Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model

In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L-energy methods.


Introduction
In this paper, we consider the one-dimensional viscous micropolar fluid model in Lagrangian coordinates: γ > is the adiabatic exponent, s is the entropy of fluid, R and B are the positive constants. We impose the following initial and far field conditions: v u where 0 v ± > , , 0 u θ ± ± > , ω ± are given constants, and we assume It is known that the large-time behavior of solutions of the Cauchy problem (1) and (2) is closely related to the Riemann problem of the compressible Euler system: with the Riemann initial data v u It is well-known that the above system has three eigenvalues: provided that , .
The viscous contact wave ( )( )  (1) becomes smooth and behaviors as a diffusion waves due to the effect of heat conductivity. As in [2], we can define the viscous contact wave ( )( ) as follows.
Since the pressure of the profile ( )( ) , , , Then the leading part of the energy Equation (1) Using the Equation (7), c c t x V U = and (8), we get a nonlinear diffusion equation which has a unique self-similar solution On the other hand, there exists some positive constant δ  , such that for δ θ θ δ where 0 c and 1 c are two positive constants depending only on θ − and δ  .
is defined as follows: It is straightforward to check that ( ) , , and We are now in a position to state our main results. Let .
We can get main result from [5] and it is stated as the following theorem.
be the viscous contact wave defined in (11) with strength δ θ θ δ + − = − ≤  . Then there exist positive constants 0 δ and 0 ε , such that if 0 δ δ < and the initial data ( ) It is well-known [6] that there exists some suitably small 1 0 δ > such that for Since the rarefaction waves ( ) ) is the solution of the initial problem for the typical Burgers equation: be the viscous contact wave constructed in (11) and (9) with ( ) and Then our main result of this paper is as follows: (16) holds for some small 1 0 be as in (20) with strength δ θ θ δ Now, we briefly recall some related work in this aspect and make some comments on the analysis in this paper. The nonlinear stability of some basic wave patterns has been studied by many authors. The stability toward contact waves for solutions of systems of viscous conservation laws was first studied by Xin [8] who proved the nonlinear stability of a weak contact discontinuity for the compressible Euler equations with uniform viscosity. Later, Liu and Xin [9] showed the stability of contact discontinuities for a class of general systems of nonlinear conservation laws with uniform viscosity. And this result was improved by Xin Journal of Applied Mathematics and Physics and Zeng in [10]. The large-time asymptotic nonlinear stability of the supposition of viscous shock waves and contact discontinuities for system of viscous conservation laws with artificial viscosity under small initial perturbations was proved by Zeng [11]. Some interesting results have been obtained for compressible Navier-Stokes system. The asymptotics toward the rarefaction waves for compressible Navier-Stokes system is established in [7] [12] [13] [14]. For a free-boundary value problem, the asymptotic stability of a viscous contact wave of the one-dimensional compressible Navier-Stokes system was first proved by an elementary energy method by Huang, Matsumura, and Shi [15], where the initial perturbation and the strength of the contact discontinuity are suitably small. The asymptotic stability of the linear combination wave of viscous contact wave and the rarefaction waves for the Cauchy problem of the one-dimensional compressible Navier-Stokes system was obtained by Huang, Li and Matsumura in [16], and provided the strength of the combination wave is suitably small. The viscous shock profiles and viscous rarefaction waves have been shown to be asymptotically stable for quite general perturbation for the compressible Navier-Stokes system and more general systems of viscous strictly hyperbolic con- The compressible micropolar fluid model has become an important area of interest for mathematicians in the last several decades. The model for compressible flow of micropolar fluid in the one-dimensional case was first studied by N. Mujaković.
She considered the local-in-time existence and uniqueness [42], the global existence [43] and regularity of solutions [44] to an initial-boundary value problem with homogeneous boundary conditions of the compressible one-dimensional micropolar , where .

Reformation of the Problem and Preliminaries
Noticing that ( ) , , (1) 1 and (8). To make it more convenient to prove Theorem 1.2, in this section, we will reformulate the problem (1), then the system (1)-(2) be rewritten as and We derive an elementary inequality concerning the heat kernel which will play an essential role later. For 0 α > , we define Journal of Applied Mathematics and Physics It is easy to check that ( ) Then we have Lemma 2.1 (see [16]) For 0 T Then the following estimate holds: For the proof Lemma 2.1, one refers to [16]. Next, we summarize some basic ( ) Lemma 2.2 can be proved directly from Equations (10) and (11) (3) and (17) .
x L f f f ∞ ≤ (30) Since the local existence of the solution is well known (for example, see [42]), to prove the global existence part of Theorems 1.2, we only have to establish the following a priori estimates. T Once Proposition 2.1 is proved, we can extend the unique local solution ( ) , , , v u θ ω which can be obtained as in [42], to T = ∞ . Estimate (32) together with the Equation (22) which as well as (32) and the Sobolev inequality easily leads to the asymptotic behavior of the solutions, that is, (21).
From now on until the end of this paper, we always assume that 0 0 1 ε δ + ≤ . Proposition 2.1 is an easy consequence of the following lemmas.

Energy Estimates
In this section we will drive some a priori energy estimates for the solutions to the system (1). Since Theorem 1.1 has been proved by Liu and Yin that we can see the details in [5], we will give here the proof of Theorem 1.2 for brevity. We first give the following key estimate.
It follows from (20) that We can treat the other terms on the righthand side of (34) in the same way to obtain ( ) 1 .
x L Similar to (36), we have and It follows from (10)  , 1 .
Here we used the same method as in [16] and combined with [50], then, we can complete the proof of Lemma 3.2. We omit the details for simplicity.  , , , 0, X T φ ψ ζ ω ∈ satisfies (31) with suitably small 0 0 ε δ + . Then it holds for Proof. We rewrite Equation (22) 2 as Using (36), we obtain by direct calculation The Cauchy inequality leads to ( ) The estimate (10) It follows from (10) Then, by using the same argument as the previous, and integrating (80) over ( ) , using (84)-(86) and following Lemma 3.6, and first choosing η suitably small then δ suitably small, one can obtain equation (79). This completes the proof of Lemma 3.5.  The next Lemma is devoted to controlling the term   (89) and (90), and first choosing η suitably small then δ suitably small, one can obtain Equation (87). This completes the proof of Lemma 3.6. 

Conclusion
Thus, we finish the proof of Proposition 2.1, and so the proof of Theorem 1.2 is completed.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.