + b γ γ a γ 1 b for all a, b, γ > 1. By concavity of z log z, ηk 1 η k 1 log η k 1 η k dx =. η k Due to te confinement conditions, tis, Corollary 3.1, and (3.33) allow us to conclude te bounds ϱ δ, u δ, 2 b L (, T; L 1 ()), u δ, b L 2 (, T; W 1,2 ()), ϱ δ, b L (, T; L γ () L 6 ()), η δ, log η δ, b L (, T; L 1 ()). By arguments similar to tose leading to (3.3), we conclude tat η k dx (3.33) (3.34) η δ, b L 2 (, T; L 3 ()) L 1 (, T; W 1, 3 2 ()), (3.35) independent of bot and δ. Utilizing te above -independent bounds, it is a simple exercise to obtain from te system (3.1) (3.3) te weak time difference bounds d t [ϱ δ,] b L (, T; W 1, 3 2 ()), d t [η δ,] b L 1 (, T; W 1, 3 2 ()), d t [ϱ δ,u δ, ] b L 1 (, T; W 1,1 ()). (3.36) Convergence In view of (3.34) and (3.35), we ave te existence of functions (ϱ δ, u δ, η δ ) suc tat, along a subsequence as, ϱ δ, ϱδ in L (, T; L γ ()), u δ, u δ in L 2 (, T; W 1,2 ()), η δ, η δ in L 2 (, T; L 3 ()). By virtue of (3.36), Lemma 3.1 can be applied to yield ϱ δ, u δ, ϱ δ u δ, ϱ δ, u δ, u δ, ϱ δ u δ u δ, (3.37) in te sense of distributions on (, T) as. Here, ϱ δ, u δ, u δ, ϱ δ u δ u δ follows from setting g = ϱ δ, u δ, and f = u,δ in Lemma 3.1, were g g = ϱ δ u δ from te result second to last in (3.37). Next, since η δ, b L 1 (, T; L 2 ()) L 1 (, T; W 3 2 ()), t η δ, b L 1 (, T; W 1,1 ()) and W 3 2 is compactly embedded in L 2, we can apply Lemma 3.9 to conclude tat η δ, η δ in L 2 ϵ (, T; L 2 ()), loc ηδ, η δ in L 2 (, T; L 2 loc ()). (3.38) Now, by a straigtforward application of te Hölder inequality, we deduce x η δ, b L 2 (, T; L 1 ()) L 1 (, T; L 3 2 ()). Te standard interpolation inequality provides an estimate of te form x η δ, b L α 1 (, T; L α 2()). Tus, in particular x η δ, x η δ, in te sense of distributions on (, T). (3.39)

17 2794 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Now, equipped wit (3.37), (3.39), and te bounds (3.34), tere is no problem wit taking te limit in (3.1) and (3.3) to discover tat T ϱ δ (ψ t u δ x ψ) dxdt = ϱ δ, ψ(, x) dx, (3.4) for all ψ C c ([, T) ) and T η δ (ψ t + (u δ x Φ) x ψ) x η δ x ψ dxdt = η δ, ψ(, x) dx (3.41) for all ψ C c ([, T) ). Hence, te limiting functions satisfy bot te continuity equation and te particle equation in te sense of Definition 3.1. Similarly, we can go to te limit in (3.2) to discover in te limit T ϱ δ u δ v t + ϱ δ u δ u δ : x v δ + η δ div x v dxdt T = µ x u δ x v + λdiv x u δ div x v + (βϱ δ + η δ ) x Φv dxdt T lim p δ (ϱ δ, )div x v dxdt m v(, x) dx, for all v C c ([, T) ). Tus, in order to conclude existence of an artificial pressure solution in te sense of Definition 3.1, we must prove tat in fact lim T p δ (ϱ δ, )div x v dxdt = T p δ (ϱ δ )div x v dxdt. (3.42) Lemma Fix any δ > and let {ϱ δ,, u δ,, η δ, } > be a sequence of time discretized solutions constructed according to Definition 3.2 and (3.4). Ten, tere exists a triple (ϱ δ, u δ, η δ ) suc tat as, ϱ δ, ϱ δ in L (, T; L γ () L 6 ()), ϱ δ, ϱ δ a.e. in (, T), u δ, u δ in L 2 (, T; W 1,2 ()), ϱ δ, u δ, ϱ δ u δ in te sense of distributions on (, T), η δ, η δ in L 1 (, T; W 1, 3 2 ()), and η δ, η δ a.e. in (, T), were (ϱ δ, u δ, η δ ) is a weak solution to te artificial pressure approximation sceme in te sense of Definition 3.1. Proof. In view of te ig integrability, and strong convergence properties, of η δ,, (3.42) can be proved by te same arguments as tose leading to Teorem 7.2 in [18]. Some minor modifications are needed to treat te unbounded potential Φ. Te arguments needed are identical to tose of Section and will not be given ere. From tis and te previous results of tis section we can conclude te existence of an artificial pressure solution. It remains to prove te energy inequality (1.8). We start wit te following calculation: t lim 4 x ηδ, x η δ, x Φ + η δ, x Φ 2 dxdt t = t η δ x Φ x η δ x Φ + 4 x ηδ 2 dxdt 2 x ηδ + η δ x Φ 2 dxdt, (3.43) were we ave used Lemma 3.3 and (3.39). By taking te limit in (3.33) (using convexity of z z γ and z z log z), we obtain t t E(ϱ δ, u δ, η δ )(t) + µ x u δ 2 + λ div x u δ 2 dxdt + 2 x ηδ + η δ x Φ 2 dxdt E(ϱ δ,, u δ,, η δ, ), (3.44) for any t (, T) Vanising artificial pressure limit (δ ) In te previous subsection we proved tat, for eac fixed δ >, tere exists an artificial pressure solution in te sense of Definition 3.1. Trougout tis section, we let {ϱ δ, u δ, η δ } δ> be a sequence of suc solutions. Te aim is now to prove tat tis sequence converges as δ to a weak solution of te fluid particle interaction model (1.2) (1.4) in te sense of Definition 2.2. Tis will ten conclude te proof of Lemma 3.1.

18 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Energy bounds Te energy inequality (1.8), togeter wit Sobolev embedding and Corollary 3.1, allow us to conclude te following δ- independent bounds: ϱ δ u δ 2 b L (, T; L 1 ()) L 2 (, T; L m 1 ()), m 1 > 1, u δ b L 2 (, T; W 1,2 ()), ϱ δ b L (, T; L γ ()), ϱ δ u δ b L, T; L 2γ γ +1 (), (3.45) η δ log η δ b L (, T; L 1 ()). By straigtforward applications of te Hölder inequality, using (3.45) and te Sobolev embedding, we deduce te bounds were ϱ δ u δ b L 2 (, T; L m 2 ()), m 2 = 6γ 6 + γ, c 2 = 3γ 3 + γ. From (3.35), we also ave tat η δ b L 2 (, T; L 3 ()) L 1 (, T; W 3 2 ()). ϱ δ u δ u δ b L 2 (, T; L c 2 ()), Using te above δ-independent estimates, we easily deduce te weak time control bounds t η δ b L 1 (, T; W 1, 3 2 ()), t ϱ δ b L 2γ 1, (, T; W γ +1 ()), t (ϱ δ u δ ) b L 1 (, T; W 1,1 ()) Convergence Te bounds in te previous subsection assert te existence of functions (ϱ, u, η) suc tat, passing to a subsequence if necessary, ϱ δ ϱδ in L (, T; L γ ()), u δ u δ in L 2 (, T; W 1,2 ()), η δ η δ in L 2 (, T; L 3 ()). As in te previous subsection, Lemma 3.1 can be applied to conclude ϱ δ u δ ϱu, ϱ δ u δ u δ ϱu u, (3.46) in te sense of distributions on (, T). By arguments similar to tose leading to (3.38), we deduce and η δ η in L 2 ϵ (, T; L 2 loc ()), ηδ η in L 2 (, T; L 2 loc ()) x η δ x η, in te sense of distributions on (, T). Using (3.46), te bounds (3.45), (3.47), and strong convergence of te initial conditions, we can take te limit δ in (3.4) and (3.41) to discover tat T ϱ(ψ t u x ψ) dxdt = ϱ ψ(, x) dx, for all ψ C c ([, T) ) and T η (ψ t + (u x Φ) x ψ) x η x ψ dxdt = η ψ(, x) dx for all ψ C c ([, T) ). Hence, te limiting functions satisfy bot te continuity equation and te particle equation in te sense of Definition 2.2. (3.47)

19 2796 J.A. Carrillo et al. / Nonlinear Analysis 74 (211) Similarly, we can go to te limit δ in (3.2) to discover in te limit T = T ϱuv t + ϱu u : x v δ + ηdiv x v dxdt T lim δ µ x u x v + λdiv x udiv x v + (βϱ + η) x Φv dxdt p(ϱ δ )div x v dxdt m v(, x) dx, for all v C c ([, T) ). Tus, in order to conclude te existence of a weak solution in te sense of Definition 2.2, it remains to prove tat T T lim p(ϱ δ )div x v dxdt = p(ϱ)div x v dxdt, δ togeter wit te energy inequality (1.8). Consequently, we are faced wit a situation similar to tat in te previous subsection. Te main difference is tat we now only ave γ > N 2. Since η δ enjoys bot ig integrability and compactness, te proof follows by a small extension of Feireisl s arguments in [25]. First, we establis iger integrability of te density on te entire domain. Lemma Let {ϱ δ, u δ, η δ } δ> be a sequence of artificial pressure solutions in te sense of Definition 3.1. Ten, tere exists a constant c(t), independent of δ, suc tat T dxdt c(t), ϱ γ +θ δ were Θ = 2 min γ 1, Proof. Since η δ b L 2 (, T; L 3 ()) L, 1 T; W 1, 3 2 (), te addition of η δ in te equations does not impose any potential problems. Consequently, if is bounded, te proof follows by te classical arguments [18,25]. If is unbounded, ten x Φ is no longer integrable and we cannot simply apply existing results. To prove te bound in tis case, let 1 be te inverse Laplacian realized using Fourier multipliers (see [18,17] for details). For eac fixed δ >, let te test function v δ be given as v δ = x 1 ϱ θ δ. By te requirements on θ, we ave in particular ϱδ θ b L (, T; L s ()), 3γ s = max 2γ 3, 4γ. Tus, v δ b L (, T; W 1,s ()) L (, T; L ()). Next, since (ϱ δ, u δ ) is a renormalized solution to te continuity equations, (2.3) wit B(ϱ δ ) = ϱ θ δ states t ϱ θ δ = div x(ϱ θ δ u) (θ 1)ϱθ δ div xu, in te sense of distributions on (, T). For notational convenience, we observe tat t v δ L p (,T;L q ()) = x 1 t ϱ θ δ L p (,T;L q ()) ϱ θ δ u δ L p (,T;L q ()) + ϱ θ div x u L p (,T;L r ()), for appropriate 1 p, q and r = q. Next, we apply v δ as a test function for te momentum equation to obtain T T dxdt = (ϱ δ u δ ) t v δ + ϱ δ u δ u δ : x v δ dxdt aϱ γ +θ δ + T T := I 1 + I 2 + I 3. µ x u δ x v δ + λdiv x u δ div x v δ dxdt η δ ϱδ θ (ϱ δβ + η δ ) x Φ δ v δ dxdt m v δ (, ) dx