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

Abstract

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)=𝑃₀𝜌^𝛾), 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:𝜌𝑢⃗_𝑡𝜌_𝑡+∇⋅𝜌𝑢⃗=0,+∇⋅𝜌𝑢⃗⊗𝑢⃗+∇𝑃(𝜌)=−𝛼𝜌𝑢⃗−𝛽𝜌||𝑢||⃗^𝑞−1𝑢.⃗(1.1) This model represents the compressible flow through porous media with nonlinear external force field. Here 𝜌, 𝑢⃗, and 𝑀=𝜌𝑢⃗ 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)=𝑃₀𝜌^𝛾(𝑃₀>0,𝛾>1) satisfies this condition. (∇⋅) denotes the divergence in 𝐑^𝟑 and the symbol ⊗ denotes the Kronecker tensor product. The external term −𝛼𝜌𝑢⃗−𝛽𝜌|𝑢|⃗^𝑞−1𝑢⃗ appears in the momentum equation, where 𝛼 is a positive constant, 𝛽 is another constant but can be either negative or positive. The term −𝛼𝜌𝑢⃗ is called the linear damping and throughout this paper we assume 𝛼≡1. The term −𝛽𝜌|𝑢|⃗^𝑞−1𝑢⃗ with 𝑞>1 is regarded as a nonlinear source to the linear damping, where the symbol |𝑢|⃗ denotes 𝑢²¹+𝑢²²+𝑢²³ if we assume 𝑢⃗=(𝑢₁,𝑢₂,𝑢₃). When 𝛽>0, the term −𝛽𝜌|𝑢|⃗^𝑞−1𝑢⃗ 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:𝜌,𝑢⃗𝑥,⃗0=𝑢𝜌⃗⋅₀,𝑛𝑢∣⃗₀⃗_𝜕Ω𝑥⃗,𝑥⃗=𝑥₁,𝑥₂,𝑥₃∈Ω,=0,𝑡≥0,_Ω𝜌₀𝑑𝑥⃗=𝜌^∗>0,(1.2) where Ω∈𝐑^𝟑 is a bounded domain with smooth boundary 𝜕Ω, 𝑛⃗ is the unit outward normal vector of 𝜕Ω, and the last condition is imposed to avoid the trivial case, 𝜌≡0.

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 [1–10] and the references therein. [11–13] studied the 𝑝-system with nonlinear damping (𝛽≠0). 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 [14–16]. In the direction of nonlinear damping (𝛽≠0), 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 system𝑀=𝜌̃_𝑡𝜌=̃𝑢̃⃗=Δ𝑃−∇(𝜌𝑃̃),(𝜌̃),(1.3) where the first equation is the famous porous medium equation with 𝑃(𝜌)=𝜌^𝛾/𝛾, while the second one states Darcy’s law. The corresponding initial-boundary conditions change into∇𝑃𝜌(𝜌̃̃)𝑥⋅,⃗𝑛0∣⃗_𝜕Ω=𝜌̃₀𝑥⃗,=0,𝑡≥0,_Ω𝜌̃₀𝑑𝑥⃗=𝜌^∗>0.(1.4) When 𝛽≠0, 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 𝐻^𝑠(Ω), respectively. For any vector valued function 𝐹=(𝑓₁,𝑓₂,𝑓₃)∶Ω↦𝐑^𝟑, we denote‖𝐹‖^2𝑠=‖‖𝑓₁‖‖^2𝑠+‖‖𝑓₂‖‖^2𝑠+‖‖𝑓₃‖‖^2𝑠,‖𝐹‖_𝐿^∞=‖‖𝑓₁‖‖_𝐿∞+‖‖𝑓₂‖‖_𝐿∞+‖‖𝑓₃‖‖_𝐿∞,(1.5) furthermore, for any functional matrix 𝑀=(𝑀_𝑖𝑗)_𝑛×𝑛∶Ω→𝐑^𝐧×𝐧, we denote‖𝑀‖^2𝑠=^𝑛_𝑖_,𝑗=1‖‖𝑀_𝑖𝑗‖‖^2𝑠,‖𝑀‖_𝐿^∞=^𝑛_𝑖_,𝑗=1‖‖𝑀_𝑖𝑗‖‖_𝐿∞.(1.6) The energy space under consideration is𝑋₃(Ω,[0,𝑇])=𝐹∶Ω×[0,𝑇]⟶𝐑³(or𝐑)(1.7) equipped with norm‖𝐹‖_3,𝑇=esssup_{0≤𝑡≤𝑇}||‖𝐹‖||=esssup_{0≤𝑡≤𝑇}³_𝑙=0‖‖𝜕^𝑙𝑡𝐹(⋅,𝑡)‖‖^23−𝑙^1/2,(1.8) for any 𝐹∈𝑋₃(Ω,[0,𝑇]). The notation |Ω| denotes the measure of the 𝐑^𝟑 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 0≤𝑙≤2. Then there exists a constant 𝛿>0 such that if (𝜌₀−𝜌^∗/|Ω|,𝑢₀⃗)∈𝐻³(Ω) and ‖(𝜌₀−𝜌^∗/|Ω|,𝑢₀⃗)‖₃≤𝛿, 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 ||||‖‖‖𝜌−𝜌^∗||Ω||(⋅,𝑡)‖‖‖||||+||‖‖𝑢⃗(⋅,𝑡)‖‖||≤𝐶¹‖‖‖𝜌₀−𝜌^∗||Ω||,𝑢⁰⃗‖‖‖₃exp^−𝜂1𝑡,(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), (𝜌̃,𝑢̃)⃗ be the global solution of (1.3)(1.4). Then there exist constants 𝐶₂ and 𝜂₂ such that ‖(𝜌−𝜌̃)(⋅,𝑡)‖₁+‖‖𝑢⃗−𝑢̃⃗(⋅,𝑡)‖‖≤𝐶₂exp^−𝜂2𝑡(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) into𝑢_𝑡⃗𝜌_𝑡+∇⋅𝜌𝑢⃗=0,+𝑢⃗⋅∇𝑢⃗+𝑢⃗+∇𝑃(𝜌)𝜌=−𝛽||𝑢||⃗^𝑞−1𝑢,⃗(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)=𝜌^𝛾/𝛾, then the authors can introduce a nonlinear transformation 𝜎=(𝜌^𝜃/𝜃)(𝜃=(𝛾−1)/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 𝐑^𝟑 with smooth boundary. Then ‖𝑓‖_𝐿^∞(Ω)≤𝐶‖𝑓‖_𝐻²(Ω),‖𝑓‖_𝐿^𝑝(Ω)≤𝐶‖𝑓‖_𝐻¹_(Ω),2≤𝑝≤6(1.12) for some constant 𝐶>0 depending only on Ω.

Lemma 1.4. Let 𝑢⃗∈𝐻^𝑠(Ω) be a vector-valued function satisfying 𝑢⃗⋅𝑛∣⃗_𝜕Ω=0, where 𝑛⃗ is the unit outer normal of 𝜕Ω. Then ‖‖𝑢‖‖⃗_𝑠≤𝐶‖‖∇×𝑢‖‖⃗_𝑠−1+‖‖∇⋅𝑢‖‖⃗_𝑠−1+‖‖𝑢‖‖⃗_𝑠−1,(1.13) for 𝑠≥1, 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 expect𝜌,𝑢⃗⟶||𝜌Ω^∗||,𝟎as𝑡⟶∞,(2.1) where (𝜌,𝑢)⃗ is the solution of problem (1.11) and (1.2), and |Ω| is the measure of Ω in 𝐑^𝟑. Without loss of generality, we assume |Ω|=1.

For this purpose, we consider the perturbation system:𝑢_𝑡⃗𝜑_𝑡+∇⋅𝜑+𝜌^∗𝑢⃗=0,+𝑢⃗⋅∇𝑢⃗+𝑢⃗+∇𝑃𝜑+𝜌^∗[𝜑+𝜌^∗]=−𝛽||𝑢||⃗^𝑞−1𝑢,⃗(2.2) where 𝜑=𝜌−𝜌^∗. Let 𝑔(𝜌)=𝑄^(𝜌)=𝑃^(𝜌)/𝜌, then 𝑄^(𝜌)∇𝜌=∇𝑃(𝜌)/𝜌 and system (2.2) changes into𝑢_𝑡⃗𝜑_𝑡+𝜑+𝜌^∗∇⋅𝑢⃗+∇𝜑⋅𝑢⃗=0,+𝑢⃗⋅∇𝑢⃗+𝑢⃗+𝑔𝜑+𝜌^∗∇𝜑=−𝛽||𝑢||⃗^𝑞−1𝑢,⃗(2.3) that is, 𝑢_𝑖𝑡⃗𝜑_𝑡+𝜑+𝜌^∗𝑢^𝑖,𝑖+𝜑_,𝑖𝑢^𝑖⃗=0,+𝑢^𝑗⃗𝑢^𝑖,𝑗+𝑢^𝑖⃗+𝑔𝜑+𝜌^∗𝜑_,𝑖=−𝛽||𝑢||⃗^𝑞−1𝑢^𝑖⃗.(2.4) In matrix notation, we have⎡⎢⎢⎢⎢⎢⎢⎣𝜑_𝑡𝑢^1𝑡⃗𝑢^2𝑡⃗𝑢^3𝑡⃗⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣𝑔𝜑𝑢+¹⃗𝜌_∗𝜑+𝜌^∗000𝑢0¹⃗𝑢0₁⃗0000𝑢0₁⃗⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝜑_,1𝑢^1,1𝑢^2,1𝑢^3,1⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣𝑢0²⃗𝑢0₂⃗𝜑+𝜌^∗0𝑔𝜑+𝜌_∗00000𝑢0²⃗𝑢0₂⃗⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝜑_,2𝑢^1,2𝑢^2,2𝑢^3,2+⎡⎢⎢⎢⎢⎢⎢⎣𝑢₃⃗⎤⎥⎥⎥⎥⎥⎥⎦00𝜑+𝜌^∗00𝑢0³⃗𝑢0₃⃗0𝑔𝜑+𝜌_∗000𝑢³⃗⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣𝜑_,3𝑢^1,3𝑢^2,3𝑢^3,3⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣𝑢₁⃗0+𝛽||𝑢||⃗^𝑞−1𝑢¹⃗𝑢²⃗+𝛽||𝑢||⃗^𝑞−1𝑢²⃗𝑢³⃗+𝛽||𝑢||⃗^𝑞−1𝑢³⃗⎤⎥⎥⎥⎥⎥⎥⎦=0.(2.5) Denoting 𝑤=^𝜑⃗𝑢 and multiplying (2.5) on the left by the symmetrizer𝐷=diag𝑔𝜑+𝜌^∗,𝜑+𝜌^∗,𝜑+𝜌^∗,𝜑+𝜌^∗,(2.6) we can rewrite the result to be𝐷𝑤_𝑡+𝐴¹𝑤_,1+𝐴²𝑤_,2+𝐴³𝑤_,3+𝑓=0,(2.7) where𝐴¹=⎡⎢⎢⎢⎢⎢⎢⎣𝑔𝜑+𝜌^∗𝜑+𝜌_∗𝑢¹⃗𝜑+𝜌^∗𝑔𝜑+𝜌^∗00𝑔0𝜑+𝜌^∗𝜑+0𝜌^∗𝑢¹⃗𝜑+0𝜌_∗0000𝑢¹⃗𝜑+0𝜌_∗𝑢¹⃗𝐴₂⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣𝑔𝜑+𝜌^∗0𝑢²⃗𝜑+0𝜌_∗𝜑+𝜌^∗𝑔𝜑+𝜌^∗0𝜑+𝜌_∗𝑢²⃗00𝑔0𝜑+𝜌^∗00𝜑+0𝜌^∗𝑢²⃗𝜑+0𝜌_∗𝑢²⃗𝐴₃⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣𝑔𝜑+𝜌^∗0𝑢³⃗𝜑+0𝜌_∗0𝜑+𝜌^∗𝑔𝜑+𝜌^∗00𝑢³⃗𝜑+0𝜌_∗0𝜑+𝜌_∗𝑢³⃗0𝑔𝜑+𝜌^∗00𝜑+𝜌^∗𝑢³⃗𝜑+𝜌_∗⎤⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎢⎢⎣𝑢¹⃗0+𝛽||𝑢||⃗^𝑞−1𝑢¹⃗𝑢²⃗+𝛽||𝑢||⃗^𝑞−1𝑢²⃗𝑢³⃗+𝛽||𝑢||⃗^𝑞−1𝑢³⃗⎤⎥⎥⎥⎥⎥⎥⎦.(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 𝑑𝑑𝑡_Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤𝑑𝑥⃗+𝜌^∗_Ω||𝜕^𝑘𝑡𝑢𝑥,⃗𝑡||²𝑑𝑥⃗≤𝐶₃||‖𝑤‖||³,(2.9) for 𝑘=0,1,2,3.

Proof. (1) Zero Order Estimate.
Multiplying (2.7) by 𝑤 and integrating over Ω, we have _Ω𝑤⋅𝐷𝑤_𝑡𝑑𝑥⃗+³_𝑖=1_Ω𝑤⋅𝐴^𝑖𝑤_,𝑖𝑑𝑥⃗+_Ω𝑤⋅𝑓𝑑𝑥⃗=0.(2.10) Using Lemma 1.3 and Cauchy-Schwarz inequality, we have _Ω𝑤⋅𝐷𝑤_𝑡𝑑𝑥⃗=12𝑑𝑑𝑡_Ω𝑤⋅𝐷𝑤𝑑𝑥⃗−12_Ω≤1𝑤⋅𝐷_𝑡𝑤𝑑𝑥⃗2𝑑𝑑𝑡_Ω𝑤⋅𝐷𝑤𝑑𝑥⃗+𝐶‖‖𝐷_𝑡‖‖_𝐿∞_Ω≤1|𝑤|²𝑑𝑥⃗2𝑑𝑑𝑡_Ω𝑤⋅𝐷𝑤𝑑𝑥⃗+𝐶||‖𝑤‖||³.(2.11) For the second term coming from the left side of (2.10), we have ³_𝑖=1_Ω𝑤⋅𝐴^𝑖𝑤_,𝑖𝑑𝑥⃗=12_Ω3𝑖=1𝑤⋅𝐴^𝑖𝑤_𝑖𝑑𝑥⃗−12_Ω𝑤³_𝑖=1≤1𝐴^𝑖,𝑖𝑤𝑑𝑥⃗2_𝜕Ω𝑤⋅³_𝑖=1𝐴^𝑖𝑛_𝑖⃗𝑤𝑑𝑠+𝐶‖‖𝐴^𝑖,𝑖‖‖_𝐿∞_Ω|𝑤|²𝑑𝑥,⃗(2.12) where we have used the Divergence Theorem. Since 𝑤⋅³_𝑖=1𝐴^𝑖𝑛_𝑖⃗𝑤=𝑔𝜑²+2𝜌𝑔𝜑+𝜌||𝑢||⃗²𝑛⃗⋅𝑢⃗=0.(2.13) Here 𝑔=𝑔(𝜌),𝜌=𝜑+𝜌^∗, then (2.12) turns into ³_𝑖=1_Ω𝑤⋅𝐴^𝑖𝑤_,𝑖𝑑𝑥⃗≤𝐶||‖𝑤‖||³.(2.14) While the term _Ω𝑤⋅𝑓𝑑𝑥⃗=𝜌^∗+𝜑_Ω1+𝛽||𝑢||⃗^𝑞−1||𝑢||⃗²𝑑𝑥⃗≥𝜌^∗2_Ω||𝑢||⃗²𝑑𝑥,⃗(2.15) for 𝑞>1 and the smallness of 𝛿. Thus, we have 𝑑𝑑𝑡_Ω𝑤⋅𝐷𝑤𝑑𝑥⃗+𝜌^∗_Ω||𝑢||⃗²𝑑𝑥⃗≤𝐶||‖𝑤‖||³(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𝜕^𝑘𝑡𝑤_𝑡+³_𝑖=1𝐴^𝑖𝜕^𝑘𝑡𝑤_,𝑖+𝜕^𝑘𝑡𝑓=−^𝑘_𝑙=1𝑘!𝑙!(𝑘−𝑙)!𝜕^𝑙𝑡𝐷𝜕^{𝑡𝑘−𝑙}𝑤_𝑡+𝜕^𝑙𝑡𝐴^𝑖𝜕^{𝑡𝑘−𝑙}𝑤_,𝑖.(2.17) Multiply (2.17) by 𝜕^𝑘𝑡 and integrate over Ω to have _Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤_𝑡𝑑𝑥⃗+_Ω𝜕^𝑘𝑡𝑤⋅³_𝑖=1=−_𝑘𝐴^𝑖𝜕^𝑘𝑡𝑤_,𝑖𝑑𝑥⃗+_Ω𝜕^𝑘𝑡𝑤⋅𝜕^𝑘𝑡𝑓𝑑𝑥⃗_𝑙=1𝑘!𝑙!(𝑘−𝑙)!_Ω𝜕^𝑘𝑡𝑤⋅𝜕^𝑙𝑡𝐷𝜕^{𝑡𝑘−𝑙}𝑤_𝑡+𝜕^𝑘𝑡𝑤⋅𝜕^𝑙𝑡𝐴^𝑖𝜕^{𝑡𝑘−𝑙}𝑤_,𝑖𝑑𝑥.⃗(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𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤_𝑡𝑑𝑥⃗=12𝑑𝑑𝑡_Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤𝑑𝑥⃗−12_Ω≤1𝜕^𝑘𝑡𝑤⋅𝐷_𝑡𝜕^𝑘𝑡𝑤𝑑𝑥⃗2𝑑𝑑𝑡_Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤𝑑𝑥⃗+𝐶‖‖𝐷_𝑡‖‖_𝐿∞_Ω≤1||𝜕^𝑘𝑡𝑤||²𝑑𝑥⃗2𝑑𝑑𝑡_Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤𝑑𝑥⃗+𝐶||‖𝑤‖||³,(2.19)₃_𝑖=1_Ω𝜕^𝑘𝑡𝑤⋅𝐴^𝑖𝜕^𝑘𝑡𝑤_,𝑖𝑑𝑥⃗=12_Ω3𝑖=1𝜕^𝑘𝑡𝑤⋅𝐴^𝑖𝜕^𝑘𝑡𝑤_𝑖𝑑𝑥⃗−12_Ω𝜕^𝑘𝑡𝑤⋅³_𝑖=1≤1𝐴^𝑖,𝑖𝜕^𝑘𝑡𝑤𝑑𝑥⃗2_𝜕Ω𝜕^𝑘𝑡𝑤⋅³_𝑖=1𝐴^𝑖𝑛_𝑖⃗𝜕^𝑘𝑡𝑤𝑑𝑠+𝐶‖‖𝐴^𝑖,𝑖‖‖_𝐿∞_Ω=1||𝜕^𝑘𝑡𝑤||²𝑑𝑥⃗2_𝜕Ω𝑔𝜕^𝑘𝑡+𝐶‖‖𝐴_𝑖,𝑖𝜑²+𝜌||𝜕^𝑘𝑡𝑢||⃗²𝑢⃗⋅𝑛𝑑⃗𝑠+2_Ω𝜌𝑔𝜕^𝑘𝑡𝜑𝜕^𝑘𝑡𝑢⃗⋅𝑛⃗𝑑𝑠‖‖_𝐿∞_Ω||𝜕^𝑘𝑡𝑤||²𝑑𝑥⃗≤𝐶||‖𝑤‖||³,(2.20)_Ω𝜕^𝑘𝑡𝑤⋅𝜕^𝑘𝑡𝑓𝑑𝑥⃗=_Ω𝜕^𝑘𝑡𝑢⃗⋅^𝑘_𝑙=0𝜕^𝑙𝑡𝜑+𝜌^∗𝜕^{𝑡𝑘−𝑙}𝑢⃗+𝛽||𝑢||⃗^𝑞−1≥𝜌𝑢⃗𝑑𝑥_∗⃗+𝐶+_𝑘𝜑_Ω1+𝛽||𝑢||⃗^𝑞−1||𝜕^𝑘𝑡𝑢||⃗²𝑑𝑥⃗_𝑙=1_Ω+𝐶𝜌_∗𝜕^𝑙𝑡𝜑𝜕^𝑘𝑡𝑢⃗⋅𝜕^{𝑡𝑘−𝑙}1+||𝑢||⃗^𝑞−1𝑢⃗𝑑𝑥⃗+𝜑_Ω𝜕^𝑘𝑡𝑢⃗^𝑘_𝑙=1≥𝜌𝜕^𝑙𝑡1+𝛽||𝑢||⃗^𝑞−1𝜕^{𝑡𝑘−𝑙}𝑢𝑑⃗𝑥_∗⃗2_Ω||𝜕^𝑘𝑡𝑢||⃗²𝑑𝑥⃗+𝐶||‖𝑤‖||³.(2.21) For the term on the right-hand side of (2.18) we have, ||||_Ω≤𝜕^𝑘𝑡𝐶𝑤‖‖⋅𝜕𝜕^𝑙𝑡_𝑘𝑡𝐷𝜕^{𝑡𝑘−𝑙}𝑤_𝑡+𝜕^𝑘𝑡𝑤⋅𝜕^𝑙𝑡𝐴^𝑖𝜕^{𝑡𝑘−𝑙}𝑤_,𝑖𝑑𝑥||||⃗𝑤‖‖_𝐿2‖‖𝜕^𝑙𝑡𝐷‖‖_𝐿4‖‖𝜕^{𝑡𝑘−𝑙}𝑤_𝑡‖‖_𝐿4+𝐶‖‖𝜕^𝑘𝑡𝑤‖‖_𝐿2‖‖𝜕^𝑙𝑡𝐴^𝑖‖‖_𝐿4‖‖𝜕^{𝑡𝑘−𝑙}𝑤_,𝑖‖‖_𝐿4≤𝐶||‖𝑤‖||³,(2.22) for 𝑙=1,2 and 𝑙≤𝑘≤3. When 𝑙=𝑘=3, we get ||||_Ω≤𝜕^𝑘𝑡𝐶𝑤‖‖⋅𝜕𝜕^𝑙𝑡_𝑘𝑡𝐷𝜕^{𝑡𝑘−𝑙}𝑤_𝑡+𝜕^𝑘𝑡𝑤⋅𝜕^𝑙𝑡𝐴^𝑖𝜕^{𝑡𝑘−𝑙}𝑤_,𝑖𝑑𝑥||||⃗𝑤‖‖_𝐿2‖‖𝜕^𝑙𝑡𝐷‖‖_𝐿2‖‖𝑤_𝑡‖‖_𝐿∞+𝐶‖‖𝜕^𝑘𝑡𝑤‖‖_𝐿2‖‖𝜕^𝑙𝑡𝐴^𝑖‖‖_𝐿2‖‖𝑤_,𝑖‖‖_𝐿∞≤𝐶||‖𝑤‖||³.(2.23) Combining (2.18) and (2.23), we get 𝑑𝑑𝑡_Ω𝜕^𝑘𝑡𝑤⋅𝐷𝜕^𝑘𝑡𝑤𝑑𝑥⃗+𝜌^∗_Ω||𝜕^𝑘𝑡𝑢(𝑥,𝑡)||²𝑑𝑥⃗≤𝐶||‖𝑤‖||³,(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 ||‖𝑤‖||²≤𝐶₄²_𝑙=0‖‖𝜕^𝑙𝑡∇×𝑢⃗‖‖^22−𝑙+³_𝑙=0‖‖𝜕^𝑙𝑡𝑤‖‖².(2.25)

Proof. From the velocity equation (2.3)₂ we have ∇𝜑=−1𝑔(𝜌^∗+𝜑)𝑢⃗+𝑢^𝑡⃗+𝑢⃗⋅∇𝑢⃗+𝛽||𝑢||⃗^𝑞−1𝑢⃗.(2.26) Taking the 𝐿² inner product of (2.26) with ∇𝜑, we get ‖∇𝜑‖²=−_Ω1𝑔(𝜌^∗+𝜑)𝑢⃗+𝑢^𝑡⃗≤𝐶‖‖𝑢‖‖⃗₂+𝑢⃗⋅∇𝑢⃗+𝛽||𝑢||⃗^𝑞−1𝑢⃗⋅∇𝜑𝑑𝑥⃗≤𝐶‖‖𝑢‖‖⃗₂+‖‖𝑢_𝑡⃗‖‖²+‖‖𝑢‖‖⃗_𝐿^∞‖‖∇𝑢‖‖⃗²+𝛽‖‖𝑢‖‖⃗^𝐿𝑞−1_∞‖‖𝑢‖‖⃗²+‖‖𝑢_𝑡⃗‖‖²+𝐶||‖𝑤‖||^𝑟,(2.27) where 𝑟=min{3,𝑞+1}>2.
The continuity equation (2.3)₁ implies ∇⋅𝑢⃗=−1𝜑+𝜌^∗𝜑_𝑡+𝑢⃗⋅∇𝜑.(2.28) Therefore, we have ‖‖∇⋅𝑢‖‖⃗²≤𝐶‖‖𝜑_𝑡‖‖²+||‖𝑤‖||³.(2.29) Using Lemma 1.4 with 𝑠=1, we have ‖‖𝑢‖‖⃗²¹≤≤𝐶𝐶‖‖‖‖∇∇××𝑢𝑢‖‖‖‖⃗⃗²₂+‖‖∇⋅𝑢‖‖⃗²+‖‖𝑢‖‖⃗²+‖‖𝑢‖‖⃗²+‖‖𝜑_𝑡‖‖²+||‖𝑤‖||³.(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 ∇⋅𝑢⃗ is bounded by ∑^3𝑙=0‖𝜕^𝑙𝑡𝑤‖². 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 ∇⋅𝑢⃗. 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)⃗‖^22−𝑙 and ‖𝜕^𝑖𝑡𝑤‖² for 𝑙=0,1,2,𝑖=0,1,2,3. The following Lemma is contributed to the estimate of ∇×𝑢⃗.

Lemma 2.3. Let |‖𝑤‖|≤𝛿 is sufficiently small, then for any solution 𝑤=(𝜑,𝑢)⃗ of problem (1.11)(1.2), one has 12𝑑𝑑𝑡²_𝑙=0‖‖𝜕^𝑙𝑡∇×𝑢⃗‖‖²+²_𝑙=0‖‖𝜕^𝑙𝑡∇×𝑢⃗‖‖²≤𝐶||‖𝑤(𝑡)‖||³.(2.31)

Proof. Taking ∇× of the velocity equation in (2.3) and denoting the vorticity 𝑣⃗=∇×𝑢⃗, we have 𝑣_𝑡⃗+𝑣⃗==−𝑣⃗∇⋅∇×𝑢⃗−𝑢𝑢⃗⋅⃗⋅∇∇𝑢𝑣⃗⃗−−𝑣∇∇⃗×⋅𝑢(𝑔⃗−(𝜌𝛽)||𝑢∇||⃗𝜑)_𝑞−1−𝛽∇×||𝑢||⃗^𝑞−1𝑢⃗𝑣⃗−𝛽∇||𝑢||⃗^𝑞−1×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 𝜕𝑣_𝑡⃗+𝜕𝑣⃗==−−𝜕𝜕−∇𝑣×⃗⋅∇𝑢𝑢⃗⃗⋅+∇𝑢𝑢⃗⋅⃗∇+𝑣𝛽⃗+∇𝑣×∇⃗⋅||𝑢𝑢⃗||⃗−^𝑞−1𝑢𝜕⃗𝛽||𝑢||⃗_𝑞−1𝑣⃗+𝛽∇||𝑢||⃗^𝑞−1×𝑢⃗.(2.33) Multiplying the above equation by 𝜕𝑣⃗ and integrating the result over Ω, similar the calculations in (2.22) and (2.23), we get 12𝑑𝑑t‖‖𝜕𝑣‖‖⃗²+‖‖𝜕𝑣⃗(𝑡)‖‖²≤𝐶||‖𝑤‖||³,(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 −𝑑𝑑𝑡³_𝑙=1_Ω𝜕^𝑡𝑙−1𝜑𝜕^𝑙𝑡𝜑𝑑𝑥⃗+³_𝑙=0‖‖𝜕^𝑙𝑡𝜑‖‖²≤𝐶₅‖‖𝑢‖‖⃗²+‖‖𝑢_𝑡⃗‖‖²+||‖𝑤‖||^𝑟,(2.35) where 𝑟=min{3,𝑞+1}.

Proof. Due to the conservation of total mass, we know that ∫_Ω(𝜌−𝜌^∗)𝑑𝑥⃗=0, that is, ∫_Ω𝜑𝑑𝑥⃗=0. Using Poincaré^'s inequality and (2.27), we have ‖𝜑‖²≤𝐶‖∇𝜑‖²≤𝐶‖‖𝑢‖‖⃗²+‖‖𝑢_𝑡⃗‖‖+||‖𝑤‖||^𝑟.(2.36) Taking the 𝑡 derivative of (2.3)₁ and applying (∇⋅) to (2.3)₂, we get ∇⋅𝑢⃗_𝑡𝜑_𝑡𝑡+∇𝜑⋅𝑢⃗_𝑡+𝜑_𝑡∇⋅𝑢⃗+𝜑∇⋅𝑢⃗_𝑡=0,(2.37)+∇⋅𝑢⃗⋅∇𝑢⃗+∇⋅𝑢⃗+∇⋅(𝑔(𝜌)∇𝜑)=−𝛽∇⋅||𝑢||⃗^𝑞−1𝑢⃗.(2.38)(2.37) − (2.38)×𝜑 yields 𝜑_𝑡𝑡+∇𝜑⋅𝑢⃗_𝑡+𝜑_𝑡∇⋅𝑢⃗−𝜑∇⋅𝑢⃗⋅∇𝑢⃗+𝑔(𝜌)∇𝜑+𝑢⃗+𝛽||𝑢||⃗^𝑞−1𝑢⃗=0,(2.39) that is, 𝜑_𝑡𝑡+∇𝜑⋅𝑢⃗_𝑡+𝜑_𝑡∇⋅𝑢⃗+𝜑∇⋅𝑢_𝑡⃗=0.(2.40) Multiplying (2.40) by 𝜑, and integrating the result over Ω, we have 𝑑𝑑𝑡_Ω+𝜑𝜑_𝑡𝑑𝑥⃗−_Ω𝜑^2𝑡𝑑𝑥⃗+_Ω∇𝜑_𝑡⋅𝑢𝜑⃗𝑑𝑥⃗+_Ω∇𝜑⋅𝑢_𝑡⃗𝜑𝑑𝑥_Ω⃗𝜑_𝑡∇⋅𝑢⃗𝜑𝑑𝑥⃗+_Ω∇⋅𝜑²𝑢_𝑡⃗−∇𝜑²⋅𝑢_𝑡⃗𝑑𝑥⃗=0.(2.41) Using the Divergence Theorem, we obtain 𝑑𝑑𝑡_Ω+𝜑𝜑_𝑡𝑑𝑥⃗−_Ω𝜑^2𝑡𝑑𝑥⃗+_Ω∇𝜑_𝑡⋅𝑢𝜑⃗𝑑𝑥+_Ω∇𝜑⋅𝑢_𝑡⃗𝜑𝑑𝑥_Ω⃗𝜑_𝑡∇⋅𝑢⃗𝜑𝑑𝑥⃗+_Ω𝜑²𝑢_𝑡⃗⋅𝑛𝑑⃗𝑠−2_Ω𝜑∇𝜑⋅𝑢_𝑡⃗𝑑𝑥⃗=0.(2.42) Then we have −𝑑𝑑𝑡_Ω𝜑𝜑_𝑡𝑑𝑥⃗+_Ω𝜑^2𝑡𝑑𝑥⃗≤𝐶||‖𝑤‖||³.(2.43) Furthermore, applying the derivative 𝜕_𝑡 to (2.40) we have 𝜑_𝑡𝑡𝑡+∇𝜑_𝑡𝑡⋅𝑢⃗+∇𝜑⋅𝑢_𝑡𝑡+2∇𝜑_𝑡⋅𝑢_𝑡⃗+𝜑_𝑡𝑡∇⋅𝑢⃗+𝜑_𝑡∇⋅𝑢⃗_𝑡+𝜑∇⋅𝑢_𝑡𝑡+𝜑_𝑡∇⋅𝑢_𝑡⃗=0.(2.44) Taking 𝐿² inner product of (2.44) with 𝜑_𝑡, we get −𝑑𝑑𝑡_Ω𝜑_𝑡𝑡𝜑_𝑡𝑑𝑥⃗+_Ω𝜑^2𝑡𝑡𝑑𝑥⃗−_Ω∇⋅𝜑_𝑡𝑡𝑢⃗+𝜑𝑢_𝑡𝑡𝜑_𝑡𝑑𝑥⃗−_Ω∇⋅𝜑^2𝑡𝑢_𝑡⃗𝑑𝑥⃗−_Ω𝜑^2𝑡∇𝑢⃗_𝑡𝑑𝑥⃗=0,(2.45) that is, −𝑑𝑑𝑡_Ω𝜑_𝑡𝑡𝜑_𝑡𝑑𝑥⃗+_Ω𝜑^2𝑡𝑡𝑑𝑥⃗≤𝐶||‖𝑤‖||³,(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 𝜑_𝑡𝑡 to get −𝑑𝑑𝑡_Ω𝜑_𝑡𝑡𝑡𝜑_𝑡𝑡𝑑𝑥⃗+_Ω𝜑^{2𝑡𝑡𝑡}𝑑𝑥⃗≤𝐶||‖𝑤‖||³,(2.47) in which we have used 𝜑_𝑡𝑡∇⋅𝜑_𝑡𝑡𝑡𝑢⃗+𝜑_𝑡𝑡∇⋅𝜑𝑢_𝑡𝑡𝑡=∇⋅𝜑_𝑡𝑡𝑡𝜑_𝑡𝑡𝑢⃗−∇𝜑_𝑡𝑡𝜑_𝑡𝑡𝑡𝑢⃗+∇⋅𝜑𝜑_𝑡𝑡𝑢_𝑡𝑡𝑡−∇𝜑_𝑡𝑡𝜑𝑢_𝑡𝑡𝑡,(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‖𝜕^𝑙𝑡𝑤‖². Let𝜆=min𝜌^∗2𝐶₅,𝜌^∗4,𝑔𝜌^∗/2𝐸(𝑡)=_Ω4,³_𝑙=0𝜕^𝑙𝑡𝑤⋅𝐷𝜕^𝑙𝑡𝑤−𝜆³_𝑙=1𝜕^𝑡𝑙−1𝜑𝜕^𝑙𝑡𝜑𝑑𝑥.⃗(2.49) We can easily see that ∑^3𝑙=0‖𝜕^𝑙𝑡𝑤‖² and 𝐸(𝑡) are equivalent, that is, there exist constants 𝐴 and 𝐵 such that𝐴³_𝑙=0‖‖𝜕^𝑙𝑡𝑤‖‖²≤𝐸(𝑡)≤𝐵³_𝑙=0‖‖𝜕^𝑙𝑡𝑤‖‖².(2.50)

Calculating (2.9) +𝜆(2.35), we have𝑑𝑑𝑡_Ω³_𝑙=0𝜕^𝑙𝑡𝑤⋅𝐷𝜕^𝑙𝑡𝑤−𝜆³_𝑙=0𝜕^𝑡𝑙−1𝜑𝜕^𝑙𝑡𝜑𝑑𝑥⃗+𝜆³_𝑙=0‖‖𝜕^𝑙𝑡𝑤‖‖²≤𝐶||‖𝑤‖||𝑟,̃(2.51) where 𝑟̃=min{3,𝑟}, that is,𝑑𝑑𝑡𝐸(𝑡)+𝐵𝜆𝐸(𝑡)≤𝐶||‖𝑤‖||𝑟.̃(2.52) Then Gronwall’s inequality and the equivalent relationship (2.50) deduce that³_𝑙=0‖‖𝜕^𝑙𝑡𝑤‖‖²≤𝐶³_𝑙=0‖‖𝜕^𝑙𝑡𝑤(0)‖‖²𝑒^{−(𝜆/𝐵)𝑡}+𝐶^𝑡0𝑒^𝑠−𝑡||‖𝑤(𝑠)‖||𝑟𝑑̃𝑠.(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 if_Ω𝜌̃₀𝑑𝑥⃗=_Ω𝜌₀𝑑𝑥⃗=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).

Consider𝜌̃𝜌𝑥̃_𝑡,⃗=0Δ=𝑃𝜌(̃₀𝜌̃),∇𝑃(𝜌̃)⋅𝑛∣⃗_𝜕Ω(𝑥),=0,𝑡≥0,(3.2) with the initial data_Ω𝜌̃₀𝑑𝑥⃗=𝜌^∗,𝜌̃₀𝑥⃗∈𝐿^∞(Ω).(3.3) Here we assume 𝜌̃₀ is uniform bounded, that is, there exists a constant 𝜌^∗<𝜌<∞ such that0≤𝜌̃₀𝑥⃗≤𝜌.(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 𝜌̃ be the global solution of problem (3.2) and (3.3), 𝑀=−∇𝑃(𝜌̃). Then there exist positive constants 𝐶>0,𝜂>0, independent of time 𝑡, such that ‖‖𝜌̃−𝜌^∗‖‖₁+‖‖𝑀‖‖≤𝐶𝑒^−𝜂𝑡,(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 𝜌̃(𝑥,⃗𝑡) for 𝑡>𝑇, and 𝜌̃𝑥,⃗𝑡>𝜌^∗2,for𝑡>𝑇.(3.6) On the other hand, due to the comparison principle 0≤𝜌̃𝑥,⃗𝑡≤𝜌,forany𝑥,⃗𝑡∈Ω×[0,𝐿^∞).(3.7)
For 𝑡>𝑇, We consider 𝜌̃−𝜌^∗_𝑡=Δ𝑃(𝜌̃)−𝑃𝜌^∗.(3.8) Taking 𝐿² inner product of (3.8) with (𝜌̃−𝜌^∗), we have 0=12𝑑𝑑𝑡_Ω𝜌̃−𝜌^∗²=1𝑑𝑥⃗−_ΩΔ𝑃(𝜌̃)−𝑃𝜌^∗𝜌̃−𝜌^∗𝑑𝑥⃗2𝑑𝑑𝑡‖𝜌̃−𝜌^∗‖²+_Ω||∇𝑃(𝜌̃)−𝑃𝜌^∗||²𝑃^(𝜌̃)𝑑𝑥,⃗(3.9) where we have used boundary condition (3.2)₃. Combining the increasing property of 𝑃^(𝜌̃) and the inequality (3.7), we have 12𝑑𝑑𝑡‖𝜌̃−𝜌^∗‖²+1𝑃^𝜌‖‖∇𝑃(𝜌̃)−𝑃𝜌^∗‖‖²≤0.(3.10)
Multiplying (3.2)₁ with 𝑃^(𝜌̃), we have 𝑃(𝜌̃)_𝑡−𝑃^(𝜌̃)Δ𝑃(𝜌̃)=0,(3.11) that is, 𝑃(𝜌̃)−𝑃𝜌^∗_𝑡−𝑃^(𝜌̃)Δ𝑃(𝜌̃)−𝑃𝜌^∗=0.(3.12) Define 𝜓=𝜌̃−𝜌^∗,𝜙=𝑃(𝜌̃)−𝑃𝜌^∗.(3.13) Taking 𝐿² inner product of (3.12) with Δ𝜙, we get 12𝑑𝑑𝑡‖∇𝜙‖²+𝑃^𝜌2^∗‖Δ𝜙‖²≤0.(3.14)(3.10) plus (3.14) deduce 12𝑑𝑑𝑡‖𝜓‖²+‖∇𝜙‖²+𝐶‖∇𝜙‖²+‖Δ𝜙‖²≤0,(3.15) where 𝐶=min{𝑃^(𝜌^∗/2),1/𝑃^(𝜌)}. Since the conservation of total mass and the Poincaré’s inequality, we have ‖𝜓‖²≤𝐶‖∇𝜓‖²≤𝐶1𝑃^(𝜌^∗/2)²‖∇𝜙‖².(3.16) Then inequality (3.15) turns into 12𝑑𝑑𝑡‖𝜓‖²+‖∇𝜙‖²+𝐶‖𝜓‖²+‖∇𝜙‖²+‖Δ𝜙‖²≤0.(3.17) The Gronwall inequality and (3.16) deduce Lemma 3.1.

Acknowledgments

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).