Global weak solutions and asymptotic behavior to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

Abstract

This paper is concerned with existence of global weak solutions to a class of compressible Navier–Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρ θ with 1 2 < θ < max { 3 − γ , 3 2 } , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ = 1 , γ = 2 ).