969 research outputs found
Mathematical derivation of viscous shallow-water equations with zero surface tension
The purpose of this paper is to derive rigorously the so called viscous
shallow 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 of
equations is similar to compressible Navier-Stokes equations for a barotropic
fluid with a non-constant viscosity. To do that, we consider a layer of
incompressible and Newtonian fluid which is relatively thin, assuming no
surface tension at the free surface. The motion of the fluid is described by 3d
Navier-Stokes equations with constant viscosity and free surface. We prove that
for a set of suitable initial data (asymptotically close to "shallow water
initial data"), the Cauchy problem for these equations is well-posed, and the
solution converges to the solution of viscous shallow water equations. More
precisely, we build the solution of the full problem as a perturbation of the
strong solution to the viscous shallow water equations. The method of proof is
based on a Lagrangian change of variable that fixes the fluid domain and we
have to prove the well-posedness in thin domains: we have to pay a special
attention to constants in classical Sobolev inequalities and regularity in
Stokes problem
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities
In this paper, we prove the local well-posedness in critical Besov spaces for
the compressible Navier-Stokes equations with density dependent viscosities
under the assumption that the initial density is bounded away from zero.Comment: 27page
Optimized Schwarz waveform relaxation for Primitive Equations of the ocean
In this article we are interested in the derivation of efficient domain
decomposition methods for the viscous primitive equations of the ocean. We
consider the rotating 3d incompressible hydrostatic Navier-Stokes equations
with free surface. Performing an asymptotic analysis of the system with respect
to the Rossby number, we compute an approximated Dirichlet to Neumann operator
and build an optimized Schwarz waveform relaxation algorithm. We establish the
well-posedness of this algorithm and present some numerical results to
illustrate the method
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