We construct global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients when the initial data is large, discontinuous, and spherically symmetric. We focus on the case where those coefficients vanish on vacuum. The solutions are obtained as limits of solutions in annular regions between two balls, and the equations hold in the sense of distribution in the entire space-time domain. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water.

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Existence of weak solutions Asymptotic behavior of solutionsThis paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with densitydependent viscosity and vacuum. When the viscosity coefficienta 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).

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