Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier–Stokes equations with degenerate viscosity coefficient

Abstract

This paper is concerned with the existence of global weak solutions to the 1D compressible Navier–Stokes equations with density-dependent viscosity and initial density that is connected to vacuum with discontinuities. When the viscosity coefficient is proportional to ρ^θ with 0<θ<max{3−γ, 3/2} where ρ is the density, we prove a global existence theorem, improving thus the result in Meth. Appl. Anal. 12 (2005), 239–252, where 0<θ<1 is required. Moreover, we show that the domain occupied by the fluid expands into vacuum at an algebraic rate as time grows up due to the dispersion effect of the 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).