Global behavior of 1D compressible isentropic Navier-Stokes equations with a non-autonomous external force
© Huang and Lian; licensee Springer. 2011
Received: 14 June 2011
Accepted: 3 November 2011
Published: 3 November 2011
In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force. The viscosity coefficient μ is proportional to ρ θ with 0 < θ < 1, where ρ is the density. Under certain assumptions imposed on the initial data and external force f, we obtain the global existence and regularity. Some ideas and more delicate estimates are introduced to prove these results.
The investigation in  showed that the continuous dependence on the initial data of the solutions to the compressible Navier-Stokes equations with vacuum failed. The main reason for the failure at the vacuum is because of kinematic viscosity coefficient being independent of the density. On the other hand, we know that the Navier-Stokes equations can be derived from the Boltzmann equation through Chapman-Enskog expansion to the second order, and the viscosity coefficient is a function of temperature. For the hard sphere model, it is proportional to the square-root of the temperature. If we consider the isentropic gas flow, this dependence is reduced to the dependence on the density function by using the second law of thermal dynamics.
For simplicity of presentation, we consider only the polytropic gas, i.e. P(ρ) = Aρ γ with A > 0 being constants. Our main assumption is that the viscosity coefficient μ is assumed to be a functional of the density ρ, i.e. μ = cρ θ , where c and θ are positive constants. Without loss of generality, we assume A = 1 and c = 1.
Now let us first recall some previous works in this direction. When the external force f ≡ 0, there have been many works (see, e.g., [2–9]) on the existence and uniqueness of global weak solutions, based on the assumption that the gas connects to vacuum with jump discontinuities, and the density of the gas has compact support. Among them, Liu et al.  established the local well-posedness of weak solutions to the Navier-Stokes equations; Okada et al.  obtained the global existence of weak solutions when 0 < θ < 1/3 with the same property. This result was later generalized to the case when 0 < θ < 1/2 and 0 < θ < 1 by Yang et al.  and Jiang et al. , respectively. Later on, Qin et al. [8, 9] proved the regularity of weak solutions and existence of classical solution. Fang and Zhang  proved the global existence of weak solutions to the compressible Navier-Stokes equations when the initial density is a piece-wise smooth function, having only a finite number of jump discontinuities.
For the related degenerated density function and viscosity coefficient at free boundaries, see Yang and Zhao , Yang and Zhu , Vong et al. , Fang and Zhang [13, 14], Qin et al. , authors studied the global existence and uniqueness under some assumptions on initial data.
When f ≠ 0, Qin and Zhao  proved the global existence and asymptotic behavior for γ = 1 and μ = const with boundary conditions u(0,t) = u(1,t) = 0; Zhang and Fang  established the global behavior of the Equations (1.1)-(1.2) with boundary conditions u(0,t) = ρ(1,t) = 0. In this paper, we obtain the global existence of the weak solutions and regularity with boundary conditions (1.4)-(1.5). In order to obtain existence and higher regularity of global solutions, there are many complicated estimates on external force and higher derivations of solution to be involved, this is our difficulty. To overcome this difficulty, we should use some proper embedding theorems, the interpolation techniques as well as many delicate estimates. This is the novelty of the paper.
The notation in this paper will be as follows:
denote the usual (Sobolev) spaces on [0,1]. In addition, || · || B denotes the norm in the space B; we also put .
The rest of this paper is organized as follows. In Section 2, we shall prove the global existence in H1. In Section 3, we shall establish the global existence in H2. In Section 4, we give the detailed proof of Theorem 4.1.
2 Global existence of solutions in H1
In this section, we shall establish the global existence of solutions in H1.
The proof of Theorem 2.1 can be done by a series of lemmas as follows.
where C1(T) denotes generic positive constant depending only on, time T and.
which, by virtue of Gronwall's inequality and assumption f(r(x,·),·) ∈ L2n([0,T], L2n[0,1]), gives (2.1).
and (2.2) follows from (2.8) and (2.9).
, which, along with (2.11), yields (2.3). The proof of Lemma 2.1 is complete.
Using the Gronwall inequality to (2.18), we obtain (2.13).
which, by virtue of Gronwall's inequality, (2.1) and (2.14), gives (2.19).
Proof of Theorem 2.1 By Lemmas 2.1-2.3, we complete the proof of Theorem 2.1.
3 Global existence of solutions in H 2
Constant C2(T) denotes generic positive constant depending only on the H2-norm of initial data , time T and constant C1(T).
Therefore, the generic constant C2(T) depending only on the norm of initial data (ρ0,u0) in H2, the norms of f in the class of functions in (3.2)-(3.3) and time T.
The proof of Theorem 3.1 can be divided into the following several lemmas.
which, along with (3.14), gives (3.7). The proof is complete.
which, along with Lemma 2.1, gives estimate (3.15).
The proof is complete.
Proof of Theorem 3.1 By Lemmas 3.2-3.3, Theorem 2.1 and Sobolev's embedding theorem, we complete the proof of Theorem 3.1.
4 Global existence of solutions in H 4
Therefore, the generic constant C4(T) depending only on the norm of initial data (ρ0,u0) in H4, the norms of f in the class of functions in (4.2)-(4.3) and time T.
The proof of Theorem 4.1 can be divided into the following several lemmas.
Thus, estimate (4.9) follows from (4.12), (4.14), (4.17) and condition (4.1).
If we apply Gronwall's inequality to (4.19), we conclude (4.11). The proof is complete.
Integrating (4.30) with respect to t, picking ε small enough, using Theorem 2.1 and Theorem 3.1, Lemma 4.2 and assumption (4.1), we complete the proof of estimate (4.20).