STABILITY OF THE COMPOSITE WAVE FOR THE INFLOW PROBLEM ON THE MICROPOLAR FLUID MODEL

. In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar ﬂuid model in half line R + := (0 , ∞ ) . Inspired by the relationship between the micropolar ﬂuid model and Navier-Stokes system, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inﬂow problem on micropolar ﬂuid model is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method.


HAIBO CUI AND HAIYAN YIN
We assume that the initial data at the far field x = +∞ are constants, namely lim x→+∞ (ρ 0 , u 0 , ω 0 , θ 0 )(x) = (ρ + , u + , ω + , θ + ) (4) and the boundary data for ρ, u, ω and θ at x = 0 are given by where ρ − > 0, u − > 0, θ − > 0, ω − are constants and the following compatibility conditions hold Note that ρ ± , u ± , ω ± and θ ± could be distinct. The boundary condition u − > 0 means that gas blows into the region from the boundary x = 0 with the velocity u − . Thus this problem is called an inflow problem (see [18]). The inflow boundary condition u − > 0 implies that the characteristic speeds of the hyperbolic equation (1) 1 for the density ρ is positive around the boundary. Hence not only u, ω and θ but also ρ has to be imposed boundary conditions for the well-posedness of inflow problem (1). In addition, the boundary condition u − < 0 means that fluid blows out from the boundary x = 0 with the velocity u − . Thus this problem is called an outflow problem (see [18]). The outflow boundary condition u − < 0 implies that the characteristic speeds of the hyperbolic equation (1) 1 for the density ρ is negative around the boundary so that boundary conditions on u, ω and θ to parabolic equations (1) 2 , (1) 3 and (1) 4 are necessary and sufficient for the well-posedness of outflow problem.
The study of micropolar fluid model started from 1966 ( [6]). The micropolar fluid model has many potential applications not only in physics but also in mathematics. In physics, the micropolar fluid may represent fluid consisting of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored, such as liquid crystals, polymeric fluids, ferro liquids, animal blood, among others. For more background, we refer to [17] and references therein. Because of their microstructures, viscous micropolar fluids are non-Newtonian with non-symmetric tensor and cannot be described only by using Navier-Stokes equations. Now more and more mathematicians are devoted to the research of micropolar fluid. Here we only list some results related to our paper. In [1,4,17], the authors proved the existence of weak solutions and strong solutions for the compressible micropolar fluid model. Nowakowski in [28] considered the strong solutions of incompressible micropolar fluid when divu = 0. When the initial data with vacuum was allowed, the blowup criterion of solutions to the three dimensional compressible micropolar fluid model was obtained in [2,3]. The large time behavior and stability of solutions for the compressible micropolar fluid model were obtained in [22,23,25,29]. The regularity of solutions to the initial boundary value problem for compressible micropolar fluid model was proved in [21,24]. Recently, the authors in [15] obtained the optimal time decay rate of the three-dimensional compressible micropolar fluid model.
In order to study the large time behavior of solutions to the initial boundary value problem (1), (3), (4), (5), (6), we assume that the microrotation velocity asymptotically converges to zero. Then the micropolar fluid model (1) can be reduced to the following Navier-Stokes system in the form Moreover, when the dissipation effects are neglected for the large time behavior, Navier-Stokes system (7) can be reduced to the following Euler system in the form The Euler system (8) is a typical example of the hyperbolic conservation laws. The Riemann solutions for the Euler system (8) are shock waves, rarefaction waves, contact discontinuity waves and the linear combinations of these basic waves. It is well known that the large time behavior of solutions to the Cauchy problem of Navier-Stokes system are basically described by the viscous versions of three basic waves. Here we only mention several results related to Navier-Stokes system: the asymptotic stability of shock wave [7]; the asymptotic stability of rarefaction wave [16]; the asymptotic stability of contact discontinuity wave [11,12]; the asymptotic stability of combination of viscous contact discontinuity wave with rarefaction waves [9]. Later, the initial-boundary value problem (IBVP) of Navier-Stokes system has attracted many mathematicians's attention because it has more physical meanings and of course produces some new mathematical difficulties due to the boundary effect. Not only basic waves but also a new wave, which is called the boundary layer solution (BL-solution for brevity), may appear in the IBVP of Navier-Stokes system. So far, a great number of mathematical results about the BL-solution for the Navier-Stokes system have been studied by many authors. For details, please refer to [8,13,19,20,26,27,30] and references therein.
Recently, for one dimensional compressible micropolar fluid model (1), Liu-Yin [14] have obtained the stability of contact discontinuity wave for the Cauchy problem and Cui-Yin also [5] have obtained the stability of BL-solution to the outflow problems. However, to our knowledge, there are few results on the stability of composite wave for one dimensional compressible micropolar fluid model (1). Hence, in this paper, we pay our attention on the stability of composite wave consisting of the subsonic BL-solution, the contact discontinuity wave, and the rarefaction wave for the inflow problem on the micropolar fluid model (1) to the Riemann problem on Euler system (8). Compared with the previous results about the micropolar fluid model for the global existence of solutions near constant states, this paper is concerned with nonlinear stability of solutions near non-constant states, and thereby yields many nonlinear hard terms. In one word, the main difficulties in our proofs are how to deal with the boundary terms, the interactions of three different waves and the external term, i.e. microrotation velocity ω. Moreover, the case for ω + = ω − which leads to more complex structure is left to study in the future.
For the purpose of obtaining the large time behavior of solutions to the initial boundary value problem (1), (3), (4), (5) and (6), it is more convenient to use the Lagrangian coordinates. That is, we consider the following coordinate transformation: We still denote the Lagrangian coordinates by (x, t) for simplicity of notation. Thus the system (1) can be transformed into the following moving boundary problem for the micropolar fluid model in the Lagrangian coordinates where v(x, t) = 1 ρ(x,t) represents the specific volume of the fluid, and the boundary moves with the constant speed σ − = − u− v− < 0. For the perfect gas, we have In order to fix the moving boundary x = σ − t, we introduce a new variable ξ = x − σ − t. Then we have the half-space problem Next, in thermodynamics, we know that by any given two of the five thermodynamical variables, v, p, e, θ and entropy s, the remaining three variables can be expressed. Without loss of generality, we define the entropy s as follows which obeys the second law of thermodynamics θds = de + pdv.
Then due to (12), the entropy of the initial data is expressed as follows Thus s + = lim x→+∞ s(v 0 (x), θ 0 (x)) satisfies The rest of the paper is arranged as follows. In Section 2, we state some important results in [30] which will be used in this paper. In Section 3, we reformulate the original system (1), then introduce our main theorem concerning the global existence and asymptotic stability of solutions. The proof of Theorem 3.1 is concluded in Section 4. In Appendix, we present the details which are left in the proofs of the previous sections for the completeness of the paper.
Notation: Throughout the paper, we denote generally large and small positive constants independent of t by C and c, respectively. And the character "C" and "c" may take different values in different places.
2. Some preliminaries of the Navier-Stokes system. Since we expect the solutions of the micropolar fluid model behave as that of Navier-Stokes system, we assume ω(x, t) = 0 for the large time behavior. Therefore, when t → ∞, the micropolar fluid model (9) and (11) respectively become the following Navier-Stokes system and Notice that Navier-Stokes system (15) and (16) have been studied by Qin and Wang in [30]. They obtained the existence (or nonexistence) of the BL-solution for the inflow problem when the right end state (v + , u + , θ + ) respectively belongs to the subsonic region Ω sub , transonic region Γ trans , and supersonic region Ω super (Ω sub , Γ trans and Ω super are defined in Section 2.1). Moreover, they proved the asymptotic stability of the single contact discontinuity wave and the composite wave consisting of the subsonic BL-solution, the contact discontinuity wave, and the rarefaction wave. Hence, in order to prove the asymptotical stability of composite wave consisting of the subsonic BL-solution, the contact discontinuity wave, and the rarefaction wave for the inflow problem on the micropolar fluid model (11), we first review some known results about Navier-Stokes system in [30] which will be used repeatedly in this paper.
For any given right end state (v + , u + , θ + ), we can define wave curves (BL-solution curve , contact discontinuity wave curve and 3-Rarefaction wave curve) in terms of (v, u, θ) with v > 0 and θ > 0 in the phase space as follows where M is a center-stable manifold defined in Section 2.1(BL-solution), p + = Rθ+ v+ , and λ 3 = λ 3 (v, s) is the third characteristic speed given in (17).
Before we present our main results, we first define Ω + sub := {(u, θ)|0 < u < Rγθ + }. In this paper, we expect to prove that if the left end state Moreover, the superposition of the BL-solution, the viscous contact discontinuity wave and the 3-rarefaction wave for the inflow problem to the micropolar fluid model (11) is asymptotically stable, provided that |(u − − u * , θ − − θ * )| and |v * − v * | are suitably small and the conditions in Theorem 3.1 hold. It is remarked that the BL-solution and the viscous contact discontinuity wave must be weak, but the rarefaction wave is not necessarily weak.
2.1. BL-solution. The characteristic speeds of the hyperbolic part of (15) are The sound speed C(v, θ) and the Mach number M (v, u, θ) are defined by Let C + = C(v + , θ + ) = Rγθ + and M + = |u+| C+ be the sound speed and the Mach number at the far field x = +∞, respectively. The phase plane R + × R × R + of (v, u, θ) can be divided into three subsets: where Ω sub , Γ trans and Ω super are called the subsonic, transonic and supersonic regions, respectively. If we add the alternative condition u > 0 or u < 0, then we have six connected subsets Ω ± sub , Γ ± trans and Ω ± super .
Hence the existence of the traveling wave solution to (15) or the stationary solution (BL-solution) to (16) is expected. From (18), where Integrating the system (19) 1 over (ξ, +∞), and then taking ξ = 0 in the resulting equality, it is easy to get Then the existence and uniqueness for the ODE system (19) are given as follows. For later use, we only list some useful properties of solutions of (19).
Case III. Subsonic case: M + < 1. Then there exists a center-stable manifold M tangent to the line on the opposite directions at the point (u + , θ + ), where a 2 and c 2 are some positive constants, see [30] for their definitions. Only when (u − , θ − ) ∈ M, does there exist a unique solution 2.2. Viscous contact wave.
then the following Riemann problem of the Euler system admits a single contact discontinuity solution From [11], we know that the viscous version of the above contact discontinuity, called viscous contact discontinuity wave V CD , U CD , Θ CD (x, t), could be defined by where Θ Sim (η) (η = x √ 1+t ) is the unique self-similar solution of the nonlinear diffusion equation Thus the viscous discontinuity contact wave defined in (27) satisfies the following property where δ CD = |θ + − θ − | is the amplitude of the viscous contact discontinuity wave and c 0 is some positive constant. Note that ξ = x − σ − t. Then the viscous contact where P CD := p(V CD , Θ CD ) = RΘ CD V CD and the error termsQ 1 ,Q 2 are given bȳ 2.3. Rarefaction wave. If (v − , u − , θ − ) ∈ R 3 (v + , u + , θ + ), then there exists a 3rarefaction wave (v r , u r , θ r ) ( x t ) which is the global (in time) weak solution of the following Riemann problem For the sake of constructing the smooth approximated rarefaction wave, we consider the Riemann problem on the Burgers equation for w − < w + . It is well-known that the Riemann problem (33) admits a continuous weak solution w r ( x t ) connecting w − and w + , taking the form of

HAIBO CUI AND HAIYAN YIN
Moreover, w r ( x t ) is approximated by a smooth function w(x, t) satisfying where δ r := w + −w − , q ≥ 16 is a constant, C q is a constant such that C q ∞ 0 y q e −y dy = 1, and ≤ 1 is a small positive constant. Then the solution w(x, t) of the Burgers equation (35) have the following properties. ( Thus we construct the smooth approximated rarefaction wave Note that ξ = x − σ − t. Then the smooth 3-rarefaction wave V R , U R , Θ R (ξ, t) defined above satisfies where then the smooth approximate rarefaction wave V R , U R , Θ R (ξ, t) satisfies the following properties: (i) ∂ ξ U R ≥ 0 for ξ ∈ R + and t ≥ 0.
We first define the perturbation as Then from (11) and (39), it is easy to obtain that [ϕ, ψ, ω, ζ](ξ, t) satisfies Concerning the global existence and time-asymptotic properties of solutions to the above reformulated half-space problem (42), one has the following theorem. , t) be the composite wave consisting of the subsonic BL-solution, the viscous contact discontinuity wave, and the rarefaction wave defined in (38) with the BL-solution amplitude δ B and the contact discontinuity wave amplitude δ CD . There exist positive constants δ 0 > 0 and C 0 > 0, such that if

and
[ϕ 0 , ψ 0 , ω 0 , ζ 0 ](ξ) 2 then the micropolar fluid model to the inflow problem (9) or the half-space problem (11) admits a unique global solution [v, u, ω, θ](ξ, t) satisfying Moreover, it holds that The key to the proof of global existence part of Theorem 3.1 is to derive the uniform a priori estimates of solutions to the half-space problem (42). Our a priori assumption is defined as follows: where ε 1 is a small positive constant.
Lemma 4.1 (Boundary estimates). There exists a positive constant C such that for any t > 0,

HAIBO CUI AND HAIYAN YIN
where ν is a positive small constant to be determined later, and C ν is a positive constant depending on ν.
We now give the following estimate concerning the delicate term (ϕ 2 + ζ 2 + ψ 2 )dξdτ provided that the wave strength δ B , δ CD and the constant are small enough.