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

Boris Haspot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced formally by Van der Waals in [44]. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in [39]. 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 possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. Cauchy problem for viscous shallow water equations with a term of capillarity. Mathematical Models and Methods in Applied Sciences, 2010, 20 (7), pp.1049-1087. ⟨hal-00776874⟩