209 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
Whitham's equations for modulated roll-waves in shallow flows
This paper is concerned with the detailed behaviour of roll-waves undergoing
a low-frequency perturbation. We rst derive the so-called Whitham's averaged
modulation equations and relate the well-posedness of this set of equations to
the spectral stability problem in the small Floquet-number limit. We then fully
validate such a system and in particular, we are able to construct solutions to
the shallow water equations in the neighbourhood of modulated roll-waves proles
that exist for asymptotically large time
Nonlinear stability of viscous roll waves
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic
conservation laws, we show that spectral stability implies nonlinear stability
for spatially periodic viscous roll wave solutions of the one-dimensional St.
Venant equations for shallow water flow down an inclined ramp. The main new
issues to be overcome are incomplete parabolicity and the nonconservative form
of the equations, which leads to undifferentiated quadratic source terms that
cannot be handled using the estimates of the conservative case. The first is
resolved by treating the equations in the more favorable Lagrangian
coordinates, for which one can obtain large-amplitude nonlinear damping
estimates similar to those carried out by Mascia--Zumbrun in the related shock
wave case, assuming only symmetrizability of the hyperbolic part. The second is
resolved by the observation that, similarly as in the relaxation and detonation
cases, sources occurring in nonconservative components experience greater than
expected decay, comparable to that experienced by a differentiated source.Comment: Revision includes new appendix containing full proof of nonlinear
damping estimate. Minor mathematical typos fixed throughout, and more
complete connection to Whitham averaged system added. 42 page
Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications
Recently, A. Vasseur and C. Yu have proved the existence of global
entropy-weak solutions to the compressible Navier-Stokes equations with
viscosities and and a pressure
law under the form with and
constants. In this note, we propose a non-trivial relative entropy for such
system in a periodic box and give some applications. This extends, in some
sense, results with constant viscosities initiated by E. Feiersl, B.J. Jin and
A. Novotny.
We present some mathematical results related to the weak-strong uniqueness,
convergence to a dissipative solution of compressible or incompressible Euler
equations. As a by-product, this mathematically justifies the convergence of
solutions of a viscous shallow water system to solutions of the inviscid
shall-water system
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