Cauchy problem for viscous shallow water equations with a term of capillarity

Details Cauchy problem for viscous shallow water equations with a term of capillarity Cauchy problem for viscous shallow water equations with a term of capillarity

2008-03-13

arxiv.org/abs/0803.1939v1

Mathematics

Analysis of PDEs

Scientific paper

In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions with general initial data. Uniqueness is also obtained.

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