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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1


Volume 13 (2015)

Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential

Yukun Song / Fengming Liu
Published Online: 2018-06-27 | DOI: https://doi.org/10.1515/math-2018-0064


This paper is concerned with a compressible shear thickening fluid-particle interaction model for the evolution of particles dispersed in a viscous non-Newtonian fluid. Taking the influence of non-Newtonian gravitational potential into consideration, the existence and uniqueness of strong solutions are established.

Keywords: Existence; Strong solutions; Compressible; Non-Newtonian fluid

MSC 2010: 76A05; 76A10

1 Introduction

We consider a compressible non-Newtonian fluid-particle interaction model which reads as follows ρt+(ρu)x=0,(ρu)t+(ρu2)x+ρΨxλ[(ux2+μ1)p22ux]x+(P+η)x=ηΦx,(x,t)ΩT(|Ψx|q2Ψx)x=4πg(ρ1|Ω|Ωρdx),ηt+[η(uΦx)]x=ηxx(1)

with the initial and boundary conditions (ρ,u,η)|t=0=(ρ0,u0,η0),xΩ,u|Ω=Ψ|Ω=0,t[0,T],(2)

and the no-flux condition for the density of particles (ηx+ηΦx)|Ω=0,t[0,T].(3)

where ρ, u, η, P(ρ) = γ denote the fluid density, velocity, the density of particle in the mixture and pressure respectively, Ψ denotes the non-Newtonian gravitational potential and the given function Φ(x) denotes the external potential. a > 0, γ > 1, μ1 > 0, p > 2, 1 < q < 2, λ > 0 is the viscosity coefficient and β ≠ 0 is a constant. Ω is a one-dimensional bounded interval, for simplicity we only consider Ω = (0, 1), ΩT = Ω × [0, T].

In fact, there are extensive studies concerning the theory of strong and weak solutions for the multi-dimensional fluid-particle interaction models for the newtonian case. In [1], Carrillo et al. discussed the global existence and asymptotic behavior of the weak solutions for a fluid-particle interaction model. Subsequently, Fang et al. [2] obtained the global classical solution in dimension one. In dimension three, Ballew and Trivisa [3, 4] established the global existence of weak solutions and the existence of weakly dissipative solutions under reasonable physical assumptions on the initial data. In addition, Constantin and Masmoudi [5] obtained the global existence of weak solutions for a coupled incompressible fluid-particle interaction model in 2D case followed the spirit of reference [6].

The non-Newtonian fluid is an important type of fluid because of its immense applications in many fields of engineering fluid mechanics such as inks, paints, jet fuels etc., and biological fluids such as blood (see [7]). Many researchers turned to the study of this type of fluid under different conditions both theoretically and experimentally. For details, we refer the readers to [8, 9, 10, 11, 12] and the references therein. To our knowledge, there seems to be a very few mathematical results for the case of the fluid-interaction model systems with non-Newtonian gravitational potential. There are still no existence results to problem (1)-(3) when p > 2, 1 < q < 2 which describes that the motion of the compressible viscous isentropic gas flow is driven by a non-Newtonian gravitational force.

We are interested in the existence and uniqueness of strong solutions on a one dimensional bounded domain. The strong nonlinearity of (1) bring us new difficulties in getting the upper bound of ρ and the method used in [2] is not suitable for us. Motivated by the work of Cho et al. [13, 14] on Navier-Stokes equations, we establish local existence and uniqueness of strong solutions by the iteration techniques.

Throughout the paper we assume that a = 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)).

We state the definition of strong solution as follows:

Definition 1.1

The (ρ, u, Φ, η) is called a strong solution to the initial boundary value problem (1)-(3), if the following conditions are satisfied:

  1. ρL(0,T;H1(Ω)),uL(0,T;W01,p(Ω)H2(Ω)),ηL(0,T;H2(Ω)),ΨL(0,T;H2(Ω)),ρtL(0,T;L2(Ω)),utL2(0,T;H01(Ω)),ΨtL(0,TH1(Ω)),ρutL(0,T;L2(Ω)),ηtL(0,T;L2(Ω)),((ux2+μ1)p22ux)xL2(0,T;L2(Ω))

  2. For all φL(0, T*;H1(Ω)), φtL(0, T*;L2(Ω)), for a.e. t ∈ (0, T), we have Ωρφ(x,t)dx0tΩ(ρφt+ρuφx)(x,s)dxds=Ωρ0φ(x,0)dx(4)

  3. For all ϕL(0,T;W01,p(Ω)H2(Ω)),ϕtL2(0,T;H01(Ω)), for a.e. t ∈ (0, T), we have Ωρuϕ(x,t)dx0tΩ{ρu2ϕxρΨxϕλ(ux2+μ1)p22uxϕx+(P+η)ϕx+ηΦxϕ}(x,s)dxds=Ωρ0u0ϕ(x,0)dx(5)

  4. For all ϑL(0, T*;H2(Ω)), ϑtL(0, T*H1(Ω)), for a.e. t∈ (0, T), we have 0tΩ|Ψx|q2Ψxϑx(x,s)dxds=0tΩ4πg(ρ1|Ω|Ωρdx)ϑ(x,0)dxds(6)

  5. For all ψL(0, T*;H2(Ω)), ψtL(0, T*;L2(Ω)), for a.e. t ∈ (0, T), we have Ωηψ(x,t)dx0tΩ[η(uΦx)ηx]ψx(x,s)dxds=Ωη0ψ(x,0)dx(7)

1.1 Main results

Theorem 1.2

Let μ1 > 0 be a positive constant and ΦC2(Ω), and assume that the initial data (ρ0, u0, η0) satisfy the following conditions 0ρ0H1(Ω),u0H01(Ω)H2(Ω),η0H2(Ω)

and the compatibility condition [(u0x2+μ1)p22u0x]x+(P(ρ0)+η0)x+η0Φx=ρ012(g+βΦx),(8)

for some gL2(Ω). Then there exist a T*∈(0, +∞) and a unique strong solution (ρ, u, η) to (1)-(3) such that ρL(0,T;H1(Ω)),ρtL(0,T;L2(Ω)),uL(0,T;W01,p(Ω)H2(Ω)),utL2(0,T;H01(Ω)),ηL(0,T;H2(Ω)),ηtL(0,T;L2(Ω)),ΨL(0,T;H2(Ω)),ΨtL(0,TH1(Ω)),ρutL(0,T;L2(Ω)),((ux2+μ1)p22ux)xL2(0,T;L2(Ω)).(9)

2 A priori Estimates for Smooth Solutions

In this section, we will prove the local existence of strong solutions. By virtue of the continuity equation (1)1, we deduce the conservation of mass Ωρ(t)dx=Ωρ0dx:=m0,(t>0,m0>0).

Provided that (ρ, u, η) is a smooth solution of (1)-(3) and ρ0δ, where 0 < δ ≪ 1 is a positive number. We denote by M0=1+μ1+μ11+|ρ0|H1+|g|L2, and introduce an auxiliary function Z(t)=sup0st(1+|u(s)|W01,p+|ρ(s)|H1+|ηt(s)|L2+|η(s)|H1+|ρut(s)|L2).

Then we estimate each term of Z(t) in terms of some integrals of Z(t), apply arguments of Gronwall-type and thus prove that Z(t) is locally bounded.

2.1 Estimate for |u|W01,p

By using (1)1, we rewrite the (1)2 as ρut+ρuux+ρΨx[(ux2+μ1)p22ux]x+(P+η)x=ηΦx.(10)

Multiplying (10) by ut, integrating (by parts) over ΩT, we have ΩTρ|ut|2dxds+ΩT(ux2+μ1)p22uxuxtdxds=ΩT(ρuux+ρΨx+Px+ηx+ηΦx)utdxds.(11)

We deal with each term as follows: Ω(ux2+μ1)p22uxuxtdx=12Ω(ux2+μ1)p22(ux2)tdx=12ddtΩ(0ux2(s+μ1)p22ds)dx0ux2(s+μ1)p22ds=μ1ux2+μ1tp22dt=2p[(ux2+μ1)p2μ1p2]2p|ux|p2pμ1p2 ΩTPxutdxds=ΩTPuxtdxds=ddtΩTPuxdxdsΩTPtuxdxds.

Since from (1)1 we get Pt=γPuxPxu(12) ΩT(ηx+ηΦx)utdxds=ddtΩT(ηx+ηΦx)udxdsΩT(ηx+ηΦx)tudxds ΩT(ηx+ηΦx)tudxds=ΩTηt(uxΦxu)dxds=ΩT[ηxη(uΦx)](uxΦxu)xdxds.

Substituting the above into (11), we obtain 0t|ρut(s)|L22ds+1pΩ|ux(t)|pdxC+Ω|Pux|dx+ΩT(|ρuuxut|+|ρΨxut|+|γPux2|+|Pxuux|)dxds+ΩT(|ηxuxx|+|ηxΦxux|+|ηxΦxxu|+|ηuuxx|+|ηu2Φxx|+|ηuΦxux|+|ηΦxuxx|+|ηΦxΦxxu|+|ηΦx2ux|)dxds.

Using Young’s inequality, we obtain 0t|ρut(s)|L22ds+|ux(t)|LppC+C0t(|ρ|L|u|L2|ux|Lp2+|ρ|L|Ψxx|L22+|P|L|ux|Lp2+|Px|L2|u|L|ux|Lp+|ηx|L2|uxx|L2+|ηx|L2|ux|Lp+|ηx|L2|u|L+|η|L|u|L|uxx|L2+|η|L|u|L2+|η|L|u|L|ux|Lp+|η|L|uxx|L2+|η|L|u|L+|η|L|ux|Lp)ds+C|P(t)|L2pp1.(13)

On the other hand, multiplying (1)3 by Ψ and integrating over Ω, we get Ω|Ψx|qdx=Ω(|Ψx|q2Ψx)xΨdx=4πg(ΩρΨdxm0ΩΨdx)8πgm0|Ψ|L8πgm0|Ψx|Lq1q|Φx|Lqq+1p(8π.gm0)p

Then we have Ω|Ψx|qdxC(m0),1<q<2.

Differentiating (1)3 with respect to x, multiplying it by Ψx and integrating over Ω, we have Ω(|Ψx|q2Ψx)xΨxxdx=4πgΩρxΨxdx.

By virtue of Ω(|Ψx|q2Ψx)xΨxxdx=Ω(|Ψx|q4[(q2)Ψx2+Ψx2]Ψxx2dxCΩ|Ψx|q2Ψxx2dxC|Ψx|Lq2|Ψxx|L22C|Ψxx|L2q2|Ψxx|L22C|Ψxx|L2q

and 4πgΩρxΨxdxCΩ|ρx||Ψx|dxC|ρx|L2p+C|Ψx|L2qC|ρx|L2p+C(ε)|Ψxx|L2q.

Therefore, |Ψxx|L2CZpq(t).(14)

We deal with the term of |uxx|L2. Notice that |[(ux2+μ1)p22ux]x|μ1p22|uxx|.

Then |uxx|C|ρut+ρuux+ρΨx+(P+η)x+ηΦx|.

Taking the above inequality by L2 norm, we get |uxx|L2C|ρut+ρuux+ρΨx+(P+η)x+ηΦx|L2C(|ρ|L12|ρut|L2+|ρ|L|u|L|ux|L2+|ρ|L2|Ψxx|L2+|Px|L2+|ηx|L2+|η|L2|Φx|L2).

Hence, we deduce that |uxx|L2CZmax(pq+1,3)(t).(15)

Moreover, using (1)1, we have |P(t)|L2pp1Ω|P(t)|2dx=Ω|P(0)|2dx+0ts(ΩP(s)2dx)dsΩ|P(0)|2dx+20tΩP(s)γργ1(ρxuρux)dxdsC+C0t|P|L|ρ|Lγ1|ρ|H1|ux|LpdsC(1+0tZ2γ+1(s)ds).(16)

Combining (13)-(16), yields 0t|ρut(s)|L22(s)ds+|ux(t)|LppC(1+0tZmax(2pq,2γ+3)(s)ds),(17)

where C is a positive constant, depending only on M0.

2.2 Estimate for |ρ|H1

From (1)3, taking it by L2 norm, we get |ηxx|L2|ηt+(η(uΦx))x|L2|ηt|L2+|ηx|L2|u|L+C|ηx|L2+|η|H1|ux|Lp+C|η|H1CZ2(t).(18)

Multiplying (1)1 by ρ, integrating over Ω, we have 12ddtΩ|ρ|2ds+Ω(ρu)xρdx=0.

Integrating by parts, using Sobolev inequality, we deduce that ddt|ρ(t)|L22Ω|ux||ρ|2dx|uxx|L2|ρ|L22.(19)

Differentiating (1)1 with respect to x, and multiplying it by ρx, integrating over Ω, using Sobolev inequality, we have ddtΩ|ρx|2dx=Ω[32ux(ρx)2+ρρxuxx](t)dxC[|ux|L|ρx|L22+|ρ|L|ρx|L2|uxx|L2]C|ρ|H12|uxx|L2.(20)

From (19) and (20), by Gronwall’s inequality, it follows that sup0tT|ρ(t)|H12|ρ0|H12exp{C0t|uxx|L2ds}Cexp(C0tZmax(pq+1,3)(s)ds).(21)

Besides, by using (1)1, we can also get the following estimates: |ρt(t)|L2|ρx(t)|L2|u(t)|L+|ρ(t)|L|ux(t)|L2CZ2(t).(22)

2.3 Estimate for |ηt|L2 and |η|H1

Multiplying (1)4 by η, integrating the resulting equation over ΩT, using the boundary conditions (3), Young’s inequality, we have 0t|ηx(s)|L22ds+12|η(t)|L22ΩT(|ηuηx|+|ηΦxηx|)dxds140t|ηx(s)|L22ds+C0t|ux|Lp2|η|H12ds+C0t|η|H12+C140t|ηx(s)|L22ds+C(1+0tZ4(t)ds).(23)

Multiplying (1)4 by ηt, integrating (by parts) over ΩT, using the boundary conditions (3), Young’s inequality, we have 0t|ηt(s)|L22ds+12|ηx(t)|L22ΩT|η(uΦx)ηxt|dxds140t|ηxt(s)|L22ds+C0t|η|H12|ux|Lp2ds+C0t|η|H12ds+C140t|ηxt(s)|L22ds+C(1+0tZ4(t)ds).(24)

Differentiating (1)4 with respect to t, multiplying the resulting equation by ηt, integrating (by parts) over ΩT, we get 0t|ηxt(s)|L22ds+12|ηt(t)|L22=ΩT(η(uΦx))tηxtdxdsC+ΩT(|ηtuηxt|+|ηtΦxηxt|+|ηxutηt|+|ηuxtηt|)dxdsC(1+0t(|ηt|L22||ux|Lp2+|ηt|L22+|ηx|L22|ηt|L22+|η|H12|ηt|L22)dx)+120t|ηxt|L22+120t|uxt|L22C(1+0tZ2γ+6(s)ds).(25)

Combining (23)-(25), we get |η|H12+|ηt|L22+0t(|ηx|L22+|ηt|L22+|ηxt|L22)(s)dsC(1+0tZ2γ+6(s)ds).(26)

2.4 Estimate for |ρut|L2

Differentiating equation (10) with respect to t, multiplying the result equation by ut, and integrating it over Ω with respect to x, we have 12ddtΩρ|ut|2dx+Ω[(ux2+μ1)p22ux]tuxtdx=Ω[(ρu)x(ut2+uuxut+Ψxut)ρuxut2ρΨxtut(P+η)tuxtηtΦxut]dx.(27)

Note that [(ux2+μ1)p22ux]tuxt=(ux2+μ1)p42(p1)(ux2+μ1)uxt2μ1p22uxt2.

Combining (12), (27) can be rewritten into 12ddtΩρ|ut|2dx+Ω|uxt|2dx2Ωρ|u||ut||uxt|dx+Ωρ|u||ux|2|ut|dx+Ωρ|u|2|uxx||ut|dx+Ωρ|u|2|ux||uxt|dx+Ωρ|u||Ψxx||ut|dx+Ωρ|u||Ψx||uxt|dx+Ωρ|ux||ut|2dx+Ωγ|P||ux||uxt|dx+Ω|Px||u||uxt|dx+Ω|ηt||uxt|dx+Ω|ηt||Φx||ut|dx+Ωρ|Ψxt||ut|dx=j=112Ij.(28)

By using Sobolev inequality, Hölder inequality and Young’s inequality, (14), (15), we estimate each term of Ij as follows I1=2Ωρ|u||ut||uxt|dx2|ρ|L12|u|L|ρut|L2|uxt|L2CZ5(t)+17|uxt|L22I2=Ωρ|u||ux|2|ut|dx|ρ|L12|u|L2|ux|Lp2|ρut|L2CZ5(t)I3=Ωρ|u|2|uxx||ut|dx|ρ|L12|u|L2|uxx|L2|ρut|L2CZmax(pq+5,7)(t)I4=Ωρ|u|2|ux||uxt|dx|ρ|L|u|L2|ux|L2|uxt|L2CZ8(t)+17|uxt|L22I5=Ωρ|u||Ψxx||ut|dx|ρ|L12|u|L|Ψxx|L2|ρut|L2CZqp+3(t)I6=Ωρ|u||Ψx||uxt|dx|ρ|L|u|L|Ψxx|L2|uxt|L2CZ2pq+4(t)+17|uxt|L22I7=Ωρ|ux||ut|2dx|ux|L|ρut|L22CZmax(pq+3,5)(t)I8=Ωγ|P||ux||uxt|dxC|P|L|ux|Lp|uxt|L2CZ2γ+2(t)+17|uxt|L22I9=Ω|Px||u||uxt|dx|Px|L2|u|L|uxt|L2CZ2γ+2(t)+17|uxt|L22I10=Ω|ηt||uxt|dx|ηt|L2|uxt|L2CZ2(t)+17|uxt|L22I11=Ω|ηt||Φx||ut|dx|ηt|L2|Φx|L2|ut|LCZ2(t)+17|uxt|L22I12=Ωρ|Ψxt||ut|dxC|ρ|L12|Ψxt|L2|ρut|L2,

where C is a positive constant, depending only on M0.

Next, we deal with the term |Ψxt|L2 of I12. Differentiating (1)3 with respect to t, multiplying it by Ψt, integrating over Ω and using Young’s inequality, we obtain Ω(|Ψx|q2Ψx)tΨxtdx=4πgΩρtΨtdx.

By virtue of Ω(|Ψx|q2Ψx)tΨxtdx=Ω(|Ψx|q4[(q2)Ψx2+Ψx2]Ψxt2dxCΩ|Ψx|q2Ψxt2dxC|Ψx|Lq2|Ψxt|L22C|Ψxx|L2q2|Ψxt|L22

and 4πgΩρtΨtdxCΩ|ρt||Ψt|dxC|ρt|L22+C|Ψt|L22C|ρt|L22+C(ε)|Ψxt|L22,

then |Ψxt|L2CZ2p(2q)q(t). Therefore, I12=Ωρ|Ψxt||ut|dxC|ρ|L12|Ψxt|L2|ρut|L2Z4+p(2q)q(t).

Substituting Ij(j = 1, 2, …, 12) into (28), and integrating over (τ, t) ⊂ (0, T) on the time variable, we have |ρut(t)|L22+τt|uxt|L22(s)ds|ρut(τ)|L22+CτtZmax(2pq+4,8)(s)ds.(29)

To obtain the estimate of |ρut(t)|L22, we need to estimate limτ0|ρut(τ)|L22. Multiplying (10) by ut and integrating over Ω, we have Ωρ|ut|2dx2Ω(ρ|u|2|ux|2+ρ|Ψx|2+ρ1|[(ux2+μ1)p22ux]x+(P+η)x+ηΦx|2)dx.

According to the smoothness of (ρ, u, η), we obtain limτ0Ω(ρ|u|2|ux|2+ρ|Ψx|2+ρ1|[(ux2+μ1)p22ux]x+(P+η)x+ηΦx|2)dx=Ω(ρ|u0|2|u0x|2+ρ0|Ψx|2+ρ01|[(u0x2+μ1)p22u0x]x+(P0+η0)x+η0Φx|2)dx|ρ0|L|u0|L2|u0x|L22+|ρ0|L|Ψx|L22+|g|L22+β|Φx|L22C.

Therefore, taking a limit on τ in (29), as τ → 0, we conclude that |ρut(t)|L22+0t|uxt|L22(s)dsC(1+0tZmax(2pq+4,8)(s)ds),(30)

where C is a positive constant, depending only on M0.

Combining the estimates of (15), (18), (21), (22), (17), (26), (30) and the definition of Z(t), we conclude that Z(t)C~exp(C~~0tZmax(2pq+4,8)(s)ds),(31)

where , c͂͂ are positive constants, depending only on M0. This means that there exist a time T1 > 0 and a constant C > 0, such that esssup0tT1(|ρ|H1+|u|W01,pH2+|η|H2+|ηt|L2+|ρut|L2+|ρt|L2)+0T1(|ρut|L22+|uxt|L22+|ηx|L22+|ηt|L22+|ηxt|L22)dsC.(32)

3 Proof of the Main Theorem

In this section, our proof will be based on the usual iteration argument and some ideas developed in [13, 14]. Precisely, 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: ρtk+ρxkuk1+ρkuxk1=0ρkutk+ρkuk1uxk+ρkΨxk[((uxk)2+μ1)p22uxk]x+Pxk+ηxk=ηkΦx(|Ψxk|q2Ψxk)x=4πg(ρkm0)ηtk+(ηk(uk1Φx))x=ηxxk

with the initial and boundary conditions (ρk,uk,ηk)|t=0=(ρ0,u0,η0)uk|Ω=(ηxk+ηkΦx)|Ω=0

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 esssup0tT1(|ρk|H1+|uk|W01,pH2+|ηk|H2+|ηtk|L2+|ρkutk|L2+|ρtk|L2)+0T1(|ρkutk|L22+|uxtk|L22+|ηxk|L22+|ηtk|L22+|ηxtk|L22)dsC,(33)

where C is a generic constant depending only on M0, but independent of k.

In addition, we first find ρk from the initial problem ρtk+uk1ρxk+uxk1ρk=0,andρk|t=0=ρ0

with smooth function uk−1, obviously, there is a unique solution ρk to the above problem and also by a standard argument, we could obtain that ρk(x,t)δexp[0T1|uxk1(.,s)|Lds]>0,for all t(0,T1).

Next, we have to prove that the approximate solution (ρk, uk, ηk) converges to a solution to the original problem (1) in a strong sense. To this end, let us define ρ¯k+1=ρk+1ρk,u¯k+1=uk+1uk,Ψ¯k+1=Ψk+1Ψk,η¯k+1=ηk+1ηk,

then we can verify that the functions ρk+1, uk+1, ηk+1 satisfy the system of equations ρ¯tk+1+(ρ¯k+1uk)x+(ρku¯k)x=0(34) ρk+1u¯tk+1+ρk+1uku¯xk+1([((uxk+1)2+μ1)p22uxk+1]x[((uxk)2+μ1)p22uxk]x)=ρ¯k+1(utk+ukuxk+Ψxk+1)(Pk+1Pk)x+ρk(u¯kuxkΨ¯xk+1)η¯xk+1η¯k+1Φx(35) (|Ψxk+1|q2Ψxk+1)x(|Ψxk|q2Ψxk)x=4πgρ¯k+1(36) η¯tk+1+(ηku¯k)x+(η¯k+1(ukΦx))x=η¯xxk+1.(37)

Multiplying (34) by ρk+1, integrating over Ω and using Young’s inequality, we obtain ddt|ρ¯k+1|L22C|ρ¯k+1|L22|uxk|L+|ρk|H1|u¯xk|L2|ρ¯k+1|L2C|uxxk|L2|ρ¯k+1|L22+Cζ|ρk|H12|ρ¯k+1|L22+ζ|u¯xk|L22Cζ|ρ¯k+1|L22+ζ|u¯xk|L22,(38)

where Cζ is a positive constant, depending on M0 and ζ for all t < T1 and k ≥ 1.

Multiplying (35) by uk+1, integrating over Ω and using Young’s inequality, we obtain 12ddtΩρk+1|u¯k+1|2dx+Ω([((uxk+1)2+μ1)p22uxk+1]x[((uxk)2+μ1)p22uxk]x)u¯k+1dxCΩ(|ρ¯k+1|(|utk|+|ukuxk|+|Ψxk+1|)|u¯k+1|+|Pxk+1Pxk||u¯k+1|+|ρk|u¯k||uxk||u¯k+1|+|ρk||Ψ¯xk+1||u¯k+1|+|η¯xk+1||u¯k+1|+|η¯k+1Φx||u¯k+1|)dx.(39)

Let σ(s)=(s2+μ1)p22s,

then σ(s)=((s2+μ1)p22s)=(s2+μ1)p42((p1)s2+μ1)μ1p22.

We estimate the second term of (39) as follows Ω([((uxk+1)2+μ1)p22uxk+1]x[((uxk)2+μ1)p22uxk]x)u¯k+1dx=Ω01σ(θuxk+1+(1θ)uxk)dθ|u¯xk+1|2dxμ1p22Ω|u¯xk+1|2dx.(40)

Similarly, multiplying (36) by Ψk+1, integrating over Ω, we get Ω[(|Ψxk+1|q2Ψxk+1)x(|Ψxk|q2Ψxk)x]Ψ¯k+1dx=4πgΩρ¯k+1Ψ¯k+1dx,(41)

since Ω[|Ψxk+1|q2Ψxk+1|Ψxk|q2Ψxk]Ψ¯xk+1dx=(q1)Ω(01|θΨxk+1+(1θ)Ψxk|q2dθ)(Ψ¯xk+1)2dx

and 01|θΨxk+1+(1θ)Ψxk|q2dθ=011|θΨxk+1+(1θ)Ψxk|2qdθ011(|Ψxk+1|+|Ψxk|2q)dθ=1(|Ψxk+1|+|Ψxk|)2q.

Then Ω[|Ψxk+1|q2Ψxk+1|Ψxk|q2Ψxk]Ψ¯xk+1dx1(|Ψxk+1(t)|L+|Ψxk(t)|L)2qΩ(Ψ¯xk+1)2dx.

That means (41) turns into Ω(Ψ¯xk+1)2dxC|ρ¯k+1|L22.(42)

Substituting (40) and (42) into (39), using Young’s inequality, yields ddtΩρk+1|u¯k+1|2dx+Ω|u¯xk+1|2dxC(|ρ¯k+1|L2|uxtk|L2|u¯xk+1|L2+|ρ¯k+1|L2|uxk|Lp|uxxk|L2|u¯xk+1|L2+|ρ¯k+1|L2|Ψxk+1|L2|u¯xk+1|L2+|Pk+1Pk|L2|u¯xk+1|L2+|ρk|L212|ρku¯k|L2|uxxk|L2|u¯xk+1|L2+|ρk|L|Ψ¯xk+1|L2|u¯xk+1|L2+|η¯k+1|L2|u¯xk+1|L2+|η¯k+1|L2|u¯xk+1|L2)Bζ(t)|ρ¯k+1|L22+C(|ρku¯k|L22+|η¯k+1|L22)+ζ|u¯xk+1|L22,(43)

where Bζ(t)=C(1+|uxtk(t)|L22), for all tT1 and k ≥ 1. Using (33) we derive 0tBζ(s)dsC+Ct.

Multiplying (37) by ηk+1, integrating over Ω, using (33) and Young’s inequality, we have 12ddtΩ|η¯k+1|2dx+Ω|η¯xk+1|2dxΩ|η¯k+1||ukΦx||η¯xk+1|dx+Ω(|ηk||u¯k|)x|η¯k+1|dx|η¯k+1|L2|ukΦx|L|η¯xk+1|L2+|ηxk|L2|u¯k|L|η¯k+1|L2+|ηk|L|u¯xk|L2|η¯k+1|L2Cζ|η¯k+1|L22+ζ|η¯xk+1|L22+ζ|u¯xk|L22.(44)

Collecting (38), (43) and (44), we obtain ddt(|ρ¯k+1(t)|L22+|ρk+1u¯k+1(t)|L22+|η¯k+1(t)|L22)+|u¯xk+1(t)|L22+|η¯xk+1|L22Eζ(t)|ρ¯k+1(t)|L22+C|ρku¯k|L22+Cζ|η¯k+1|L22+ζ|u¯xk|L22,(45)

where Eζ(t) depends only on Bζ(t) and Cζ, for all tT1 and k ≥ 1. Using (33), we have 0tEζ(s)dsC+Cζt.

Integrating (45) over (0, t) ⊂ (0, T1) with respect to t, using Gronwall’s inequality, we have |ρ¯k+1(t)|L22+|ρk+1u¯k+1(t)|L22+|η¯k+1(t)|L22+0t|u¯xk+1(t)|L22ds+0t|η¯xk+1|L22dsCexp(Cζt)0t(|ρku¯k(s)|L22+|u¯xk(s)|L22)ds.(46)

From the above recursive relation, choose ζ > 0 and 0 < T* < T1 such that C exp (CζT)<12, using Gronwall’s inequality, we deduce that k=1K[sup0tT(|ρ¯k+1(t)|L22+|ρk+1u¯k+1(t)|L22+|η¯k+1(t)|L22dt+0T|u¯xk+1(t)|L22+0T|η¯xk+1(t)|L22dt]<C.(47)

Since 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 ρρδin L(0,T;L2(Ω)),(48) uuδinL(0,T;L2(Ω))L2(0,T;H01(Ω)),(49) ηηδinL(0,T;L2(Ω))L2(0,T;H1(Ω)),(50)

and there also holds esssup0tT1(|ρδ|H1+|uδ|W01,pH2+|ηδ|H2+|ηtδ|L2+|ρδutδ|L2+|ρtδ|L2)+0T(|ρδutδ|L22+|uxtδ|L22+|ηxδ|L22+|ηtδ|L22+|ηxtδ|L22)dsC.(51)

For each small δ > 0, let ρ0δ=Jδρ0+δ,Jδ is a mollifier on Ω, and u0δH01(Ω)H2(Ω) is a smooth solution of the boundary value problem [((u0xδ)2+μ1)p22u0xδ]x+(P(ρ0δ)+η0δ)x+η0δΦx=(ρ0δ)12(gδ+βΦx),u0δ(0)=u0δ(1)=0,(52)

where gδC0 and satisfies |gδ|L2|g|L2,limδ0+|gδg|L2=0.

We deduce that (ρδ, uδ, ηδ) is a solution of the following initial boundary value problem ρt+(ρu)x=0,(ρu)t+(ρu2)x+ρΨxλ[(ux2+μ1)p22ux]x+(P+η)x=ηΦx,(|Ψx|q2Ψx)x=4πg(ρ1|Ω|Ωρdx),ηt+(η(uΦx))x=ηxx,(ρ,u,η)|t=0=(ρ0δ,u0δ,η0δ),u|Ω=Ψ|Ω=(ηx+ηΦx)|Ω=0,

where ρ0δδ,p>2,1<q<2.

By the proof of Lemma 2.3 in [11], there exists a subsequence {u0δj} of {u0δ}, as δj0+,u0δu0 in H01(Ω)H2(Ω),(|u0xδj|p2u0xδj)x(|u0x|p2u0x)x in L2(Ω), Hence, u0 satisfies the compatibility condition (8) of Theorem 1.2. By virtue of the lower semi-continuity of various norms, we deduce that (ρ, u, η) satisfies the following uniform estimate esssup0tT1(|ρ|H1+|u|W01,pH2+|η|H2+|ηt|L2+|ρut|L2+|ρt|L2)+0T(|ρut|L22+|uxt|L22+|ηx|L22+|ηt|L22+|ηxt|L22)dsC,(53)

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.


The authors would like to thank the anonymous referees for their valuable suggestions and comments which improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (Nos.11526105; 11572146), the founds of education department of Liaoning Province (Nos.JQL201715411; JQL201715409).


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About the article

Received: 2018-02-15

Accepted: 2018-05-06

Published Online: 2018-06-27

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 704–717, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0064.

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© 2018 Song and Liu, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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