Global stability of rarefaction waves for the 1D compressible micropolar fluid model with density-dependent viscosity and microviscosity coefficients

Abstract

This paper is concerned with the global existence and large time behavior of strong solutions to the Cauchy problem of one-dimensional compressible isentropic micropolar fluid model with density-dependent viscosity and microviscosity coefficients, where the far-fields of the initial data are prescribed to be different. The pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R are considered. For the case when the corresponding Riemann problem of the resulting Euler equations admits two rarefaction waves solutions, it is shown that if the parameters α and γ satisfy some conditions and the initial data is sufficiently regular, without vacuum and mass concentrations, then the Cauchy problem of the one-dimensional compressible micropolar fluid model has a unique global strong nonvacuum solution, which tends to a superposition of these two rarefaction waves as time goes to infinity. This result holds for arbitrarily large initial perturbation and large-amplitudes rarefaction waves. Moreover, the exponential time decay rate of the microrotation velocity ω(t,x) under large initial perturbation is also derived. The proof is given by an elaborate energy method and the key ingredient is to deduce the uniform-in-time lower and upper bounds on the specific volume.