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CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

Abstract : In this paper, we consider the compressible Navier-Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. 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 Ref. 46. 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.
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Submitted on : Tuesday, May 1, 2012 - 7:32:38 PM
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Boris Haspot. CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2010, 20 (7), pp.1049--1087. ⟨10.1142/S0218202510004532⟩. ⟨hal-00693019⟩

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