Stability of rarefaction waves for 1-D compressible viscous micropolar fluid model

In this paper, we study the large time behavior of the Cauchy problem of the compressible micropolar fluids in one dimensional space. According to Darcy's law, we see from the equation describing the evolution of the microrotation ω that ω→0 as t→∞. Actually, if the microstructure of the fluid is not taken into account, that is to say the effect of the angular velocity fields of the particle's rotation is omitted, i.e., ω=0, then the micropolar fluids equations reduce to the classical Navier-Stokes equations. Therefore the system should tend time-asymptotically to the corresponding classical full Navier-Stokes equations. We consider the case that the far field of the initial data for the microrotation velocity ω is zero, and the far fields of the initial data for other variables, such as the specific volume v, velocity u and entropy s, are connected by rarefaction waves to the corresponding Euler equations. In this case, we prove the stability of rarefaction waves for this compressible micropolar fluids model. Compared with the classical Navier-Stokes equations, the angular velocity ω in this model brings both benefit and trouble. The benefit lies in the fact that the term −vω is a damping term which provides extra regularity of ω, while the trouble is brought by the term vω2 which increases the nonlinearity of the system.