20,473 research outputs found
Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension
We prove weak-strong uniqueness results for the compressible Navier-Stokes
system with degenerate viscosity coefficient and with vacuum in one dimension.
In other words, we give conditions on the weak solution constructed in
\cite{Jiu} so that it is unique. The novelty consists in dealing with initial
density which contains vacuum. To do this we use the notion of
relative entropy developed recently by Germain, Feireisl et al and Mellet and
Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the
compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce
a new effective velocity which makes the system parabolic on the density and
hyperbolic on this velocity).Comment: arXiv admin note: text overlap with arXiv:1411.550
Hyperbolic systems of conservation laws in one space dimension
Aim of this paper is to review some basic ideas and recent developments in
the theory of strictly hyperbolic systems of conservation laws in one space
dimension. The main focus will be on the uniqueness and stability of entropy
weak solutions and on the convergence of vanishing viscosity approximations
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
Scale interactions in compressible rotating fluids
We study a triple singular limit for the scaled barotropic Navier-Stokes
system modeling the motion of a rotating, compressible, and viscous fluid,
where the Mach and Rossby numbers are proportional to a small parameter, while
the Reynolds number becomes infinite. If the fluid is confined to an infinite
slab bounded above and below by two parallel planes, the limit behavior is
identified as a purely horizontal motion of an incompressible inviscid fluid,
the evolution of which is described by an analogue of the Euler system
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