# Strong Solutions for the Fluid-Particle Interaction Model with Non-Newtonian Potential

Authors: Yukun Song; Heyuan Wang; Fengming Liu
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

This paper deals with a mathematical fluid-particle interaction model used to describing the evolution of particles dispersed in a viscous compressible non-Newtonian fluid. It is proved that the initial boundary value problems with vacuum admits a unique local strong solution in the dimensional case. The strong nonlinearity of the system brings us difficulties due to the fact that the viscosity term and non-Newtonian gravitational potential term are fully nonlinear.


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## Body

## 1. Introduction

Fluid-particle interaction model arises in many practical applications in science and engineering [1–4] and is of primarily importance in the sedimentation analysis of disperse suspensions of particles in fluids. We focus on the fluid-particle interaction model that describes the evolution of particles dispersed in a viscous fluid.Carrillo and Goudon first derived a fluid-particle interaction system by formal asymptotics from a mesoscopic description (see [5]). In the microscopic description, the cloud of the particles is related to its distribution function f(t,x,ξ), which is the solution to a dimensionless Vlasov-Fokker-Planck equation. On the other hand, the fluid is described by its density ρ(x,t)≥0 and its velocity field u(x,t). We assume that the fluid is compressible and isentropic, then (ρ,u) solves the compressible Euler equations for the inviscid case or the Navier-Stokes equations for the viscous case, respectively.In [5], for the inviscid case, the coupling between the kinetic and the fluid equations is obtained through the friction forces that the fluid and the particles exert mutually. The friction force is assumed to follow the Stokes law and thus is proportional to the relative velocity of the fluid and the particles given by ξ-uϵ(t,x). Furthermore, both phases are affected by external forces, which are supposed to derive from a time independent potential Φ(x). The system was given as follows:(1)ft+θξ·∇xf-κ∇xΦ·∇ξf=1ϵdivξ⁡ξ-1θuf+∇ξfρt+divx⁡ρu=0ρut+divx⁡ρu⊗u+χ∇xPρ+αθκρ∇xΦ=1ϵρPρFJ-ηuwhere (2)ηt,x=∫Rft,x,ξdξ,Jt,x=θ∫Rξft,x,ξdξθ,κ,ϵ,α,χ are some related dimensionless parameters, P(ρ),ρP,ρF are the pressure, the mass density of particles and fluid, respectively. Setting ρP/ρF=1/θ2, κ=θ, θ=1/ϵ, α=sign⁡(α)ϵ with sign⁡(α)=±1, then (1) becomes(3)ftϵ+1ϵξ·∇xfϵ-∇xΦ·∇ξfϵ=1ϵdivξ⁡ξ-ϵuϵfϵ+∇ξfϵρtϵ+divx⁡ρϵuϵ=0ρϵuϵt+divx⁡ρϵuϵ⊗uϵ+χ∇xPρϵ+αθκρϵ∇xΦ=Jϵ-ηϵuϵFinally, letting ϵ→0, then (3) converges to system(4)ηt+divx⁡ηu-∇xΦ-∇xη=0ρt+divx⁡ρu=0ρut+divx⁡ρu⊗u+∇xη+χPρ+sign⁡αρ+η∇xΦ=0For the viscous case, with the dynamic viscosity terms taken into consideration, then (4)3 will be have an additional term of divx⁡S. In [6], the viscous stress tensor S=S(∇xu) is assumed to satisfy Newton’s Law for viscosity which requires that (5)S=μ∇u+∇uT+λdivx⁡uI,where μ and λ are constant viscosity coefficients satisfying (6)μ>0,λ+23μ≥0.Then (7)divx⁡S∇xu=μΔxu+λ∇xdiv⁡uand thus the system turns into the following equation(8)ρt+divx⁡ρu=0ρut+divx⁡ρu⊗u+∇xPρ+η-μΔxu-λ∇xdivx⁡u=-η+βρ∇xΦηt+divx⁡ηu-∇xΦ-Δxη=0Moreover, if the influence of gravitational potential Ψ=Ψ(x,t) was taken into consideration, there will be a Poisson equation coupled to the above system, as for the Navier-Stokes-Poisson equation in [7].Carrillo et al. obtained the global existence and asymptotic behavior of the weak solutions and stability properties to (8). Subsequently, Fang et al. [8] studied the existence of global classical solutions in dimension one. In [9, 10], Balew and Trivisa obtained the existence of global weak solutions and weakly dissipative solutions by entropy method in dimension three. The two-phase flow hydrodynamic models have been proposed in [3]. For some mathematical results on the Navier-Stokes coupled equations such as the nematic liquid crystal flows models where viscous effects are included, for more details, we refer to [11–13] and references therein.On the other hand, as we know, the viscous stress tensorS is depends on the rate of strain Eij(∇xu), where (9)Eij∇xu=∂ui∂xj+∂uj∂xiIf the stress and rate of strain satisfy the following linear relation (10)S=μEij∇u.then the fluid is called Newtonian. The coefficient of proportionality μ is called the viscosity coefficient, and it is a characteristic material quantity for the fluid concerned, which in general depends on density, temperature, and pressure. The governing equations of motions of them will be the Navier-Stokes equations. If the relation is not linear, the fluid is called non-Newtonian. Examples of non-Newtonian fluids are molten plastics, polymer solutions, dyes, varnishes, suspensions, adhesives, paints, greases, paper pulp, and biological fluids like blood. The simplest model of the stress-strain relation for such fluids given by the power laws, which states that (11)S=μ∂ui∂xj+∂uj∂xiq,for 0<q<1 (see [14]). Ladyzhenskaya (see [15]) proposed a special form for S on the incompressible model: (12)Sij=μ0+μ1E∇xup-2Eij∇xuThese models are called (13)Newtonian,forμ0>0,μ1=0;Rabinowitsch,forμ0,μ1>0,p=4;Eills,forμ0,μ1>0,p>2;Ostwald-deWaele,forμ0=0,μ1>0,p>1;Bingham,forμ0,μ1>0,p=1.For μ0=0, if p<2 then it is a pseudo-plastic fluid, and if p>2 then it is a dilatant fluid (see [14]). In the view of physics, the model captures the shear thinning fluid for the case of 1<p<2, and captures the shear thickening fluid for the case of p>2.Followed by the Ladyzhenskaya model, in this paper, we investigate the compressible non-Newtonian fluid-particle interaction model in one-dimensional case, then system (8) changes to be(14)ρt+ρux=0,ρut+ρu2x+ρΨx-λux2+μ1p-2/2uxx+P+ηx=-ηΦx,x,t∈ΩTΨx2+μ2q-2/2Ψxx=4πgρ-1Ω∫Ωρdx,ηt+ηu-Φxx=ηxxwith the initial and boundary conditions(15)ρ,u,ηt=0=ρ0,u0,η0,x∈Ω,u∂Ω=Ψ∂Ω=0,t∈0,T,and the no-flux condition for the density of particles(16)ηx+ηΦx∂Ω=0,t∈0,T.where Ω is a bounded interval, ΩT=Ω×[0,T]. ρ,u,η denote the fluid density, velocity, and the density of particle in the mixture, respectively. P(ρ)=aργ is the pressure where a>0,γ>1, Ψ(x,t) stands for the non-Newtonian gravitational potential, and the given function Φ(x) denotes the external potential. λ>0 is the viscosity coefficient and μ1>0, μ2>0, p>2, q>2 are constants.As to the non-Newtonian fluids, there has been much research both theoretically and experimentally, see ([15–23]). Indeed, we have investigated the existence results of solutions for p>2 and 1<p<2 in the absence of term Ψ(x) for (14) in [22, 23]. However, the influence of non-Newtonian gravitational potential for a practical model was not taken into consideration there. To our knowledge, there seems very few mathematical results for the case of the fluid-particle interaction model systems with non-Newtonian gravitational potential, even in dimension one. The existence results to problem (14)-(16) when p,q>2 which describes the motion of the compressible viscous isentropic gas flow is driven by a non-Newtonian gravitational force is still open up to now. We are interested in the existence and uniqueness of strong solutions on a one-dimensional bounded domain. In fact, the strong nonlinearity of (14) brings us new difficulties in getting the upper bound of ρ and the method used in [8] does not suitable for us. Motivated by Choetal’s [24, 25] work on Navier-Stokes equations, we establish local existence and uniqueness of strong solutions by the iteration techniques.Throughout the paper we assume thata=λ=1. In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as Lp=Lp(Ω), H01=H01(Ω), C(0,T;H1)=C(0,T;H1(Ω)).


### 1.1. Main Results

Theorem 1.
Letμ1>0 be a positive constant and Φ∈C2(Ω), and assume that the initial data (ρ0,u0,η0) satisfy the following conditions:(17)0≤ρ0∈H1Ω,u0∈H01Ω∩H2Ω,η0∈H2Ωand(18)-u0x2+μ1p-2/2u0xx+Pρ0+η0x+η0Φx=ρ01/2g+βΦx,for some g∈L2(Ω), where β≠0 is a constant. Then there exist a time T∗∈(0,+∞) and a unique strong solution (ρ,u,η) to (14)-(16) such that(19)ρ∈L∞0,T∗;H1Ω,u∈L∞0,T∗;W01,pΩ∩H2Ω,η∈L∞0,T∗;H2Ωρt∈L∞0,T∗;L2Ω,ut∈L20,T∗;H01Ω,ηt∈L∞0,T∗;L2Ωρut∈L∞0,T∗;L2Ω,ux2+μ1p-2/2uxx∈L20,T∗;L2Ω,


## 1.1. Main Results

Theorem 1.
Letμ1>0 be a positive constant and Φ∈C2(Ω), and assume that the initial data (ρ0,u0,η0) satisfy the following conditions:(17)0≤ρ0∈H1Ω,u0∈H01Ω∩H2Ω,η0∈H2Ωand(18)-u0x2+μ1p-2/2u0xx+Pρ0+η0x+η0Φx=ρ01/2g+βΦx,for some g∈L2(Ω), where β≠0 is a constant. Then there exist a time T∗∈(0,+∞) and a unique strong solution (ρ,u,η) to (14)-(16) such that(19)ρ∈L∞0,T∗;H1Ω,u∈L∞0,T∗;W01,pΩ∩H2Ω,η∈L∞0,T∗;H2Ωρt∈L∞0,T∗;L2Ω,ut∈L20,T∗;H01Ω,ηt∈L∞0,T∗;L2Ωρut∈L∞0,T∗;L2Ω,ux2+μ1p-2/2uxx∈L20,T∗;L2Ω,


## 2. A Priori Estimates for Smooth Solutions

In this section, we will prove the local existence of strong solutions. Provided that(ρ,u,η) is a smooth solution of (14)-(16) and ρ0≥δ, where 0<δ≪1 is a positive number. We denote by M0=1+μ1+μ1-1+ρ0H1+gL2 and introduce an auxiliary function (20)Zt=sup0≤s≤t⁡1+ρsH1+usW01,p+ηtsL2+ηsH1+ρutsL2Then we estimate each term of Z(t) in terms of some integrals of Z(t) and apply arguments of Gronwall-type and thus prove that Z(t) is locally bounded.


### 2.1. Estimate forρH1

First we need to do the following estimates. Using(14)1, we rewritten (14)2 as(21)ρut+ρuux+ρΨx-ux2+μ1p-2/2uxx+P+ηx=-ηΦxBy virtue of (22)ux2+μ1p-2/2uxx≥μ1p-2/2uxxThen (23)uxx≤Cρut+ρuux+ρΨx+P+ηx+ηΦxTaking the above inequality by L2 norm, we get(24)uxxL2≤Cρut+ρuux+ρΨx+P+ηx+ηΦxL2≤CρL∞1/2ρutL2+ρL∞uL∞uxL2+ρL2ΨxL2+PxL2+ηxL2+ηL2ΦxL2We deal with the term of ΨxL2.Multiplying(14)3 by Ψ and integrating over Ω, we obtain (25)∫ΩΨx2+μ2q-2/2ΨxxΨdx=4πg∫ΩρΨdx-m0∫ΩΨdx∫ΩΨxqdx≤∫ΩΨx2+μ2q-2/2Ψx2dx=-∫ΩΨx2+μ2q-2/2ΨxxΨdx=-4πg∫ΩρΨdx-m0∫ΩΨdx≤ΨL∞8πgm0≤ΨxLq8πgm0≤1qΦxLqq+1p8πgm0pthen we have(26)∫ΩΨxqdx≤Cm0,q>2.Hence, we deduce that(27)uxxL2≤CZγ+2tFrom (14)3, taking it by L2 norm, we get(28)ηxxL2≤ηt+ηu-ΦxxL2≤ηtL2+ηxL2uL∞+CηxL2+ηH1uxLp+CηH1≤CZ2tMultiplying (14)1 by ρ, integrating over Ω, we have (29)12ddt∫Ωρ2ds+∫Ωρuxρdx=0Integrating by parts, using Sobolev inequality, we deduce that(30)ddtρtL22≤∫Ωuxρ2dx≤uxxL2ρL22Differentiating (14)1 with respect to x, and multiplying it by ρx, integrating over Ω, using Sobolev inequality, we have(31)ddt∫Ωρx2dx=-∫Ω32uxρx2+ρρxuxxtdx≤CuxL∞ρxL22+ρL∞ρxL2uxxL2≤CρH12uxxL2From (30) and (31), by Gronwall’s inequality, it follows that(32)sup0≤t≤T⁡ρtH12≤ρ0H12exp⁡C∫0tuxxL2ds≤Cexp⁡C∫0tZγ+2sdsBesides, we can also get the following estimates. By using (14)1,(33)ρttL2≤ρxtL2utL∞+ρtL∞uxtL2≤CZ2tFrom (14)3, we have (34)μ1q-2/2Ψxx≤Ψx2+μ1q-2/2Ψxx=4πgρ-m0then(35)ΨxxL2≤Cm0Differentiating (14)3 with respect to time t, multiplying it by Ψt, integrating over Ω to x, and using Young’s inequality, we have (36)μ1q-2/2∫ΩΨxt2dx≤∫ΩΨx2+μ1q-2/2ΨxtΨxtdx=-4πg∫ΩρtΨtdx≤∫ΩρtL2ΨxtL2≤CρtL22+ηΨxtL22thus, we get(37)ΨxtL22≤CZ4twhere C is a positive constant, depending only on M0.


### 2.2. Estimate foruW01,p

Multiplying (21) by ut, integrating (by parts) over ΩT, we have(38)∫∫ΩTρut2dxds+∫∫ΩTux2+μ1p-2/2uxuxtdxds=-∫∫ΩTρuux+ρΨx+Px+ηx+ηΦxutdxdsWe deal with each term as follows: (39)∫Ωux2+μ1p-2/2uxuxtdx=12∫Ωux2+μ1p-2/2ux2tdx=12ddt∫Ω∫0ux2s+μ1p-2/2dsdx∫0ux2s+μ1p-2/2ds=∫μ1ux2+μ1tp-2/2dt=2pux2+μ1p/2-μ1p/2≥2puxp-2pμ1p/2-∫∫ΩTPxutdxds=∫∫ΩTPuxtdxds=ddt∫∫ΩTPuxdxds-∫∫ΩTPtuxdxdsFrom (14)1 we get(40)Pt=-γPux-Pxu(41)-∫∫ΩTηx+ηΦxutdxds=ddt∫∫ΩTηx+ηΦxudxds-∫∫ΩTηx+ηΦxtudxds-∫∫ΩTηx+ηΦxtudxds=∫∫ΩTηtux-Φxudxds=-∫∫ΩTηx-ηu-Φxux-ΦxuxdxdsSubstituting the above into (38), and using Young’s inequality, we obtain(42)∫0tρutsL22ds+uxtLpp≤C+∫ΩPuxdx+∫∫ΩTρuuxut+ρΨxut+γPux2+Pxuuxdxds+∫∫ΩTηxuxx+ηxΦxux+ηxΦxxu+ηuuxx+ηu2Φxx+ηuΦxux+ηΦxuxx+ηΦxΦxxu+ηΦx2uxdxds≤C+C∫0tρL∞uL∞2uxLp2+ρL∞ΨxL22+PL∞uxLp2+PxL2uL∞uxLp+ηxL2uxxL2+ηxL2uxLp+ηxL2uL∞+ηL∞uL∞uxxL2+ηL∞uL∞2+ηL∞uL∞uxLp+ηL∞uxxL2+ηL∞uL∞+ηL∞uxLpds+CPtL22+12∫0tρutL22ds+12uxtLpp≤C1+∫0tZγ+4sds+PtL22Using (14)1, we have(43)∫ΩPt2dx=∫ΩP02dx+∫0t∂∂s∫ΩPs2dxds≤∫ΩP02dx+2∫0t∫ΩPsγργ-1-ρxu-ρuxdxds≤C+C∫0tPL∞ρL∞γ-1ρH1uxLpds≤C1+∫0tZ2γ+1sdsCombining (42)-(43), yields(44)∫0tρutsL22sds+uxtLpp≤C1+∫0tZ2γ+3sdswhere C is a positive constant, depending only on M0.


### 2.3. Estimate forηtL2 and ηH1

Multiplying(14)4 by η, integrating the resulting equation over ΩT, and using the boundary conditions (16), Young’s inequality, we have(45)∫0tηxsL22ds+12ηtL22≤∫∫ΩTηuηx+ηΦxηxdxds≤14∫0tηxsL22ds+C∫0tuxLp2ηH12ds+C∫0tηH12+C≤14∫0tηxsL22ds+C1+∫0tZ4tdsMultiplying (14)4 by ηt, integrating (by parts) over ΩT, and using the boundary conditions (16), Young’s inequality, we have(46)∫0tηtsL22ds+12ηxtL22≤∫∫ΩTηu-Φxηxtdxds≤14∫0tηxtsL22ds+C∫0tηH12uxLp2ds+C∫0tηH12ds+C≤14∫0tηxtsL22ds+C1+∫0tZ4tdsDifferentiating (14)4 with respect to t, multiplying the resulting equation by ηt, and integrating (by parts) over ΩT, we get(47)∫0tηxtsL22ds+12ηttL22=∫∫ΩTηu-Φxtηxtdxds≤C+∫∫ΩTηtuηxt+ηtΦxηxt+ηxutηt+ηuxtηtdxds≤C1+∫0tηtL22uxLp2+ηtL22+ηxL22ηtL22+ηH12ηtL22dx+12∫0tηxtL22+12∫0tuxtL22≤C1+∫0tZ2γ+6sdsCombining (45)-(47), we get(48)ηH12+ηtL22+∫0tηxL22+ηtL22+ηxtL22sds≤C1+∫0tZ2γ+6sds


### 2.4. Estimate forρutL2

Differentiating (21) with respect to t, multiplying the result equation by ut, and integrating it over Ω with respect to x, we have(49)12ddt∫Ωρut2dx+∫Ωux2+μ1p-2/2uxtuxtdx=∫Ωρuxut2+uuxut+Ψxut-ρuxut2-ρΨxtut-P+ηtuxt-ηtΦxutdxNote that (50)ux2+μ1p-2/2uxtuxt=ux2+μ1p-4/2p-1ux2+μ1uxt2≥μ1p-2/2uxt2Combining (40), (49) can be rewritten into(51)12ddt∫Ωρut2dx+∫Ωuxt2dx≤2∫Ωρuutuxtdx+∫Ωρuux2utdx+∫Ωρu2uxxutdx+∫Ωρu2uxuxtdx+∫ΩρuΨxxutdx+∫ΩρuΨxuxtdx+∫Ωρuxut2dx+∫ΩγPuxuxtdx+∫ΩPxuuxtdx+∫Ωηtuxtdx+∫ΩηtΦxutdx+∫ΩρΨxtutdx=∑j=112IjBy using Sobolev inequality, Hölder inequality and Young’s inequality, (35), (37), we estimate each term of Ij as follows: (52)I1=2∫Ωρuutuxtdx≤2ρL∞1/2uL∞ρutL2uxtL2≤CZ5t+17uxtL22I2=∫Ωρuux2utdx≤ρL∞1/2uL22uxL∞2ρutL2≤CZ2γ+7tI3=∫Ωρu2uxxutdx≤ρL∞1/2uL∞2uxxL2ρutL2≤CZγ+6tI4=∫Ωρu2uxuxtdx≤ρL∞uL∞2uxL2uxtL2≤CZ8t+17uxtL22I5=∫ΩρuΨxxutdx≤ρL∞1/2uL2ΨxxL2ρutL2≤CZ3tI6=∫ΩρuΨxuxtdx≤ρL∞uL∞ΨxL2uxtL2≤CZ4t+17uxtL22I7=∫Ωρuxut2dx≤uxL∞ρutL22≤CZγ+4tI8=∫ΩγPuxuxtdx≤CPL∞uxLpuxtL2≤CZ2γ+2t+17uxtL22I9=∫ΩPxuuxtdx≤PxL2uL∞uxtL2≤CZ2γ+2t+17uxtL22I10=∫Ωηtuxtdx≤ηtL2uxtL2≤CZ2t+17uxtL22I11=∫ΩηtΦxutdx≤CZ2t+17uxtL22I12=∫ΩρΨxtutdx≤CρL∞1/2ΨxtL2ρutL2≤CZ6tSubstituting Ij(j=1,2,…,12) into (51), and integrating over (τ,t)⊂(0,T) on the time variable, we have(53)ρuttL22+∫τtuxtL22sds≤ρutτL22+C∫τtZ2γ+7sds.To obtain the estimate of ρuttL22, we need to estimate limτ→0⁡ρutτL22. Multiplying (21) by ut and integrating over Ω, we get (54)∫Ωρut2dx≤2∫Ωρu2ux2+ρΨx2+ρ-1-ux2+μ1p-2/2uxx+P+ηx+ηΦx2dx.According to the smoothness of (ρ,u,η), we obtain (55)limτ→0⁡∫Ωρu2ux2+ρΨx2+ρ-1-ux2+μ1p-2/2uxx+P+ηx+ηΦx2dx=∫Ωρu02u0x2+ρ0Φx2+ρ0-1-u0x2+μ1p-2/2u0xx+P0+η0x+η0Φx2dx≤ρ0L∞u0L∞2u0xL22+ρ0L∞ΦxL22+gL22+βΦxL22≤C.Therefore, taking a limit on τ in (53), as τ→0, we conclude that(56)ρuttL22+∫0tuxtL22sds≤C1+∫0tZ2γ+7sds,where C is a positive constant, depending only on M0.Combining the estimates of (27), (28), (32), (33), (44), (48), (56) and the definition of Z(t), we conclude that(57)Zt≤C~exp⁡C~~∫0tZ2γ+7sdswhere C~,C~~ are positive constants, depending only on M0. This means that there exist a time T1>0 and a constant C>0, such that(58)ess⁡sup0≤t≤T1⁡ρH1+uW01,p∩H2+ηH2+ηtL2+ρutL2+ρtL2+∫0T1ρutL22+uxtL22+ηxL22+ηtL22+ηxtL22ds≤C


## 2.1. Estimate forρH1

First we need to do the following estimates. Using(14)1, we rewritten (14)2 as(21)ρut+ρuux+ρΨx-ux2+μ1p-2/2uxx+P+ηx=-ηΦxBy virtue of (22)ux2+μ1p-2/2uxx≥μ1p-2/2uxxThen (23)uxx≤Cρut+ρuux+ρΨx+P+ηx+ηΦxTaking the above inequality by L2 norm, we get(24)uxxL2≤Cρut+ρuux+ρΨx+P+ηx+ηΦxL2≤CρL∞1/2ρutL2+ρL∞uL∞uxL2+ρL2ΨxL2+PxL2+ηxL2+ηL2ΦxL2We deal with the term of ΨxL2.Multiplying(14)3 by Ψ and integrating over Ω, we obtain (25)∫ΩΨx2+μ2q-2/2ΨxxΨdx=4πg∫ΩρΨdx-m0∫ΩΨdx∫ΩΨxqdx≤∫ΩΨx2+μ2q-2/2Ψx2dx=-∫ΩΨx2+μ2q-2/2ΨxxΨdx=-4πg∫ΩρΨdx-m0∫ΩΨdx≤ΨL∞8πgm0≤ΨxLq8πgm0≤1qΦxLqq+1p8πgm0pthen we have(26)∫ΩΨxqdx≤Cm0,q>2.Hence, we deduce that(27)uxxL2≤CZγ+2tFrom (14)3, taking it by L2 norm, we get(28)ηxxL2≤ηt+ηu-ΦxxL2≤ηtL2+ηxL2uL∞+CηxL2+ηH1uxLp+CηH1≤CZ2tMultiplying (14)1 by ρ, integrating over Ω, we have (29)12ddt∫Ωρ2ds+∫Ωρuxρdx=0Integrating by parts, using Sobolev inequality, we deduce that(30)ddtρtL22≤∫Ωuxρ2dx≤uxxL2ρL22Differentiating (14)1 with respect to x, and multiplying it by ρx, integrating over Ω, using Sobolev inequality, we have(31)ddt∫Ωρx2dx=-∫Ω32uxρx2+ρρxuxxtdx≤CuxL∞ρxL22+ρL∞ρxL2uxxL2≤CρH12uxxL2From (30) and (31), by Gronwall’s inequality, it follows that(32)sup0≤t≤T⁡ρtH12≤ρ0H12exp⁡C∫0tuxxL2ds≤Cexp⁡C∫0tZγ+2sdsBesides, we can also get the following estimates. By using (14)1,(33)ρttL2≤ρxtL2utL∞+ρtL∞uxtL2≤CZ2tFrom (14)3, we have (34)μ1q-2/2Ψxx≤Ψx2+μ1q-2/2Ψxx=4πgρ-m0then(35)ΨxxL2≤Cm0Differentiating (14)3 with respect to time t, multiplying it by Ψt, integrating over Ω to x, and using Young’s inequality, we have (36)μ1q-2/2∫ΩΨxt2dx≤∫ΩΨx2+μ1q-2/2ΨxtΨxtdx=-4πg∫ΩρtΨtdx≤∫ΩρtL2ΨxtL2≤CρtL22+ηΨxtL22thus, we get(37)ΨxtL22≤CZ4twhere C is a positive constant, depending only on M0.


## 2.2. Estimate foruW01,p

Multiplying (21) by ut, integrating (by parts) over ΩT, we have(38)∫∫ΩTρut2dxds+∫∫ΩTux2+μ1p-2/2uxuxtdxds=-∫∫ΩTρuux+ρΨx+Px+ηx+ηΦxutdxdsWe deal with each term as follows: (39)∫Ωux2+μ1p-2/2uxuxtdx=12∫Ωux2+μ1p-2/2ux2tdx=12ddt∫Ω∫0ux2s+μ1p-2/2dsdx∫0ux2s+μ1p-2/2ds=∫μ1ux2+μ1tp-2/2dt=2pux2+μ1p/2-μ1p/2≥2puxp-2pμ1p/2-∫∫ΩTPxutdxds=∫∫ΩTPuxtdxds=ddt∫∫ΩTPuxdxds-∫∫ΩTPtuxdxdsFrom (14)1 we get(40)Pt=-γPux-Pxu(41)-∫∫ΩTηx+ηΦxutdxds=ddt∫∫ΩTηx+ηΦxudxds-∫∫ΩTηx+ηΦxtudxds-∫∫ΩTηx+ηΦxtudxds=∫∫ΩTηtux-Φxudxds=-∫∫ΩTηx-ηu-Φxux-ΦxuxdxdsSubstituting the above into (38), and using Young’s inequality, we obtain(42)∫0tρutsL22ds+uxtLpp≤C+∫ΩPuxdx+∫∫ΩTρuuxut+ρΨxut+γPux2+Pxuuxdxds+∫∫ΩTηxuxx+ηxΦxux+ηxΦxxu+ηuuxx+ηu2Φxx+ηuΦxux+ηΦxuxx+ηΦxΦxxu+ηΦx2uxdxds≤C+C∫0tρL∞uL∞2uxLp2+ρL∞ΨxL22+PL∞uxLp2+PxL2uL∞uxLp+ηxL2uxxL2+ηxL2uxLp+ηxL2uL∞+ηL∞uL∞uxxL2+ηL∞uL∞2+ηL∞uL∞uxLp+ηL∞uxxL2+ηL∞uL∞+ηL∞uxLpds+CPtL22+12∫0tρutL22ds+12uxtLpp≤C1+∫0tZγ+4sds+PtL22Using (14)1, we have(43)∫ΩPt2dx=∫ΩP02dx+∫0t∂∂s∫ΩPs2dxds≤∫ΩP02dx+2∫0t∫ΩPsγργ-1-ρxu-ρuxdxds≤C+C∫0tPL∞ρL∞γ-1ρH1uxLpds≤C1+∫0tZ2γ+1sdsCombining (42)-(43), yields(44)∫0tρutsL22sds+uxtLpp≤C1+∫0tZ2γ+3sdswhere C is a positive constant, depending only on M0.


## 2.3. Estimate forηtL2 and ηH1

Multiplying(14)4 by η, integrating the resulting equation over ΩT, and using the boundary conditions (16), Young’s inequality, we have(45)∫0tηxsL22ds+12ηtL22≤∫∫ΩTηuηx+ηΦxηxdxds≤14∫0tηxsL22ds+C∫0tuxLp2ηH12ds+C∫0tηH12+C≤14∫0tηxsL22ds+C1+∫0tZ4tdsMultiplying (14)4 by ηt, integrating (by parts) over ΩT, and using the boundary conditions (16), Young’s inequality, we have(46)∫0tηtsL22ds+12ηxtL22≤∫∫ΩTηu-Φxηxtdxds≤14∫0tηxtsL22ds+C∫0tηH12uxLp2ds+C∫0tηH12ds+C≤14∫0tηxtsL22ds+C1+∫0tZ4tdsDifferentiating (14)4 with respect to t, multiplying the resulting equation by ηt, and integrating (by parts) over ΩT, we get(47)∫0tηxtsL22ds+12ηttL22=∫∫ΩTηu-Φxtηxtdxds≤C+∫∫ΩTηtuηxt+ηtΦxηxt+ηxutηt+ηuxtηtdxds≤C1+∫0tηtL22uxLp2+ηtL22+ηxL22ηtL22+ηH12ηtL22dx+12∫0tηxtL22+12∫0tuxtL22≤C1+∫0tZ2γ+6sdsCombining (45)-(47), we get(48)ηH12+ηtL22+∫0tηxL22+ηtL22+ηxtL22sds≤C1+∫0tZ2γ+6sds


## 2.4. Estimate forρutL2

Differentiating (21) with respect to t, multiplying the result equation by ut, and integrating it over Ω with respect to x, we have(49)12ddt∫Ωρut2dx+∫Ωux2+μ1p-2/2uxtuxtdx=∫Ωρuxut2+uuxut+Ψxut-ρuxut2-ρΨxtut-P+ηtuxt-ηtΦxutdxNote that (50)ux2+μ1p-2/2uxtuxt=ux2+μ1p-4/2p-1ux2+μ1uxt2≥μ1p-2/2uxt2Combining (40), (49) can be rewritten into(51)12ddt∫Ωρut2dx+∫Ωuxt2dx≤2∫Ωρuutuxtdx+∫Ωρuux2utdx+∫Ωρu2uxxutdx+∫Ωρu2uxuxtdx+∫ΩρuΨxxutdx+∫ΩρuΨxuxtdx+∫Ωρuxut2dx+∫ΩγPuxuxtdx+∫ΩPxuuxtdx+∫Ωηtuxtdx+∫ΩηtΦxutdx+∫ΩρΨxtutdx=∑j=112IjBy using Sobolev inequality, Hölder inequality and Young’s inequality, (35), (37), we estimate each term of Ij as follows: (52)I1=2∫Ωρuutuxtdx≤2ρL∞1/2uL∞ρutL2uxtL2≤CZ5t+17uxtL22I2=∫Ωρuux2utdx≤ρL∞1/2uL22uxL∞2ρutL2≤CZ2γ+7tI3=∫Ωρu2uxxutdx≤ρL∞1/2uL∞2uxxL2ρutL2≤CZγ+6tI4=∫Ωρu2uxuxtdx≤ρL∞uL∞2uxL2uxtL2≤CZ8t+17uxtL22I5=∫ΩρuΨxxutdx≤ρL∞1/2uL2ΨxxL2ρutL2≤CZ3tI6=∫ΩρuΨxuxtdx≤ρL∞uL∞ΨxL2uxtL2≤CZ4t+17uxtL22I7=∫Ωρuxut2dx≤uxL∞ρutL22≤CZγ+4tI8=∫ΩγPuxuxtdx≤CPL∞uxLpuxtL2≤CZ2γ+2t+17uxtL22I9=∫ΩPxuuxtdx≤PxL2uL∞uxtL2≤CZ2γ+2t+17uxtL22I10=∫Ωηtuxtdx≤ηtL2uxtL2≤CZ2t+17uxtL22I11=∫ΩηtΦxutdx≤CZ2t+17uxtL22I12=∫ΩρΨxtutdx≤CρL∞1/2ΨxtL2ρutL2≤CZ6tSubstituting Ij(j=1,2,…,12) into (51), and integrating over (τ,t)⊂(0,T) on the time variable, we have(53)ρuttL22+∫τtuxtL22sds≤ρutτL22+C∫τtZ2γ+7sds.To obtain the estimate of ρuttL22, we need to estimate limτ→0⁡ρutτL22. Multiplying (21) by ut and integrating over Ω, we get (54)∫Ωρut2dx≤2∫Ωρu2ux2+ρΨx2+ρ-1-ux2+μ1p-2/2uxx+P+ηx+ηΦx2dx.According to the smoothness of (ρ,u,η), we obtain (55)limτ→0⁡∫Ωρu2ux2+ρΨx2+ρ-1-ux2+μ1p-2/2uxx+P+ηx+ηΦx2dx=∫Ωρu02u0x2+ρ0Φx2+ρ0-1-u0x2+μ1p-2/2u0xx+P0+η0x+η0Φx2dx≤ρ0L∞u0L∞2u0xL22+ρ0L∞ΦxL22+gL22+βΦxL22≤C.Therefore, taking a limit on τ in (53), as τ→0, we conclude that(56)ρuttL22+∫0tuxtL22sds≤C1+∫0tZ2γ+7sds,where C is a positive constant, depending only on M0.Combining the estimates of (27), (28), (32), (33), (44), (48), (56) and the definition of Z(t), we conclude that(57)Zt≤C~exp⁡C~~∫0tZ2γ+7sdswhere C~,C~~ are positive constants, depending only on M0. This means that there exist a time T1>0 and a constant C>0, such that(58)ess⁡sup0≤t≤T1⁡ρH1+uW01,p∩H2+ηH2+ηtL2+ρutL2+ρtL2+∫0T1ρutL22+uxtL22+ηxL22+ηtL22+ηxtL22ds≤C


## 3. Proof of the Main Theorem

In this section, our proof will be based on the usual iteration argument and some ideas developed in [24, 25]. We construct the approximate solutions, by using the iterative scheme, inductively, as follows: first define u0=0 and assuming that uk-1 was defined for k≥1, let ρk,uk,ηk be the unique smooth solution to the following problems:(59)ρtk+ρxkuk-1+ρkuxk-1=0(60)ρkutk+ρkuk-1uxk+ρkΨxk+Lpuk+Pxk+ηxk=-ηkΦx(61)LqΨk=-4πgρk-m0(62)ηtk+ηkuk-1-Φxx=ηxxkwith the initial and boundary conditions(63)ρk,uk,ηkt=0=ρ0,u0,η0uk∂Ω=Ψk∂Ω=0ηxk+ηkΦx∂Ω=0where (64)Lpuk=-uxk2+μ1p-2/2uxkxLqΨk=-Ψxk2+μ2q-2/2ΨxkxWe directly construct approximate solutions of the problem (59)–(63). More precisely, we first find ρk from (59) and (63) with smooth function uk-1, i.e., (65)ρtk+ρxkuk-1+ρkuxk-1=0,and(66)ρkt=0=ρ0,ρ0≥δ.It follows from the classical linear hyperbolic theory that there is a unique solution ρk on this above initial problem. Using the method of characteristics, we have(67)dρkdt=-ρkuxk-1,(68)dxdt=uk-1x,t,(69)xt=0=x0,ρkt=0=ρ0.By (68) and (69), we have (70)xt=x0+∫0tuk-1xs,sds=Ux0,tUsing (67), then (71)dln⁡ρk=-uxk-1dtwhich means(72)ρkx,t≥δexp⁡-∫0T1uxk-1·,sL∞ds>0,forallt∈0,T1.Next, combining the classical stableness results of the elliptic equation, the existence of Ψk can be obtained by (61) and (63), then by (62) and (63) we get ηk. The last, with ρk,Ψk,ηk being given, by virtue of (72), from (60) and (63), according to the classical theorem of quasi-linear parabolic equation (see [17], Chapter VI, Theorem 5.2), there exists a unique smooth solution uk. With the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold:(73)ess⁡sup0≤t≤T1⁡ρkH1+ukW01,p∩H2+ηkH2+ηtkL2+ρkutkL2+ρtkL2+∫0T1ρkutkL22+uxtkL22+ηxkL22+ηtkL22+ηxtkL22ds≤Cwhere C is a generic constant depending only on M0, but independent of k.Next, we have to prove that the approximate solution(ρk,uk,ηk) converges to a solution to the original problem (14) in a strong sense. To this end, let us define (74)ρ¯k+1=ρk+1-ρk,u¯k+1=uk+1-uk,Ψ¯k+1=Ψk+1-Ψk,η¯k+1=ηk+1-ηkthen we can verify that the functions ρ¯k+1,u¯k+1,η¯k+1 satisfy the system of equations(75)ρ¯tk+1+ρ¯k+1ukx+ρku¯kx=0(76)ρk+1u¯tk+1+ρk+1uku¯xk+1+Lpuk+1-Lpuk=-ρ¯k+1utk+ukuxk+Ψxk+1-Pxk+1-Pxk+ρku¯kuxk-Ψ¯xk+1-η¯xk+1-η¯k+1Φx(77)LqΨk+1-LqΨk=-4πgρ¯k+1(78)η¯tk+1+ηku¯kx+η¯k+1uk-Φxx=η¯xxk+1Multiplying (75) by ρ¯k+1, integrating over Ω, and using Young’s inequality, we obtain(79)ddtρ¯k+1L22≤Cρ¯k+1L22uxkL∞+ρkH1u¯xkL2ρ¯k+1L2≤CuxxkL2ρ¯k+1L22+CζρkH12ρ¯k+1L22+ζu¯xkL22≤Cζρ¯k+1L22+ζu¯xkL22where Cζ is a positive constant, depending on M0 and ζ for all t<T1 and k≥1.Multiplying (76) by u¯k+1, integrating over Ω, and using Young’s inequality, we obtain(80)12ddt∫Ωρk+1u¯k+12dx+∫ΩLpuk+1-Lpuku¯k+1dx≤C∫Ωρ¯k+1utk+ukuxk+Ψxk+1u¯k+1+Pxk+1-Pxku¯k+1+ρku¯kuxku¯k+1+ρkΨ¯xk+1u¯k+1+η¯xk+1u¯k+1+η¯k+1Φxu¯k+1dxLet (81)σs=s2+μ1p-2/2sand then (82)σ′s=s2+μ1p-2/2s′=s2+μ1p-4/2p-1s2+μ1≥μ1p-2/2We estimate the second term of (80) as follows:(83)∫ΩLpuk+1-Lpuku¯k+1dx=∫Ω∫01σ′θuxk+1+1-θuxkdθu¯xk+12dx≥μ1p-2/2∫Ωu¯xk+12dxSimilarly, we have (84)μ2q-2/2∫ΩΨ¯xk+12dx≤∫ΩLqΨk+1-LqΨkΨ¯k+1dx=4πg∫Ωρ¯k+1Ψ¯k+1dxand then we have (85)Ψ¯xk+1L22≤Cρ¯k+1L22Substituting (83) into (80) and using Young’s inequality, we have(86)ddt∫Ωρk+1u¯k+12dx+∫Ωu¯xk+12dx≤Cρ¯k+1L2uxtkL2u¯xk+1L2+ρ¯k+1L2uxkLpuxxkL2u¯xk+1L2+ρ¯k+1L2Ψxk+1L2u¯xk+1L2+Pk+1-PkL2u¯xk+1L2+ρkL21/2ρku¯kL2uxxkL2u¯xk+1L2+ρkL∞Ψ¯xk+1L2u¯xk+1L2+η¯k+1L2u¯xk+1L2+η¯k+1L2u¯xk+1L2≤Bζtρ¯k+1L22+Cρku¯kL22+η¯k+1L22+ζu¯xk+1L22where Bζ(t)=C(1+uxtktL22), for all t≤T1 and k≥1. Using (73) we derive (87)∫0tBζsds≤C+CtMultiplying (78) by η¯k+1, integrating over Ω, and using (73) and Young’s inequality, we have(88)12ddt∫Ωη¯k+12dx+∫Ωη¯xk+12dx≤∫Ωη¯k+1uk-Φxη¯xk+1dx+∫Ωηku¯kxη¯k+1dx≤η¯k+1L2uk-ΦxL∞η¯xk+1L2+ηxkL2u¯kL∞η¯k+1L2+ηkL∞u¯xkL2η¯k+1L2≤Cζη¯k+1L22+ζη¯xk+1L22+ζu¯xkL22Collecting (79), (86), and (88), we obtain(89)ddtρ¯k+1tL22+ρk+1u¯k+1tL22+η¯k+1tL22+u¯xk+1tL22+η¯xk+1L22≤Eζtρ¯k+1tL22+Cρku¯kL22+Cζη¯k+1L22+ζu¯xkL22where Eζ(t) depends only on Bζ(t) and Cζ, for all t≤T1 and k≥1. Using (73), we have (90)∫0tEζsds≤C+CζtIntegrating (89) over (0,t)⊂(0,T1) with respect to t, using Gronwall’s inequality, we have (91)ρ¯k+1tL22+ρk+1u¯k+1tL22+η¯k+1tL22+∫0tu¯xk+1tL22ds+∫0tη¯xk+1L22ds≤Cexp⁡Cζt∫0tρku¯ksL22+u¯xksL22dsfrom the above recursive relation, choose ζ>0 and 0<T∗<T1 such that Cexp⁡(CζT∗)<1/2, using Gronwall’s inequality, we deduce that(92)∑k=1Ksup0≤t≤T∗⁡ρ¯k+1tL22+ρk+1u¯k+1tL22+η¯k+1tL22+∫0T∗u¯xk+1tL22dt+∫0T∗η¯xk+1tL22dt<CSince all of the constants do not depend on δ, as k→∞, we conclude that sequence (ρk,uk,ηk) converges to a limit (ρδ,uδ,ηδ) in the following convergence:(93)ρ→ρδinL∞0,T∗;L2Ω,(94)u→uδinL∞0,T∗;L2Ω∩L20,T∗;H01Ω,(95)η→ηδinL∞0,T∗;L2Ω∩L20,T∗;H1Ω,and there also holds(96)ess⁡sup0≤t≤T1⁡ρδH1+uδW01,p∩H2+ηδH2+ηtδL2+ρδutδL2+ρtδL2+∫0T∗ρδutδL22+uxtδL22+ηxδL22+ηtδL22+ηxtδL22ds≤C.For each small δ>0, let ρ0δ=Jδ∗ρ0+δ, Jδ is a mollifier on Ω, and u0δ∈H01(Ω)∩H2(Ω) is a smooth solution of the boundary value problem:(97)Lpu0δ+Pρ0δ+η0δx+η0δΦx=ρ0δ1/2gδ+βΦx,u0δ0=u0δ1=0,where gδ∈C0∞ and satisfies gδL2≤gL2, limδ→0+⁡gδ-gL2=0.We deduce that(ρδ,uδ,ηδ) is a solution of the following initial boundary value problem: (98)ρt+ρux=0,ρut+ρu2x+ρΨx-λux2+μ1p-2/2uxx+P+ηx=-ηΦx,Ψx2+μ2q-2/2Ψxx=4πgρ-1Ω∫Ωρdx,ηt+ηu-Φxx=ηxx,ρ,u,ηt=0=ρ0δ,u0δ,η0δ,u∂Ω=Ψ∂Ω=ηx+ηΦx∂Ω=0.where ρ0δ≥δ,p,q>2.By the proof of Lemma 2.3 in [20], there exists a subsequence u0δj of u0δ, as δj→0+, u0δ→u0 in H01(Ω)∩H2(Ω), -(u0xδjp-2u0xδj)x→-(u0xp-2u0x)x in L2(Ω), Hence, u0 satisfies (18) of Theorem 1. By virtue of the lower semi-continuity of various norms, we deduce that (ρ,u,η) satisfies the following uniform estimate:(99)ess⁡sup0≤t≤T1⁡ρH1+uW01,p∩H2+ηH2+ηtL2+ρutL2+ρtL2+∫0T∗ρutL22+uxtL22+ηxL22+ηtL22+ηxtL22ds≤C,where C is a positive constant, depending only on M0.The uniqueness of solution can be obtained by the same method as the above proof of convergence; we omit the details here. This completes the proof.



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*Source: 2028657-2018-07-17.xml*