Mathematics – Analysis of PDEs
Scientific paper
2010-02-12
Mathematics
Analysis of PDEs
Scientific paper
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.
Bresch Didier
Noble Pascal
No associations
LandOfFree
Mathematical derivation of viscous shallow-water equations with zero surface tension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Mathematical derivation of viscous shallow-water equations with zero surface tension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Mathematical derivation of viscous shallow-water equations with zero surface tension will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-420731