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Journal of Applied Mathematics
Volume 2012, Article ID 584680, 16 pages
Research Article

Asymptotic Behavior of the 3D Compressible Euler Equations with Nonlinear Damping and Slip Boundary Condition

Department of Mathematics, Shandong Normal University, Jinan 250014, China

Received 7 January 2012; Accepted 1 March 2012

Academic Editor: Laurent Gosse

Copyright © 2012 Huimin Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas (𝑃(𝜌)=𝑃0𝜌𝛾), time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy's law. In this paper, we use a more general method and extend this result to the 3D compressible Euler equations with nonlinear damping and a more general pressure term. Comparing with linear damping, nonlinear damping can be ignored under small initial data.

1. Introduction and Main Results

We study the 3D compressible Euler equations with nonlinear damping:𝜌𝑡+𝜌𝑢=0,𝜌𝑢𝑡||||+𝜌𝑢𝑢+𝑃(𝜌)=𝛼𝜌𝑢𝛽𝜌𝑢𝑞1𝑢.(1.1) This model represents the compressible flow through porous media with nonlinear external force field. Here 𝜌, 𝑢, and 𝑀=𝜌𝑢 denote density, velocity, and momentum, respectively. The pressure 𝑃 is a smooth function of 𝜌 such that 𝑃(𝜌)>0, 𝑃(𝜌)>0 for any 𝜌>0. Obviously, for polytropic perfect gas, the pressure term 𝑃(𝜌)=𝑃0𝜌𝛾(𝑃0>0,𝛾>1) satisfies this condition. () denotes the divergence in 𝐑𝟑 and the symbol denotes the Kronecker tensor product. The external term 𝛼𝜌𝑢𝛽𝜌|𝑢|𝑞1𝑢 appears in the momentum equation, where 𝛼 is a positive constant, 𝛽 is another constant but can be either negative or positive. The term 𝛼𝜌𝑢 is called the linear damping and throughout this paper we assume 𝛼1. The term 𝛽𝜌|𝑢|𝑞1𝑢 with 𝑞>1 is regarded as a nonlinear source to the linear damping, where the symbol |𝑢| denotes 𝑢21+𝑢22+𝑢23 if we assume 𝑢=(𝑢1,𝑢2,𝑢3). When 𝛽>0, the term 𝛽𝜌|𝑢|𝑞1𝑢 is nonlinear damping, while, when 𝛽<0 this term is regarded as nonlinear accumulating. For convenience, we call both the two cases nonlinear damping.

System (1.1) is supplemented by the following initial and boundary conditions:=𝜌𝜌,𝑢𝑥,00,𝑢0𝑥𝑥,𝑥=1,𝑥2,𝑥3Ω,𝑢𝑛𝜕Ω=0,𝑡0,Ω𝜌0𝑑𝑥=𝜌>0,(1.2) where Ω𝐑𝟑 is a bounded domain with smooth boundary 𝜕Ω, 𝑛 is the unit outward normal vector of 𝜕Ω, and the last condition is imposed to avoid the trivial case, 𝜌0.

For 1D case, system (1.1) and its corresponding 𝑝-system in Lagrangian coordinates have been studied intensively during the past decades. When 𝛽=0, for various initial and initial-boundary conditions, both the existence and large time behaviors of solutions (including classical and weak solutions) were investigated, see [110] and the references therein. [1113] studied the 𝑝-system with nonlinear damping (𝛽0). The existence as well as approximate behavior of smooth solutions to the initial boundary condition in half line and Cauchy problem are considered.

From physical point of view, the 3D model (1.1) describes more realistic phenomena. Also, the 3D compressible Euler equations carry some unique features, such as the effect of vorticity, which are totally absent in the 1D case and make the problem more mathematically challenging. Thus, due to strong physical background and significant mathematical challenge, system (1.1) and its time-asymptotic behavior are of great importance and are much less understood than its 1D companion. When 𝛽=0, investigations were carried out among small smooth solutions and we refer the readers to [1416]. In the direction of nonlinear damping (𝛽0), even the global existence of classical solutions is still open, no less than the asymptotic behavior. In this paper, we consider the global existence and asymptotic behavior of classical solutions to the 3D problem (1.1)(1.2) with nonlinear damping and slip boundary condition.

When 𝛽=0, it has been proved that (see [15]) the dissipation in the momentum equations and the boundary effect make system (1.1) be approximated by the decoupled system̃𝜌𝑡̃=Δ𝑃(̃𝜌),𝑀=̃𝜌𝑢=𝑃(̃𝜌),(1.3) where the first equation is the famous porous medium equation with 𝑃(𝜌)=𝜌𝛾/𝛾, while the second one states Darcy’s law. The corresponding initial-boundary conditions change intõ𝜌𝑥,0=̃𝜌0,𝑥𝑃(̃𝜌)𝑛𝜕Ω=0,𝑡0,Ω̃𝜌0𝑑𝑥=𝜌>0.(1.4) When 𝛽0, we will prove in this paper that the effect of nonlinear damping (𝛽<0) or accumulating (𝛽>0) can be ignored comparing with the linear damping when the initial perturbation around equilibrium state is small.

Before stating our main results, we give some notations. Throughout this paper, and 𝑠 denote the norms of 𝐿2(Ω) and 𝐻𝑠(Ω), respectively. For any vector valued function 𝐹=(𝑓1,𝑓2,𝑓3)Ω𝐑𝟑, we denote𝐹2𝑠=𝑓12𝑠+𝑓22𝑠+𝑓32𝑠,𝐹𝐿=𝑓1𝐿+𝑓2𝐿+𝑓3𝐿,(1.5) furthermore, for any functional matrix 𝑀=(𝑀𝑖𝑗)𝑛×𝑛Ω𝐑𝐧×𝐧, we denote𝑀2𝑠=𝑛𝑖,𝑗=1𝑀𝑖𝑗2𝑠,𝑀𝐿=𝑛𝑖,𝑗=1𝑀𝑖𝑗𝐿.(1.6) The energy space under consideration is𝑋3[][](Ω,0,𝑇)=𝐹Ω×0,𝑇𝐑3(or𝐑)(1.7) equipped with norm𝐹3,𝑇=esssup0𝑡𝑇||||𝐹=esssup0𝑡𝑇3𝑙=0𝜕𝑙𝑡𝐹(,𝑡)23𝑙1/2,(1.8) for any 𝐹𝑋3(Ω,[0,𝑇]). The notation |Ω| denotes the measure of the 𝐑𝟑 domain Ω. In this paper, unless specified, 𝐶 will denote a generic constant which is independent of time. The followings are the main results of this paper.

Theorem 1.1. Suppose the initial data satisfy the compatibility condition 𝜕𝑙𝑡𝑢(0)𝑛𝜕Ω=0 for 0𝑙2. Then there exists a constant 𝛿>0 such that if (𝜌0𝜌/|Ω|,𝑢0)𝐻3(Ω) and (𝜌0𝜌/|Ω|,𝑢0)3𝛿, the initial boundary condition (1.1) and (1.2) exists a unique global solution (𝜌,𝑢) in 𝐶1(Ω×[0,))𝑋3(Ω,[0,)). Moreover, the global solution satisfies ||||𝜌𝜌||Ω||||||+||||(,𝑡)𝑢(,𝑡)𝐶1𝜌0𝜌||Ω||,𝑢03exp𝜂1𝑡,(1.9) for some certain positive constants 𝐶1 and 𝜂1.

Theorem 1.2. Let (𝜌,𝑢) be the unique global smooth solution of (1.1) and (1.2), ̃(̃𝜌,𝑢) be the global solution of (1.3)(1.4). Then there exist constants 𝐶2 and 𝜂2 such that (𝜌̃𝜌)(,𝑡)1+̃𝑢𝑢(,𝑡)𝐶2exp𝜂2𝑡(1.10) satisfies for big enough 𝑡>0.

Theorem 1.1 states that the global solution of (1.1) and (1.2) converges to the steady state (𝜌/|Ω|,𝟎) exponentially fast in time. To prove this theorem, we first change problem (1.1) into𝜌𝑡+𝜌𝑢=0,𝑢𝑡+𝑢𝑢+𝑢+𝑃(𝜌)𝜌||||=𝛽𝑢𝑞1𝑢,(1.11) and then we consider the existence and large time behavior of perturbation solution. It is worth pointing out that in [15] the pressure 𝑃 is assumed to be 𝑃(𝜌)=𝜌𝛾/𝛾, then the authors can introduce a nonlinear transformation 𝜎=(𝜌𝜃/𝜃)(𝜃=(𝛾1)/2) to reformulate the perturbation system as a symmetric hyperbolic system. In this paper, the pressure is more general and the transformation in [15] do not work any more. To overcome this difficulty, we use a symmetrizer to reduce the system to a symmetric hyperbolic one in the sense of Friedrichs. Due to the slip boundary condition, the basic energy estimates can not be applied directly to spatial derivatives. Inspired by [15, 17], we use time-derivatives, which still preserve the boundary conditions, to estimate the spatial derivatives.

Theorem 1.2 is our target result. To prove this theorem, we first claim system (1.3) and (1.4) decays to the steady state (𝜌/|Ω|,𝟎) exponentially, too. Then using the triangular inequality we get Theorem 1.2.

Now, we recall some inequalities which will be used in the following.

Lemma 1.3. Let Ω be any bounded domain in 𝐑𝟑 with smooth boundary. Then 𝑓𝐿(Ω)𝐶𝑓𝐻2(Ω),𝑓𝐿𝑝(Ω)𝐶𝑓𝐻1(Ω),2𝑝6(1.12) for some constant 𝐶>0 depending only on Ω.

Lemma 1.4. Let 𝑢𝐻𝑠(Ω) be a vector-valued function satisfying 𝑢𝑛𝜕Ω=0, where 𝑛 is the unit outer normal of 𝜕Ω. Then 𝑢𝑠𝐶×𝑢𝑠1+𝑢𝑠1+𝑢𝑠1,(1.13) for 𝑠1, and the constant 𝐶 depends only on 𝑠 and Ω.

2. Global Existence and Asymptotic Behavior

In this section, we will consider the global existence and the asymptotic behavior of system (1.11) and (1.2). Due to the boundary effect (1.2)2 and the dissipation in the velocity equations, the kinetic energy is conjectured to vanish and the potential energy will converge to a constant as time goes to infinity. Furthermore, since the conservation of mass and the initial condition (1.2)3, we expect𝜌𝜌,𝑢||Ω||,𝟎as𝑡,(2.1) where (𝜌,𝑢) is the solution of problem (1.11) and (1.2), and |Ω| is the measure of Ω in 𝐑𝟑. Without loss of generality, we assume |Ω|=1.

For this purpose, we consider the perturbation system:𝜑𝑡+𝜑+𝜌𝑢=0,𝑢𝑡+𝑢𝑢+𝑢+𝑃𝜑+𝜌[𝜑+𝜌]||||=𝛽𝑢𝑞1𝑢,(2.2) where 𝜑=𝜌𝜌. Let 𝑔(𝜌)=𝑄(𝜌)=𝑃(𝜌)/𝜌, then 𝑄(𝜌)𝜌=𝑃(𝜌)/𝜌 and system (2.2) changes into𝜑𝑡+𝜑+𝜌𝑢+𝜑𝑢=0,𝑢𝑡+𝑢𝑢+𝑢+𝑔𝜑+𝜌||||𝜑=𝛽𝑢𝑞1𝑢,(2.3) that is, 𝜑𝑡+𝜑+𝜌𝑢𝑖,𝑖+𝜑,𝑖𝑢𝑖=0,𝑢𝑖𝑡+𝑢𝑗𝑢𝑖,𝑗+𝑢𝑖+𝑔𝜑+𝜌𝜑,𝑖||||=𝛽𝑢𝑞1𝑢𝑖.(2.4) In matrix notation, we have𝜑𝑡𝑢1𝑡𝑢2𝑡𝑢3𝑡+𝑢1𝜑+𝜌𝑔00𝜑+𝜌𝑢10000𝑢10000𝑢1𝜑,1𝑢1,1𝑢2,1𝑢3,1+𝑢20𝜑+𝜌00𝑢2𝑔00𝜑+𝜌0𝑢20000𝑢2𝜑,2𝑢1,2𝑢2,2𝑢3,2+𝑢300𝜑+𝜌0𝑢30000𝑢30𝑔𝜑+𝜌00𝑢3𝜑,3𝑢1,3𝑢2,3𝑢3,3+0𝑢1||||+𝛽𝑢𝑞1𝑢1𝑢2||||+𝛽𝑢𝑞1𝑢2𝑢3||||+𝛽𝑢𝑞1𝑢3=0.(2.5) Denoting 𝑤=𝜑𝑢 and multiplying (2.5) on the left by the symmetrizer𝑔𝐷=diag𝜑+𝜌,𝜑+𝜌,𝜑+𝜌,𝜑+𝜌,(2.6) we can rewrite the result to be𝐷𝑤𝑡+𝐴1𝑤,1+𝐴2𝑤,2+𝐴3𝑤,3+𝑓=0,(2.7) where𝐴1=𝑔𝜑+𝜌𝑢1𝜑+𝜌𝑔𝜑+𝜌00𝜑+𝜌𝑔𝜑+𝜌𝜑+𝜌𝑢10000𝜑+𝜌𝑢10000𝜑+𝜌𝑢1,𝐴2=𝑔𝜑+𝜌𝑢20𝜑+𝜌𝑔𝜑+𝜌00𝜑+𝜌𝑢200𝜑+𝜌𝑔𝜑+𝜌0𝜑+𝜌𝑢20000𝜑+𝜌𝑢2,𝐴3=𝑔𝜑+𝜌𝑢300𝜑+𝜌𝑔𝜑+𝜌0𝜑+𝜌𝑢30000𝜑+𝜌𝑢30𝜑+𝜌𝑔𝜑+𝜌00𝜑+𝜌𝑢3,𝜑+𝜌0𝑢1||||+𝛽𝑢𝑞1𝑢1𝑢2||||+𝛽𝑢𝑞1𝑢2𝑢3||||+𝛽𝑢𝑞1𝑢3.(2.8)

The existence of global solution to (2.7) can be proved by local existence result and a priori estimates. The local existence is classical and we omit it here. As for the a priori estimates, we first have the following estimate about the temporal derivatives of 𝑤.

Lemma 2.1. Let |𝑤|𝛿 be sufficiently small. Then there exists a positive constant 𝐶3>0 such that 𝑑𝑑𝑡Ω𝜕𝑘𝑡𝑤𝐷𝜕𝑘𝑡𝑤𝑑𝑥+𝜌Ω||𝜕𝑘𝑡𝑢||𝑥,𝑡2𝑑𝑥𝐶3||||𝑤3,(2.9) for 𝑘=0,1,2,3.

Proof. (1) Zero Order Estimate.
Multiplying (2.7) by 𝑤 and integrating over Ω, we have Ω𝑤𝐷𝑤𝑡𝑑𝑥+3𝑖=1Ω𝑤𝐴𝑖𝑤,𝑖𝑑𝑥+Ω𝑤𝑓𝑑𝑥=0.(2.10) Using Lemma 1.3 and Cauchy-Schwarz inequality, we have Ω𝑤𝐷𝑤𝑡1𝑑𝑥=2𝑑𝑑𝑡Ω1𝑤𝐷𝑤𝑑𝑥2Ω𝑤𝐷𝑡1𝑤𝑑𝑥2𝑑𝑑𝑡Ω𝐷𝑤𝐷𝑤𝑑𝑥+𝐶𝑡𝐿Ω|𝑤|21𝑑𝑥2𝑑𝑑𝑡Ω||||𝑤𝐷𝑤𝑑𝑥+𝐶𝑤3.(2.11) For the second term coming from the left side of (2.10), we have 3𝑖=1Ω𝑤𝐴𝑖𝑤,𝑖1𝑑𝑥=2Ω3𝑖=1𝑤𝐴𝑖𝑤𝑖1𝑑𝑥2Ω𝑤3𝑖=1𝐴𝑖,𝑖1𝑤𝑑𝑥2𝜕Ω𝑤3𝑖=1𝐴𝑖𝑛𝑖𝐴𝑤𝑑𝑠+𝐶𝑖,𝑖𝐿Ω|𝑤|2𝑑𝑥,(2.12) where we have used the Divergence Theorem. Since 𝑤3𝑖=1𝐴𝑖𝑛𝑖𝑤=𝑔𝜑2||||+2𝜌𝑔𝜑+𝜌𝑢2𝑛𝑢=0.(2.13) Here 𝑔=𝑔(𝜌),𝜌=𝜑+𝜌, then (2.12) turns into 3𝑖=1Ω𝑤𝐴𝑖𝑤,𝑖||||𝑑𝑥𝐶𝑤3.(2.14) While the term Ω𝜌𝑤𝑓𝑑𝑥=+𝜑Ω||||1+𝛽𝑢𝑞1||||𝑢2𝜌𝑑𝑥2Ω||||𝑢2𝑑𝑥,(2.15) for 𝑞>1 and the smallness of 𝛿. Thus, we have 𝑑𝑑𝑡Ω𝑤𝐷𝑤𝑑𝑥+𝜌Ω||||𝑢2||||𝑑𝑥𝐶𝑤3(2.16) from (2.10) to (2.15).
(2) Higher Derivatives Estimate.
For 𝑘=1,2,3, taking 𝜕𝑘𝑡 of system (2.7), we get 𝐷𝜕𝑘𝑡𝑤𝑡+3𝑖=1𝐴𝑖𝜕𝑘𝑡𝑤,𝑖+𝜕𝑘𝑡𝑓=𝑘𝑙=1𝑘!𝜕𝑙!(𝑘𝑙)!𝑙𝑡𝐷𝜕𝑡𝑘𝑙𝑤𝑡+𝜕𝑙𝑡𝐴𝑖𝜕𝑡𝑘𝑙𝑤,𝑖.(2.17) Multiply (2.17) by 𝜕𝑘𝑡 and integrate over Ω to have Ω𝜕𝑘𝑡𝜕𝑤𝐷𝑘𝑡𝑤𝑡𝑑𝑥+Ω𝜕𝑘𝑡𝑤3𝑖=1𝐴𝑖𝜕𝑘𝑡𝑤,𝑖𝑑𝑥+Ω𝜕𝑘𝑡𝑤𝜕𝑘𝑡𝑓𝑑𝑥=𝑘𝑙=1𝑘!𝑙!(𝑘𝑙)!Ω𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐷𝜕𝑡𝑘𝑙𝑤𝑡+𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐴𝑖𝜕𝑡𝑘𝑙𝑤,𝑖𝑑𝑥.(2.18) Just like the estimates from (2.11) and (2.15), we estimate the three terms on the left side of (2.18) as follows: Ω𝜕𝑘𝑡𝜕𝑤𝐷𝑘𝑡𝑤𝑡1𝑑𝑥=2𝑑𝑑𝑡Ω𝜕𝑘𝑡𝑤𝐷𝜕𝑘𝑡1𝑤𝑑𝑥2Ω𝜕𝑘𝑡𝑤𝐷𝑡𝜕𝑘𝑡1𝑤𝑑𝑥2𝑑𝑑𝑡Ω𝜕𝑘𝑡𝑤𝐷𝜕𝑘𝑡𝐷𝑤𝑑𝑥+𝐶𝑡𝐿Ω||𝜕𝑘𝑡𝑤||21𝑑𝑥2𝑑𝑑𝑡Ω𝜕𝑘𝑡𝑤𝐷𝜕𝑘𝑡||||𝑤𝑑𝑥+𝐶𝑤3,(2.19)3𝑖=1Ω𝜕𝑘𝑡𝑤𝐴𝑖𝜕𝑘𝑡𝑤,𝑖1𝑑𝑥=2Ω3𝑖=1𝜕𝑘𝑡𝑤𝐴𝑖𝜕𝑘𝑡𝑤𝑖1𝑑𝑥2Ω𝜕𝑘𝑡𝑤3𝑖=1𝐴𝑖,𝑖𝜕𝑘𝑡1𝑤𝑑𝑥2𝜕Ω𝜕𝑘𝑡𝑤3𝑖=1𝐴𝑖𝑛𝑖𝜕𝑘𝑡𝐴𝑤𝑑𝑠+𝐶𝑖,𝑖𝐿Ω||𝜕𝑘𝑡𝑤||2=1𝑑𝑥2𝜕Ω𝑔𝜕𝑘𝑡𝜑2||𝜕+𝜌𝑘𝑡||𝑢2𝑢𝑛𝑑𝑠+2Ω𝜌𝑔𝜕𝑘𝑡𝜑𝜕𝑘𝑡𝐴𝑢𝑛𝑑𝑠+𝐶𝑖,𝑖𝐿Ω||𝜕𝑘𝑡𝑤||2||||𝑑𝑥𝐶𝑤3,(2.20)Ω𝜕𝑘𝑡𝑤𝜕𝑘𝑡𝑓𝑑𝑥=Ω𝜕𝑘𝑡𝑢𝑘𝑙=0𝜕𝑙𝑡𝜑+𝜌𝜕𝑡𝑘𝑙||||𝑢+𝛽𝑢𝑞1𝜌𝑢𝑑𝑥+𝜑Ω||||1+𝛽𝑢𝑞1||𝜕𝑘𝑡||𝑢2𝑑𝑥+𝐶𝑘𝑙=1Ω𝜕𝑙𝑡𝜑𝜕𝑘𝑡𝑢𝜕𝑡𝑘𝑙||||1+𝑢𝑞1𝜌𝑢𝑑𝑥+𝐶+𝜑Ω𝜕𝑘𝑡𝑢𝑘𝑙=1𝜕𝑙𝑡||||1+𝛽𝑢𝑞1𝜕𝑡𝑘𝑙𝜌𝑢𝑑𝑥2Ω||𝜕𝑘𝑡||𝑢2||||𝑑𝑥+𝐶𝑤3.(2.21) For the term on the right-hand side of (2.18) we have, ||||Ω𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐷𝜕𝑡𝑘𝑙𝑤𝑡+𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐴𝑖𝜕𝑡𝑘𝑙𝑤,𝑖||||𝜕𝑑𝑥𝐶𝑘𝑡𝑤𝐿2𝜕𝑙𝑡𝐷𝐿4𝜕𝑡𝑘𝑙𝑤𝑡𝐿4𝜕+𝐶𝑘𝑡𝑤𝐿2𝜕𝑙𝑡𝐴𝑖𝐿4𝜕𝑡𝑘𝑙𝑤,𝑖𝐿4||||𝐶𝑤3,(2.22) for 𝑙=1,2 and 𝑙𝑘3. When 𝑙=𝑘=3, we get ||||Ω𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐷𝜕𝑡𝑘𝑙𝑤𝑡+𝜕𝑘𝑡𝑤𝜕𝑙𝑡𝐴𝑖𝜕𝑡𝑘𝑙𝑤,𝑖||||𝜕𝑑𝑥𝐶𝑘𝑡𝑤𝐿2𝜕𝑙𝑡𝐷𝐿2𝑤𝑡𝐿𝜕+𝐶𝑘𝑡𝑤𝐿2𝜕𝑙𝑡𝐴𝑖𝐿2𝑤,𝑖𝐿||||𝐶𝑤3.(2.23) Combining (2.18) and (2.23), we get 𝑑𝑑𝑡Ω𝜕𝑘𝑡𝑤𝐷𝜕𝑘𝑡𝑤𝑑𝑥+𝜌Ω||𝜕𝑘𝑡||𝑢(𝑥,𝑡)2||||𝑑𝑥𝐶𝑤3,(2.24) for 𝑘=1,2,3. Thus we prove Lemma 2.1.

For slip boundary condition, spatial derivatives can be controlled by temporal derivatives and vorticity that is discussed below.

Lemma 2.2. Let |𝑤|𝛿 be sufficiently small and (𝜑,𝑢) be the solution of (1.11) and (1.2), then there exists a constant 𝐶4 such that ||||𝑤2𝐶42𝑙=0𝜕𝑙𝑡×𝑢22𝑙+3𝑙=0𝜕𝑙𝑡𝑤2.(2.25)

Proof. From the velocity equation (2.3)2 we have 1𝜑=𝑔(𝜌)+𝜑𝑢+𝑢𝑡||||+𝑢𝑢+𝛽𝑢𝑞1.𝑢(2.26) Taking the 𝐿2 inner product of (2.26) with 𝜑, we get 𝜑2=Ω1𝑔(𝜌+𝜑)𝑢+𝑢𝑡||||+𝑢𝑢+𝛽𝑢𝑞1𝑢𝜑𝑑𝑥𝐶𝑢2+𝑢𝑡2+𝑢𝐿𝑢2+𝛽𝑢𝐿𝑞1𝑢2𝐶𝑢2+𝑢𝑡2||||+𝐶𝑤𝑟,(2.27) where 𝑟=min{3,𝑞+1}>2.
The continuity equation (2.3)1 implies 1𝑢=𝜑+𝜌𝜑𝑡.+𝑢𝜑(2.28) Therefore, we have 𝑢2𝜑𝐶𝑡2+||||𝑤3.(2.29) Using Lemma 1.4 with 𝑠=1, we have 𝑢21𝐶×𝑢2+𝑢2+𝑢2𝐶×𝑢2+𝑢2+𝜑𝑡2+||||𝑤3.(2.30)
Next, we take time derivatives of (2.26) and (2.28). It is clear that every time derivative up to order two of 𝜑 and 𝑢 is bounded by 3𝑙=0𝜕𝑙𝑡𝑤2. Furthermore, by using an induction on the number of spatial derivatives, we get that the same is true for any derivative up to order of two of 𝜑 and 𝑢. Applying Lemma 1.4 with 𝑠=1,2,3, we complete the proof of Lemma 2.2.

From Lemma 2.2, to prove Theorem 1.1 we only need to estimate 𝜕𝑙𝑡(×𝑢)22𝑙 and 𝜕𝑖𝑡𝑤2 for 𝑙=0,1,2,𝑖=0,1,2,3. The following Lemma is contributed to the estimate of ×𝑢.

Lemma 2.3. Let |𝑤|𝛿 is sufficiently small, then for any solution 𝑤=(𝜑,𝑢) of problem (1.11)(1.2), one has 12𝑑𝑑𝑡2𝑙=0𝜕𝑙𝑡×𝑢2+2𝑙=0𝜕𝑙𝑡×𝑢2||||𝐶𝑤(𝑡)3.(2.31)

Proof. Taking × of the velocity equation in (2.3) and denoting the vorticity 𝑣=×𝑢, we have 𝑣𝑡+||||𝑣=×𝑢𝑢×(𝑔(𝜌)𝜑)𝛽×𝑢𝑞1=||||𝑢𝑣𝑢𝑢𝑣𝑣𝑢𝛽𝑢𝑞1||||𝑣𝛽𝑢𝑞1×𝑢.(2.32) Let 𝜕 denote any mixed time and spatial derivative of order at most 2, then by taking any mixed derivative of the above equation, we have 𝜕𝑣𝑡||||+𝜕𝑣=𝜕×𝑢𝑢+𝛽×𝑢𝑞1𝛽||||𝑢=𝜕𝑣𝑢+𝑢𝑣+𝑣𝑢𝜕𝑢𝑞1||||𝑣+𝛽𝑢𝑞1.×𝑢(2.33) Multiplying the above equation by 𝜕𝑣 and integrating the result over Ω, similar the calculations in (2.22) and (2.23), we get 12𝑑𝜕𝑣𝑑t2+𝜕𝑣(𝑡)2||||𝐶𝑤3,(2.34) by using the Sobolev inequality in Lemma 1.3. This completes the proof of Lemma 2.3.

The next Lemma gives the dissipation in density due to nonlinearity.

Lemma 2.4. There exists a positive constant 𝐶5 such that 𝑑𝑑𝑡3𝑙=1Ω𝜕𝑡𝑙1𝜑𝜕𝑙𝑡𝜑+𝑑𝑥3𝑙=0𝜕𝑙𝑡𝜑2𝐶5𝑢2+𝑢𝑡2+||||𝑤𝑟,(2.35) where 𝑟=min{3,𝑞+1}.

Proof. Due to the conservation of total mass, we know that Ω(𝜌𝜌)𝑑𝑥=0, that is, Ω𝜑𝑑𝑥=0. Using Poincaré's inequality and (2.27), we have 𝜑2𝐶𝜑2𝐶𝑢2+𝑢𝑡+||||𝑤𝑟.(2.36) Taking the 𝑡 derivative of (2.3)1 and applying () to (2.3)2, we get 𝜑𝑡𝑡+𝜑𝑢𝑡+𝜑𝑡𝑢+𝜑𝑢𝑡=0,(2.37)𝑢𝑡||||+𝑢𝑢+𝑢+(𝑔(𝜌)𝜑)=𝛽𝑢𝑞1𝑢.(2.38)(2.37) − (2.38)×𝜑 yields 𝜑𝑡𝑡+𝜑𝑢𝑡+𝜑𝑡||||𝑢𝜑𝑢𝑢+𝑔(𝜌)𝜑+𝑢+𝛽𝑢𝑞1𝑢=0,(2.39) that is, 𝜑𝑡𝑡+𝜑𝑢𝑡+𝜑𝑡𝑢+𝜑𝑢𝑡=0.(2.40) Multiplying (2.40) by 𝜑, and integrating the result over Ω, we have 𝑑𝑑𝑡Ω𝜑𝜑𝑡𝑑𝑥Ω𝜑2𝑡𝑑𝑥+Ω𝜑𝑡𝑢𝜑𝑑𝑥+Ω𝜑𝑢𝑡+𝜑𝑑𝑥Ω𝜑𝑡𝑢𝜑𝑑𝑥+Ω𝜑2𝑢𝑡𝜑2𝑢𝑡𝑑𝑥=0.(2.41) Using the Divergence Theorem, we obtain 𝑑𝑑𝑡Ω𝜑𝜑𝑡𝑑𝑥Ω𝜑2𝑡𝑑𝑥+Ω𝜑𝑡𝑢𝜑𝑑𝑥+Ω𝜑𝑢𝑡+𝜑𝑑𝑥Ω𝜑𝑡𝑢𝜑𝑑𝑥+Ω𝜑2𝑢𝑡𝑛𝑑𝑠2Ω𝜑𝜑𝑢𝑡𝑑𝑥=0.(2.42) Then we have 𝑑𝑑𝑡Ω𝜑𝜑𝑡𝑑𝑥+Ω𝜑2𝑡||||𝑑𝑥𝐶𝑤3.(2.43) Furthermore, applying the derivative 𝜕𝑡 to (2.40) we have 𝜑𝑡𝑡𝑡+𝜑𝑡𝑡𝑢+𝜑𝑢𝑡𝑡+2𝜑𝑡𝑢𝑡+𝜑𝑡𝑡𝑢+𝜑𝑡𝑢𝑡+𝜑𝑢𝑡𝑡+𝜑𝑡𝑢𝑡=0.(2.44) Taking 𝐿2 inner product of (2.44) with 𝜑𝑡, we get 𝑑𝑑𝑡Ω𝜑𝑡𝑡𝜑𝑡𝑑𝑥+Ω𝜑2𝑡𝑡𝑑𝑥Ω𝜑𝑡𝑡𝑢+𝜑𝑢𝑡𝑡𝜑𝑡𝑑𝑥Ω𝜑2𝑡𝑢𝑡𝑑𝑥Ω𝜑2𝑡𝑢𝑡𝑑𝑥=0,(2.45) that is, 𝑑𝑑𝑡Ω𝜑𝑡𝑡𝜑𝑡𝑑𝑥+Ω𝜑2𝑡𝑡||||𝑑𝑥𝐶𝑤3,(2.46) where we have used the boundary condition and the Sobolev inequality in Lemma 1.3. Next, apply the derivative 𝜕𝑡 to (2.44) and times the result with 𝜑𝑡𝑡 to get 𝑑𝑑𝑡Ω𝜑𝑡𝑡𝑡𝜑𝑡𝑡𝑑𝑥+Ω𝜑2𝑡𝑡𝑡||||𝑑𝑥𝐶𝑤3,(2.47) in which we have used 𝜑𝑡𝑡𝜑𝑡𝑡𝑡𝑢+𝜑𝑡𝑡𝜑𝑢𝑡𝑡𝑡𝜑=𝑡𝑡𝑡𝜑𝑡𝑡𝑢𝜑𝑡𝑡𝜑𝑡𝑡𝑡𝑢+𝜑𝜑𝑡𝑡𝑢𝑡𝑡𝑡𝜑𝑡𝑡𝜑𝑢𝑡𝑡𝑡,(2.48) and the similar method as (2.46). Combining (2.43), (2.46), and (2.47), we finish the proof of Lemma 2.4.

Combining Lemmas 2.1 and 2.4, we can characterize the total dissipation of 3𝑙=0𝜕𝑙𝑡𝑤2. Let𝜌𝜆=min2𝐶5,𝜌4,𝑔𝜌/24,𝐸(𝑡)=Ω3𝑙=0𝜕𝑙𝑡𝑤𝐷𝜕𝑙𝑡𝑤𝜆3𝑙=1𝜕𝑡𝑙1𝜑𝜕𝑙𝑡𝜑𝑑𝑥.(2.49) We can easily see that 3𝑙=0𝜕𝑙𝑡𝑤2 and 𝐸(𝑡) are equivalent, that is, there exist constants 𝐴 and 𝐵 such that𝐴3𝑙=0𝜕𝑙𝑡𝑤2𝐸(𝑡)𝐵3𝑙=0𝜕𝑙𝑡𝑤2.(2.50)

Calculating (2.9) +𝜆(2.35), we have𝑑𝑑𝑡Ω3𝑙=0𝜕𝑙𝑡𝑤𝐷𝜕𝑙𝑡𝑤𝜆3𝑙=0𝜕𝑡𝑙1𝜑𝜕𝑙𝑡𝜑𝑑𝑥+𝜆3𝑙=0𝜕𝑙𝑡𝑤2||||𝐶𝑤̃𝑟,(2.51) where ̃𝑟=min{3,𝑟}, that is,𝑑𝜆𝑑𝑡𝐸(𝑡)+𝐵||||𝐸(𝑡)𝐶𝑤̃𝑟.(2.52) Then Gronwall’s inequality and the equivalent relationship (2.50) deduce that3𝑙=0𝜕𝑙𝑡𝑤2𝐶3𝑙=0𝜕𝑙𝑡𝑤(0)2𝑒(𝜆/𝐵)𝑡+𝐶𝑡0𝑒𝑠𝑡||||𝑤(𝑠)̃𝑟𝑑𝑠.(2.53) This inequality combines Lemmas 2.2 and 2.3 deduces the result of Theorem 1.1.

3. About Porous Medium Equations

In this section, we investigate the large time behavior of classical solutions to problem (1.3)(1.4). As indicated in the introduction, we expect that (1.1) and (1.2) are captured by (1.3) and (1.4) time asymptotically ifΩ̃𝜌0𝑑𝑥=Ω𝜌0𝑑𝑥=𝜌.(3.1) Noticing Theorem 1.1 and the triangle inequality, we only need to show that the large time asymptotic state of (1.3) and (1.4) is also the constant state (𝜌,𝟎).

Consider̃𝜌𝑡=Δ𝑃(̃𝜌),̃𝜌𝑥,0=̃𝜌0(𝑥),𝑃(̃𝜌)𝑛𝜕Ω=0,𝑡0,(3.2) with the initial dataΩ̃𝜌0𝑑𝑥=𝜌,̃𝜌0𝑥𝐿(Ω).(3.3) Here we assume ̃𝜌0 is uniform bounded, that is, there exists a constant 𝜌<𝜌< such that0̃𝜌0𝑥𝜌.(3.4)

The global existence and the large time behavior of solutions to (3.2) and (3.3) have been established in [18]. The method we used in this section is similar with which in [15]. Yet, since the pressure is more general in this paper, we can not use the corresponding result in [15] directly.

Lemma 3.1. Let ̃𝜌 be the global solution of problem (3.2) and (3.3), 𝑀=𝑃(̃𝜌). Then there exist positive constants 𝐶>0,𝜂>0, independent of time 𝑡, such that ̃𝜌𝜌1+𝑀𝐶𝑒𝜂𝑡,(3.5) for big enough 𝑡.

Proof. From [18], we know that there exists a positive constant 𝑇>0 such that the problem (3.2) and (3.3) has a classical solution ̃𝜌(𝑥,𝑡) for 𝑡>𝑇, and >𝜌̃𝜌𝑥,𝑡2,for𝑡>𝑇.(3.6) On the other hand, due to the comparison principle 0̃𝜌𝑥,𝑡𝜌,forany𝑥,𝑡[Ω×0,𝐿).(3.7)
For 𝑡>𝑇, We consider ̃𝜌𝜌𝑡𝑃𝜌=Δ(̃𝜌)𝑃.(3.8) Taking 𝐿2 inner product of (3.8) with (̃𝜌𝜌), we have 10=2𝑑𝑑𝑡Ω̃𝜌𝜌2𝑑𝑥ΩΔ𝜌𝑃(̃𝜌)𝑃̃𝜌𝜌=1𝑑𝑥2𝑑𝑑𝑡̃𝜌𝜌2+Ω||𝜌𝑃(̃𝜌)𝑃||2𝑃(̃𝜌)𝑑𝑥,(3.9) where we have used boundary condition (3.2)3. Combining the increasing property of 𝑃(̃𝜌) and the inequality (3.7), we have 12𝑑𝑑𝑡̃𝜌𝜌2+1𝑃𝜌𝜌𝑃(̃𝜌)𝑃20.(3.10)
Multiplying (3.2)1 with 𝑃(̃𝜌), we have 𝑃(̃𝜌)𝑡𝑃(̃𝜌)Δ𝑃(̃𝜌)=0,(3.11) that is, 𝑃𝜌(̃𝜌)𝑃𝑡𝑃𝑃𝜌(̃𝜌)Δ(̃𝜌)𝑃=0.(3.12) Define 𝜓=̃𝜌𝜌𝜌,𝜙=𝑃(̃𝜌)𝑃.(3.13) Taking 𝐿2 inner product of (3.12) with Δ𝜙, we get 12𝑑𝑑𝑡𝜙2+𝑃𝜌2Δ𝜙20.(3.14)(3.10) plus (3.14) deduce 12𝑑𝑑𝑡𝜓2+𝜙2+𝐶𝜙2+Δ𝜙20,(3.15) where 𝐶=min{𝑃(𝜌/2),1/𝑃(𝜌)}. Since the conservation of total mass and the Poincaré’s inequality, we have 𝜓2𝐶𝜓21𝐶𝑃(𝜌/2)2𝜙2.(3.16) Then inequality (3.15) turns into 12𝑑𝑑𝑡𝜓2+𝜙2+𝐶𝜓2+𝜙2+Δ𝜙20.(3.17) The Gronwall inequality and (3.16) deduce Lemma 3.1.


This project is supported by the National Natural Science Foundation of China (Grant no. 10901095) and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2010SF025).


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