# Global Existence of Cylinder Symmetric Solutions for the Nonlinear Compressible Navier-Stokes Equations

Authors: Lan Huang; Fengxiao Zhai; Beibei Zhang
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We prove the global existence of cylinder symmetric solutions to the compressible Navier-Stokes equations with external forces and heat source inR3 for any large initial data. Some new ideas and more delicate estimates are used to prove this result.


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## Body

## 1. Introduction

In this paper, we study the global existence of cylinder symmetric solutions to the nonlinear compressible Navier-Stokes equations with external forces and heat source in a bounded domainG={r∈R+,0<a<r<b<+∞} of R3, where r is the radial variable. In the Eulerian coordinates, the system under consideration are expressed as (1.1)ρt+(ρu)r+ρur=0,(1.2)ρ(ut+uur-v2r)+Pr-ν(ur+ur)r=f1(r,t),(1.3)ρ(vt+uvr+uvr)-μ(vr+vr)r=f2(r,t),(1.4)ρ(wt+uwr)-μ(wrr+wrr)=f3(r,t),(1.5)CVρ(θt+uθr)-κ(θrr+θrr)+P(ur+ur)-Q=g(r,t),
where (1.6)P=γρθ,Q=λ(ur+ur)2+μ[(vr-vr)2+wr2+2ur2+2(ur)2],
and ρ is the mass density, θ is the absolute temperature, u,v,w are the radial velocity, angular velocity, and axial velocity, respectively, and λ, μ, ν, γ, CV, κ, λ are the constants satisfying γ,CV,κ,μ>0, 3λ+2μ≥0 (ν=λ+2μ). f1,f2,f3, and g represent external forces and heat source, respectively. For system (1.1)–(1.5), we consider the following initial boundary value problem: (1.7)ρ(r,0)=ρ0(r),(u,v,w)(r,0)=(u0,v0,w0)(r),θ(r,0)=θ0(r),r∈G,(1.8)(u,v,w)(a,t)=(u,v,w)(b,t)=0,θr(a,t)=θr(b,t)=0,t≥0.To show the global existence, it is convenient to transform the system (1.1)–(1.5) to that in the Lagrangian coordinates. The Eulerian coordinates (r,t) are connected to the Lagrangian coordinates (ξ,t) by the relation (1.9)r(ξ,t)=r0(ξ)+∫0tũ(ξ,τ)dτ,
where ũ(ξ,t)=u(r(ξ,t),t) and (1.10)r0(ξ)=η-1(ξ),η(r)=∫arsρ0(s)ds,r∈G.
It should be noted that if inf{ρ0(s):s∈(a,b)}>0, then η is invertible. It follows from (1.1), (1.8), and (1.10) that (1.11)∫arsρ(s,t)ds=∫ar0sρ0(s)ds=ξ,
and G is transformed into Ω=(0,L) with (1.12)L=∫absρ(s,t)ds=∫absρ0(s),∀t≥0.
Differentiating (1.11) with respect to ξ yields (1.13)∂ξr(ξ,t)=r(ξ,t)-1ρ-1(r(ξ,t),t).
In general, for a function ϕ(r,t)=ϕ̃(ξ,t)=ϕ(r(ξ,t),t), we easily get (1.14)∂tϕ̃(ξ,t)=∂tϕ(r,t)+u∂rϕ(r,t),(1.15)∂ξϕ̃(ξ,t)=∂rϕ(r,t)∂ξr(ξ,t)=∂rϕ(r,t)rρ-1(r,t).
Without danger of confusion, we denote (ρ̃,ũ,ṽ,w̃,θ̃) still by (ρ,u,v,w,θ) and (ξ,t) by (x,t). We set τ∶=1/ρ to denote the specific volume. Therefore, by virtue of (1.13)–(1.15), system (1.1)–(1.8) in the new variables (x,t) read (1.16)τt=(ru)x,(1.17)ut=r[ν(ru)x-γθτ]x+v2r+f1(r(x,t),t),(1.18)vt=μr[(rv)xτ]x-uvr+f2(r(x,t),t),(1.19)wt=μr[(rw)xτ]x+μτwr2+f3(r(x,t),t),(1.20)CVθt=κ[r2θxτ]x+1τ[ν(ru)x-γθ](ru)x+μ[(rv)x]2τ+μr2wx2τ-2μ(u2+v2)x+g(r(x,t),t),
together with (1.21)τ(x,0)=τ0(x),(u,v,w)(x,0)=(u0,v0,w0)(x),θ(x,0)=θ0(x),x∈[0,L],(1.22)(u,v,w)(0,t)=(u,v,w)(L,t)=0,θx(0,t)=θx(L,t)=0,t≥0.
By (1.9) and (1.13), we have(1.23)r(x,t)=r0(x)+∫0tu(x,s)ds,r0(x)=[a2+2∫0xτ0(y)dy]1/2,rt(x,t)=u(x,t),r(x,t)rx(x,t)=τ(x,t).Now let us first recall the related results in the literature. When there were no external forces and heat source, in two or three dimensions, the global existence and large time behavior of smooth solutions to the equations of a viscous polytropic ideal gas have been investigated for general domains only in the case of sufficiently small initial data, see, for example, [1–3]. For any large initial data, the global existence of generalized solutions was shown in [4–7]. Recently, Qin [8] proved the exponential stability in H1 and H2, and Qin and Jiang [9] studied the global existence and exponential stability in H4 with smallness of initial total energy.When there exist external forces and heat forces, for one-dimensional case, the system is isentropic compressible Navier-Stokes equations. Mucha [10] obtained the exponential stability under various boundary conditions, Yanagi [11] established the existence of classical solutions, and Qin and Zhao [12] proved the global existence and asymptotic behavior of solutions for pressure P=ργ with γ=1. Later on, Zhang and Fang [13] studied the global existence and uniqueness for γ>1. For nonisentropic compressible Navier-Stokes equations, Qin and Yu [14] proved the global existence and asymptotic behavior for perfect gas. In two- or three-dimensional case and the external force and heat source f≠0, g≠0, Qin and Wen [15] proved the global existence of spherically symmetric solutions. In this paper, we will prove the global existence of cylinder symmetric solutions with external forces and heat source in a bounded domain in R3.The notation in this paper will be as follows:Lp¯,1≤p¯≤+∞, Wm,p¯, m∈N, H1=W1,2, H01=W01,2 denote the usual (Sobolev) spaces on (0,L). In addition, ∥·∥B denotes the norm in the space B; we also put ∥·∥=∥·∥L2. We denote by Ck(I,B),k∈N0, the space of k-times continuously differentiable functions from I⊆R into a Banach space B, and likewise by Lp¯(I,B), 1≤p¯≤+∞ the corresponding Lebesgue spaces. Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use C1 to denote the generic positive constant depending on the H1-norm of the initial data and time T.We suppose thatfi(r(x,t),t) (i=1,2,3), g(r,t) satisfy, for any T>0, (1.24)fi∈L1([0,T],L∞[0,L])∩L2([0,T],L2[0,L]),(1.25)g(r,t)>0,g∈L1([0,T],L∞[0,L])∩L2([0,T],L2[0,L]).We are now in a position to state our main theorems.Theorem 1.1.
Assume that (1.24)-(1.25) hold; if (τ0,u0,v0,w0,θ0)∈H1[0,L]×H01[0,L]×H01[0,L]×H01[0,L]×H1[0,L], τ0(x)>0, θ0(x)>0 on [0,L] and the initial data are compatible with the boundary conditions (1.22), then for problem (1.16)–(1.22) there exists a unique global solution (τ,u,v,w,θ)∈C([0,T],H1[0,L]×H01[0,L]×H01[0,L]×H01[0,L]×H1[0,L]) such that, for any T>0,
(1.26)0<a≤r(x,t)≤b,(x,t)∈[0,L]×[0,T],0<C1-1≤τ(x,t)≤C1,(x,t)∈[0,L]×[0,T],‖τ(t)‖H12+‖u(t)‖H12+‖v(t)‖H12+‖w(t)‖H12+‖θ(t)‖H12+‖r(t)‖H22+∫0t(‖τ‖H12+‖u‖H22+‖v‖H22+‖w‖H22+‖θ‖H22+‖ut‖2+‖vt‖2+‖wt‖2+‖θt‖2)(τ)dτ≤C1,∀t∈[0,T].


## 2. Proof of Theorem1.1

In this section we will complete the proof of Theorem1.1. To this end, we assume that in this section all assumptions in Theorem 1.1 hold. The proof of Theorem can be divided into the following several lemmas.Lemma 2.1.
One has(2.1)a=r(0,t)≤r(x,t)≤r(L,t)=b,∀(x,t)∈[0,L]×[0,+∞).Proof.
The proof of (2.1) is borrowed from [6, 8]; please refer to (2.1) in [6] or Lemma  2.1 in [8] for detail.Lemma 2.2.
The global solution(τ(t),u(t),v(t),w(t),θ(t)) to problems (1.16)–(1.22) satisfies the following estimates:
(2.2)∫0L[12(u2+v2+w2)+CVθ](x,t)dx≤C1,(2.3)∫0LU(x,t)dx+∫0t∫0L(κr2θx2τθ2+τu2θ+ux2τθ+(ru)x2τθ+wx2τθ+(τr-1v-rvx)2τθ+gθ)(x,s)dxds≤C1,
where
(2.4)U(x,t)=12(u2+v2+w2)+γ(τ-log⁡⁡τ-1)+CV(θ-log⁡⁡θ-1).Proof.
Multiplying (1.17)–(1.19) by u, v, and w, respectively, adding up the results, and using (1.16), we have
(2.5)ddt[12(u2+v2+w2)+CVθ]=[κr2θxτ+ru(ν(ru)x-γθ)τ+μrv(rv)xτ+μr2wwxτ-2μ(u2+v2)]x+f1u+f2v+f3w+g.
Integrating (2.5) with respect to x and t over QT=[0,L]×[0,t](t∈[0,T],∀T>0), using boundary condition (1.22), we obtain
(2.6)∫0L[12(u2+v2+w2)+CVθ]dx=∫0L[12(u02+v02+w02)+CVθ0]dx+∫0t∫0L(f1u+f2v+f3w+g)(x,s)dxds≤C1+C1∫0t∫0L(u2+v2+w2)(x,s)dxds+C1∫0t∫0L(f12+f22+f32)(x,s)dxds+C1∫0t‖g‖L∞ds
which, by using Gronwall’s inequality and (1.24)-(1.25), gives (2.2).
By (1.16)–(1.20), we can easily obtain
(2.7)Ut+κr2θx2τθ2+ν(ru)x2+μ(rv)x2+μr2wx2τθ-2μ(u2+v2)xθ+gθ=[κ(θ-1)r2θxτθ+ru(ν(ru)x-γθ)τ+μrv(rv)xτ+μr2wwxτ-2μ(u2+v2)+γru]x+f1u+f2v+f3w+g.
Note that constants ν=λ+2μ and
(2.8)ν(ru)x2+μ(rv)x2τθ-2μ(u2+v2)xθ=2μ(τ2r-2u2+r2ux2)+λ(ru)x2τθ+μ(τr-1v-rvx)2τθ≥C1-1(τu2θ+ux2+λ(ru)x2τθ+μ(τr-1v-rvx)2τθ).
Integrating (2.7) with respect to x and t over QT, using (1.22), (1.24)-(1.25), and (2.8), we conclude
(2.9)∫0LU(x,t)dx+∫0t∫0L(κr2θx2τθ2+τu2θ+ux2+(ru)x2+wx2+(τr-1v-rvx)2τθ+gθ)dxds≤C1+∫0t∫0L(f1u+f2v+f3w+g)(x,s)dxds≤C1+C1∫0t∫0L(u2+v2+w2+f12+f22+f32)(x,s)dxds+C1∫0t‖g‖L∞ds≤C1.
The proof is complete.Next we adapt and modify an idea of Qin and Wen [15] for one-dimensional case to give a representation for τ.Let(2.10)σ(x,t)∶=ν(ru)x-γθτ+∫0x(v2-u2r2+f1(r(y,t),t)r)dy,(2.11)h(x,t)∶=∫0xu0r0dy+∫0tσ(x,s)dx.
Then, we infer from (1.16) and (1.17) that (2.12)hx=ur,ht=σ.
By (1.16) and (2.12), we have (2.13)(hτ)t=(ruh)x-u2+ν(ru)x-γθ+τ∫0x(v2-u2r2+f1r)dx.
Integrating (2.13) with respect to x and t over QT, we obtain (2.14)∫0Lhτdx=∫0Lh0τ0dx-∫0t∫0L(u2+γθ)dxds+∫0t∫0Lτ∫0x(v2-u2r2+f1r)dydxds=∫0Lh0τ0dx-∫0t∫0L(u2+γθ)dxds+∫0t∫0L(rrx)∫0x(v2-u2r2+f1r)dydxds=∫0Lh0τ0dx-∫0t∫0L(u2+γθ)dxds+b22∫0t∫0L(v2-u2r2+f1r)dxds-∫0t∫0L(v2-u22+rf12)dxds,
where h0(x)∶=h(x,0). It follows from integration of (1.16) over QT and use of (1.22) that (2.15)∫0Lτ(x,t)dx=∫0Lτ0(x)dx=τ*.
If we apply the mean value theorem to (2.14) and use (2.15), we conclude there is an x0(t)∈[0,L] such that (2.16)h(x0(t),t)=1τ*∫0Lh(x,t)τ(x,t)dx.
Therefore, we derive from (2.11), (2.14), and (2.16) that (2.17)∫0tσ(x0(t),s)ds=h(x0(t),t)-∫0x0(t)u0r0dx=-1τ*∫0t∫0L(v2+u22+γθ+rf12)dxds+b22τ*∫0t∫0L(v2-u2r2+f1r)dxds+1τ*∫0Lh0(x)τ0(x)dx-∫0x0(t)u0r0dx.
Using (2.17), we will show the representation of specific volume τ.Lemma 2.3.
One has the following representation:(2.18)τ(x,t)=D(x,t)B(x,t)[1+γν∫0tθ(x,s)B(x,s)D(x,s)ds],x∈[0,L],
where
(2.19)D(x,t)=τ0(x)exp⁡⁡{1ν[1τ*∫0Lτ0h0dx-∫0xu0r0dy+∫x0(t)xurdy-∫0t∫0xf1rdyds-1τ*∫0t∫0Lrf12dxds+b22τ*∫0t∫0Lf1rdxds]},B(x,t)=exp⁡⁡{1ν[1τ*∫0t∫0L(u2+v22+γθ)dxds-b22τ*∫0t∫0Lv2-u2r2dxds+∫0t∫0xv2-u2r2dxds]}.Proof.
By (1.16) and (1.17), we have
(2.20)(ur)t=σx=ν(log⁡⁡τ)tx-γ(θτ)x+v2-u2+rf1r2.
Integrating (2.20) over [x0(t),x]×[0,t] and using (2.17), we derive
(2.21)νlog⁡⁡τ(x,t)-γ∫0tθ(x,s)τ(x,s)ds=νlog⁡⁡τ0(x)+∫0tσ(x0(t),s)ds-∫0t∫0xv2-u2+rf1r2+∫x0(t)x(ur-u0r0)dy=νlog⁡⁡τ0(x)-1τ*∫0t∫0L(v2+u22+γθ+rf12)dxds+b22τ*∫0t∫0L(v2-u2r2+f1r)dxds+1τ*∫0Lh0(x)τ0(x)dx-∫0xu0r0dx-∫0t∫0xv2-u2+rf1r2dyds+∫x0(t)xurdy
which, when the exponentials are taken, turns into
(2.22)B(x,t)D(x,t)=1τ(x,t)exp⁡⁡(γν∫0tθ(x,s)τ(x,s)ds).
Multiplying (2.22) by γθ/ν and integrating the resulting equation with respect to t, we arrive at
(2.23)exp⁡⁡(γν∫0tθ(x,s)τ(x,s)ds)=1+γν∫0tθ(x,s)B(x,s)D(x,s)ds.
Substituting this into (2.22), we obtain (2.18). The proof is complete.Lemma 2.4.
There are positive constantsτ¯ and τ̲, such that, for any T>0,
(2.24)τ̲≤τ(x,t)≤τ¯,(x,t)∈[0,L]×[0,T].Proof.
Recalling the definitionD(x,t), we have by (1.24), Cauchy-Schwarz’s inequality, and Lemma 2.1 that
(2.25)|1τ*∫0t∫0Lrf12dxds-b22τ*∫0t∫0Lf1rdxds-∫0t∫0xf1rdyds|≤C1∫0t‖f1‖L∞≤C1
which, along with Lemma 2.2, gives
(2.26)0<C1-1≤D(x,t)≤C1,(x,t)∈[0,L]×[0,T].
By Lemmas 2.1 and 2.2, we easily obtain, for any 0≤s≤t,
(2.27)B(x,t)≤C1,B(x,s)B(x,t)≤exp⁡⁡{-C1(t-s)}.
Therefore, we derive from (2.2), (2.18), and (2.26)-(2.27) that
(2.28)τ(x,t)≥D(x,t)B(x,t)≥τ̲,(2.29)τ(x,t)≤C1[1+∫0tθe-C1(t-s)ds]≤C1+∫0t[(∫0Lθdx)1/2+(∫0Lr2θx2τθ2dx)1/2(∫0Lτθr2dx)1/2]2e-C1(t-s)ds≤C1+C1(1+∫0tmax⁡[0,L]τ(⋅,s)∫0Lr2θx2τθ2dxds)ds
which, by using Gronwall’s inequality and (2.28), gives (2.24). The proof is complete.Remark 1.
If the initial data or initial energy are small enough, we can obtain the uniform estimate independent of timet about specific volume τ under assumptions of external forces. Moreover, we can prove the large-time behavior of solutions.Lemma 2.5.
Under the assumptions of Theorem1.1, one has, for any T>0 and for all t∈[0,T],
(2.30)∫0L(θ2+u4+v4+w4)dx+∫0t∫0L(θx2+u2ux2+v2vx2+w2wx2)(x,s)dxds≤C1.Proof.
Multiplying (2.5) by (1/2)(u2+v2+w2)+CVθ and then integrating the result over QT, we have
(2.31)12∫0L(12(u2+v2+w2)+CVθ)2dx≤C1-CVκ2∫0t∫0Lr2θx2τdxds+C1∫0t∫0L{r2u2ux2+r2v2vx2+r2w2wx2τ+u4+v4+θ2+θ2u2}dxds+∫0t∫0L(f1u+f2v+f3w+g)(12(u2+v2+w2)+CVθ)dxds,
where
(2.32)∫0t∫0Lg(12(u2+v2+w2)+CVθ)dxds≤∫0t‖g‖L∞∫0L(12(u2+v2+w2)+CVθ)dxds≤C1,∫0t∫0L(f1u+f2v+f3w)(12(u2+v2+w2)+CVθ)dxds≤C1∫0t(‖f1‖L∞+‖f2‖L∞+‖f3‖L∞)∫0L(u2+u4+v2+v4+w2+w4)dxds+∫0t∫0Lθ2dxds.
Multiplying (1.17) by u3 and then integrating the result over QT, we get
(2.33)14∫0Lu4dx≤C1-ντ¯∫0t∫01r2u2ux2dxds+C1∫0t∫0L(u4+θ2u2)(x,s)dxds+∫0t∫0L(v2r+f1)u3dxds≤C1-ντ¯∫0t∫01r2u2ux2dxds+C1∫0t∫0L(u4+θ2u2)(x,s)dxds+C1∫0t(‖f1‖L∞+‖u‖L∞2)∫0L(v4+u2+u4)dxds.
Similarly, multiplying (1.18) and (1.19) by v3 and w3, respectively, and then integrating over QT, we have
(2.34)14∫0Lv4dx≤C1-μτ¯∫0t∫01r2v2vx2dxds+C1∫0t∫0L(v4+u2v4+f2(v2+v4))dxds≤C1-μτ¯∫0t∫01r2v2vx2dxds+C1∫0t(‖f2‖L∞+‖u‖L∞2+1)∫0L(v2+v4)dxds,14∫0Lw4dx≤C1-μτ¯∫0t∫01r2w2wx2dxds+C1∫0t∫0L(w4+f3(w2+w4))dxds≤C1-μτ¯∫0t∫01r2w2wx2dxds+C1∫0t(‖f3‖L∞+1)∫0L(w2+w4)dxds.
Multiplying (2.31) and (2.33) by μ/(2τ¯C1) and μ/ν, respectively, adding up the resulting inequalities, and using (2.34) to obtain, with the help of (2.32), the following result:
(2.35)∫0L(u4+v4+w4+θ2)dx+∫0t∫0L(θx2+u2ux2+v2vx2+w2wx2)(x,s)dxds≤C1+∫0t(‖f1‖L∞+‖f2‖L∞+‖f3‖L∞+1)∫0L(θ2+u4+v4+w4)dxds+∫0t‖u‖L∞2∫0L(θ2+u4+v4+w4)dxds.
On the other hand, by (2.3) and (2.20),
(2.36)∫0t‖u‖L∞2ds≤∫0t(∫0L|ux|2dx)2ds≤∫0t∫0Lux2τθdx∫0Lτθdxds≤C1.
In view of (1.24)-(1.25) and (2.36), we apply Gronwall’s inequality to (2.35) to obtain (2.30). The proof is complete.Lemma 2.6.
Under the assumptions of Theorem1.1, one has, for any T>0,
(2.37)∫0Lτx2(x,t)dx+∫0t∫0Lθτx2(x,s)dxds≤C1,∀t∈[0,T].Proof.
By means of (1.16), we rewrite (1.17) as
(2.38)(ur-ντxτ)t=γ(θτx-τθx)τ2+v2-u2+rf1r2.
Multiplying (2.38) by (u/r)-(ντx/τ) in L2[0,L] and using Lemmas 2.1–2.5, we arrive at
(2.39)12ddt‖ur-ντxτ‖2+νγ∫0Lθτx2τ2rdx=∫0L[γ(θτx-τθx)urτ2+γντxθxrτ2+v2-u2+rf1r2(ur-ντxτ)]dx≤12νγ∫0Lθτx2τ2rdx+C1∫0L(θu2+θx2+u2+θx2θ+u4+v4+f12+(ur-ντxτ)2)dx.
Integrating it with respect to t, using Lemmas 2.1-2.5, (1.24), and (2.36), we get
(2.40)‖ur-ντxτ‖2+∫0t∫0Lθτx2dxds≤C1+∫0t‖ur-ντxτ‖2ds+C1∫0t∫0L(θu2+f12+θx2+θx2θ)dxds≤C1+∫0t‖ur-ντxτ‖2ds+∫0t‖u‖L∞2∫0Lθdxds+∫0t∫0Lθx2(1+1θ2)dxds≤C1+∫0t‖ur-ντxτ‖2ds.
We exploit the Gronwall inequality to (2.40) to obtain (2.37). The proof is complete.Lemma 2.7.
Under the assumptions of Theorem1.1, one has, for any T>0,
(2.41)∫0Lux2(x,t)dx+∫0t∫0L(ut2+uxx2)(x,s)dxds≤C1,∀t∈[0,T],(2.42)∫0Lvx2(x,t)dx+∫0t∫0L(vt2+vxx2)(x,s)dxds≤C1,∀t∈[0,T],(2.43)∫0Lwx2(x,t)dx+∫0t∫0L(wt2+wxx2)(x,s)dxds≤C1,∀t∈[0,T].Proof.
Multiplying (1.17) by ut, integrating the result over QT, using Lemmas 2.1–2.6, and taking into account that (ru)x(rut)x/τ=(1/2)((ru)x2/τ)t-((ru)x(u2)x/τ)+((ru)x3/2τ2), we obtain
(2.44)∫0t∫0Lut2dxds=-∫0t∫0L(rut)xν(ru)xτ-∫0t∫0L(γθτ)xrutdxds+∫0t∫0L(v2r+f1)utdxds=ν2∫0L(r0u0)x2τ0dx-ν2∫0L(ru)x2τdx+γ∫0t∫0Lrτθx-θτxτ2utdxds-ν∫0t∫0Lu2((ru)xτ)xdxds+ν2∫0t∫0Lru((ru)x2τ2)xdxds+∫0t∫0L(v2r+f1)utdxds≤C1-ν2∫0L(ru)x2τdx+14∫0t∫0Lut2dxds+C1∫0t∫0L(θ2τx2+θx2+f12+v4)dxds-ν∫0t∫0Lu2[utr+1r(γθτ)x-v2+f1rr2]dxds+ν2∫0t∫0L2ru(ru)xτ[utr+1r(γθτ)x-v2+f1rr2]dxds≤C1-ν2∫0L(ru)x2τdx+12∫0t∫0Lut2dxds+C1∫0t∫0L(θ2τx2+θx2+f12+v4+u4)dxds+C1∫0t∫0Lu2(ru)x2τ(x,s)dxds.
We derive from (2.30) that
(2.45)‖θ‖L∞2≤∫0Lθ2dx+2∫0L|θθx|dx≤C1∫0L(θx2+θ2)dx≤C1+C1∫0Lθx2dx.
Therefore, using (1.24), (2.30), (2.36), (2.37), and (2.44)-(2.45), we conclude
(2.46)∫0L(ru)x2τdx+∫0t∫0Lut2dxds≤C1+C1∫0t‖θ‖L∞2∫0Lτx2dxds+C1∫0t‖u‖L∞2∫0L(ru)x2τ(x,s)dxds
which, by applying the Gronwall inequality, implies
(2.47)∫0Lux2(x,t)dx+∫0t∫0Lut2(x,s)dxds≤C1.
By (1.17), we have
(2.48)∫0t∫0Luxx2(x,s)dxds≤C1∫0t∫0L(ut2+ux2+u2+τx2ux2+θ2τx2+θx2+v4+f12)(x,s)dxds≤C1+C1∫0t(‖θ‖L∞2+‖ux‖L∞2)∫0Lτx2dxds≤C1+C1∫0t(1+‖θx‖2+‖ux‖‖uxx‖+‖ux‖)(s)ds≤C1+12∫0t∫0Luxx2(x,s)dxds.
Therefore,
(2.49)∫0t‖uxx(s)‖2ds≤C1
which, along with (2.47), gives (2.41).
Analogously, multiplying (1.18) by vt, integrating the result over QT, and using assumptions (1.24) and Lemmas 2.1–2.6, we deduce
(2.50)μ2∫0L(rv)x2τdx+∫0t‖vt(s)‖2ds=μ2∫0L(r0v0)x2τ0dx+μ2∫0t∫0Lru[(rv)x2τ]xdxds-μ∫0t∫0L[(rv)xτ]xuvdxds-∫0t∫0Luvvtrdxds+∫0t∫0Lf2vtdxds≤C1+14∫0t‖vt(s)‖2ds-12∫0t∫0Lu(rv)xτ[vt+uvr-f2]dxds-∫0t∫0Luv[vt-f2r+uvr2]dxds≤C1+12∫0t‖vt(s)‖2ds+∫0t‖u‖L∞2∫0L(rv)x2τdxds+C1∫0t∫0L(u2v2+f22)dxds≤C1+12∫0t‖vt(s)‖2ds+∫0t‖u‖L∞2∫0L(v2+(rv)x2τ)dxds.
In view of (2.36), we apply Gronwall’s inequality to (2.50) to obtain
(2.51)∫0L(rv)x2τdx+∫0t∫0Lvt2(x,s)dxds≤C1.
By (1.18) and (2.51), we easily deduce
(2.52)∫0t∫0Lvxx2(x,s)dxds≤C1
which, along with (2.51) and Lemmas 2.1–2.4, implies (2.42). The proof of (2.43) iS similar to that of (2.41) and (2.42). The proof is now complete.Lemma 2.8.
Under the assumptions of Theorem1.1, one has, for any T>0,
(2.53)∫0Lθx2(x,t)dx+∫0t∫0L(θt2+θxx2)(x,s)dxds≤C1,∀t∈[0,T].Proof.
Multiplying (1.20) by θt over QT, we have
(2.54)κ2∫0Lr2θx2τdx+CV∫0t∫0Lθt2(x,s)dxds=κ2∫0Lr02θ0x2τ0dx+κ2∫0t∫0L(r2τ)tθx2dxds+∫0t∫0L[1τ[ν(ru)x-γθ](ru)x+μ[(rv)x]2τ+μr2wx2τ-2μ(u2+v2)x+g(r(x,t),t)[(rv)x]2τ]θt(x,s)dxds.
Using Lemmas 2.1–2.7, the Cauchy-Schwarz inequality, and the interpolation inequality, we have
(2.55)|κ2∫0t∫0L(r2τ)tθx2dxds|≤C1∫0t(‖u‖L∞+‖(ru)x‖L∞)∫0Lθx2dxds≤C1∫0t(‖ux‖+‖(ru)x‖1/2‖(ru)xx‖1/2)∫0Lθx2dxds≤C1∫0t(‖ux‖+‖(ru)xx‖2+‖(ru)x‖2/3)∫0Lθx2dxds≤C1∫0t∫0Lθx2dxds+C1∫0t(∫0L(u2τx2+ux2+u2+uxx2)dx)∫0Lθx2dxds≤C1+C1∫0t(‖u‖L∞2‖τx‖2+‖u‖2+‖uxx‖2)∫0Lθx2dxds≤C1+C1∫0t(‖u‖L∞2+‖uxx‖2)∫0Lθx2dxds,(2.56)|∫0t∫0L[1τ[ν(ru)x-γθ](ru)x+μ[(rv)x]2τ+μr2wx2τ-2μ(u2+v2)x+g(r,t)]θtdxds|≤14∫0t‖θt‖2ds+C1∫0t∫0L[(ru)x4+(rv)x4+θ2(ru)x2+wx4+u2ux2+v2vx2+g2]dxds≤14∫0t‖θt‖2ds+C1∫0t[‖(ru)x‖L∞2(‖(ru)x‖2+‖θ‖2)+‖(rv)x‖L∞2‖(rv)x‖2+‖wx‖L∞2‖wx‖2](s)ds+C1∫0t∫0L(u2ux2+v2vx2+g2)(x,s)dxds.
Similarly to (2.55), by virtue of (1.25), (2.30), and (2.41)–(2.43), we arrive at
(2.57)|∫0t∫0L[1τ[ν(ru)x-γθ](ru)x+μ[(rv)x]2τ+μr2wx2τ-2μ(u2+v2)x+g(r,t)]θtdxds|≤C1+14∫0t‖θt‖2ds+C1∫0t(‖(ru)x‖L∞2+‖(rv)x‖L∞2+‖wx‖L∞2)≤C1+14∫0t‖θt‖2ds+C1∫0t(‖τx‖2+‖ux‖H12+‖vx‖H12+‖wx‖H12)≤C1+14∫0t‖θt‖2ds.
Combining (2.54)–(2.57), we conclude
(2.58)‖θx‖2+∫0t‖θt(s)‖2ds≤C1+C1∫0t(‖u‖L∞2+‖uxx‖2)‖θx‖2ds.
In view of (2.36) and (2.41), we apply Gronwall’s inequality to (2.58) to obtain
(2.59)‖θx(t)‖2+∫0t‖θt(s)‖2ds≤C1,∀t∈[0,T].
Similarly to proof of (2.41), by Lemmas 2.1–2.7, (1.20), (1.25), and (2.59), we obtain
(2.60)∫0t‖θxx(s)‖2ds≤C1
which, together with (2.59), implies (2.53). The proof is complete.Proof of Theorem1.1.
By Lemmas2.1–2.8, we complete the proof of Theorem 1.1.



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*Source: 264795-2011-11-22.xml*