227 research outputs found
New entropy for Korteweg's system, existence of global weak solution and Prodi-Serrin theorem
This work is devoted to prove new entropy estimates for a general isothermal
model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see
\cite{fDS}), which can be used as a phase transition model. More precisely we
will derive new estimates for the density and we will give a new structure for
the Korteweg system which allow us to obtain the existence of global weak
solution. The key of the proof comes from the introduction of a new effective
velocity.The proof is widely inspired from the works of A. Mellet and A.
Vasseur (see \cite{fMV}). In a second part, we shall give a Prody-Serrin
blow-up criterion for this system which widely improves the results of
\cite{Hprepa} and the known results on compressible systems
Well-posedness in critical spaces for barotropic viscous fluids
This paper is dedicated to the study of viscous compressible barotropic
fluids in dimension . We address the question of well-posedness for
{\it large} data having critical Besov regularity. %Our sole additional
assumption is that %the initial density be bounded away from zero. This
improves the analysis of \cite{DL} where the smallness of for some positive constant was needed. Our result
improve the analysis of R. Danchin by the fact that we choose initial density
more general in B^{\NN}_{p,1} with . Our result relies on a
new a priori estimate for the velocity, where we introduce a new structure to
kill the coupling between the density and the velocity. In particular our
result is the first where we obtain uniqueness without imposing hypothesis on
the gradient of the density
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
Regularity of weak solutions of the compressible isentropic Navier-Stokes equation
Regularity and uniqueness of weak solution of the compressible isentropic
Navier-Stokes equations is proven for small time in dimension under
periodic boundary conditions. In this paper, the initial density is not
required to have a positive lower bound and the pressure law is assumed to
satisfy a condition that reduces to when and
. In a second part we prove a condition of blow-up in
slightly subcritical initial data when . We finish by
proving that weak solutions in \T^{N} turn out to be smooth as long as the
density remains bounded in L^{\infty}(L^{(N+1+\e)\gamma}) with \e>0
arbitrary small
New formulation of the compressible Navier-Stokes equations and parabolicity of the density
In this paper we give a new formulation of the compressible Navier-Stokes by
introducing an suitable effective velocity v=u+\n\va(\rho) provided that the
viscosity coefficients verify the algebraic relation of \cite{BD}. We give in
particular a very simple proof of the entropy discovered in \cite{BD}, in
addition our argument show why the algebraic relation of \cite{BD} appears
naturally. More precisely the system reads in a very surprising way as two
parabolic equation on the density and the vorticity , and
as a transport equation on the divergence . We show the existence
of strong solution with large initial data in finite time when (\rho_0-1)\in
B^{\NN}_{p,1}. A remarkable feature of this solution is the regularizing
effects on the density. We extend this result to the case of global strong
solution with small initial data.Comment: 38 pages. arXiv admin note: text overlap with arXiv:1107.2332; and
with arXiv:1302.2617 by other author
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