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Abstract: The purpose of this paper is to derive rigorously the so called viscousshallow water equations given for instance page 958-959 in A. Oron, S.H.Davis, S.G. Bankoff, Rev. Mod. Phys, 69 1997, 931?980. Such a system ofequations is similar to compressible Navier-Stokes equations for a barotropicfluid with a non-constant viscosity. To do that, we consider a layer ofincompressible and Newtonian fluid which is relatively thin, assuming nosurface tension at the free surface. The motion of the fluid is described by 3dNavier-Stokes equations with constant viscosity and free surface. We prove thatfor a set of suitable initial data asymptotically close to -shallow waterinitial data-, the Cauchy problem for these equations is well-posed, and thesolution converges to the solution of viscous shallow water equations. Moreprecisely, we build the solution of the full problem as a perturbation of thestrong solution to the viscous shallow water equations. The method of proof isbased on a Lagrangian change of variable that fixes the fluid domain and wehave to prove the well-posedness in thin domains: we have to pay a specialattention to constants in classical Sobolev inequalities and regularity inStokes problem.



Author: Didier Bresch LAMA, Pascal Noble ICJ

Source: https://arxiv.org/



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