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

Lishuang Peng

doi:10.4236/jamp.2019.79143

Abstract

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 L2-energy methods.

Highlights

In this paper, we consider the one-dimensional viscous micropolar fluid model in Lagrangian coordinates: vt − ux = 0, ut e + + px u2 2 = μ +( t ux v x, pu= )x κ θx v + μ uu v x ω 2 x v + vω2
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
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

Summary

Introduction

( { }) there exist positive constants δ0 ≤ min δ1,δ and ε0 , such that if δ < δ0 and the initial data (v0 ,u0 ,θ0 ,ω0 ) satisfies (v0 (⋅) −V (0,⋅),u0 (⋅) −U (0,⋅),θ0 (⋅) − Θ(0,⋅),ω0 (⋅)) H1( ) ≤ ε0 , the Cauchy problem (1)-(2) admits a unique global solution (v,u,θ ,ω )(t, x) 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.

Results

Conclusion