20 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
A Lagrangian approach for the incompressible Navier-Stokes equations with variable density
Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes
equations in the whole -dimensional space. Under some smallness assumption
on the data, we show the existence of global-in-time unique solutions in a
critical functional framework. The initial density is required to belong to the
multiplier space of . In particular, piecewise
constant initial densities are admissible data \emph{provided the jump at the
interface is small enough}, and generate global unique solutions with piecewise
constant densities. Using Lagrangian coordinates is the key to our results as
it enables us to solve the system by means of the basic contraction mapping
theorem. As a consequence, conditions for uniqueness are the same as for
existence