37 research outputs found
Existence of global strong solutions 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 the global existence
of strong solutions for initial data close from a constant state having
critical Besov regularity. In a first time, this article show the recent
results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a
new a priori estimate for the velocity, where we introduce a new structure to
\textit{kill} the coupling between the density and the velocity as in
\cite{H2}. We study so a new variable that we call effective velocity. In a
second time we improve the results of \cite{CD} and \cite{CMZ} by adding some
regularity on the initial data in particular is in . In this
case we obtain global strong solutions for a class of large initial data on the
density and the velocity which in particular improve the results of D. Hoff in
\cite{5H4}. We conclude by generalizing these results for general viscosity
coefficients
Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system
In this paper, we first prove the unique global strong solution with vacuum
to the two dimensional nonhomogeneous incompressible MHD system, as long as the
initial data satisfies some compatibility condition. As a corollary, the global
existence of strong solution with vacuum to the 2D nonhomogeneous
incompressible Navier-Stokes equations is also established. Our main result
improves all the previous results where the initial density need to be strictly
positive. The key idea is to use some critical Sobolev inequality of
logarithmic type, which is originally due to Brezis-Wainger.Comment: 16 page
Mathematical modeling of powder-snow avalanche flows
Powder-snow avalanches are violent natural disasters which represent a major
risk for infrastructures and populations in mountain regions. In this study we
present a novel model for the simulation of avalanches in the aerosol regime.
The second scope of this study is to get more insight into the interaction
process between an avalanche and a rigid obstacle. An incompressible model of
two miscible fluids can be successfully employed in this type of problems. We
allow for mass diffusion between two phases according to the Fick's law. The
governing equations are discretized with a contemporary fully implicit finite
volume scheme. The solver is able to deal with arbitrary density ratios.
Several numerical results are presented. Volume fraction, velocity and pressure
fields are presented and discussed. Finally we point out how this methodology
can be used for practical problems.Comment: 27 pages, 13 figures. Minor changes. A few references were added.
Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
Cauchy problem for viscous shallow water equations with a term of capillarity
International audienceIn this paper, we consider the compressible Navier-Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained