114 research outputs found
Asymptotics and lower bound for the lifespan of solutions to the Primitive Equations
This article generalizes a previous work in which the author obtained a large
lower bound for the lifespan of the solutions to the Primitive Equations, and
proved convergence to the 3D quasi-geostrophic system for general and
ill-prepared (possibly blowing-up) initial data that are regularization of
vortex patches related to the potential velocity. These results were obtained
for a very particular case when the kinematic viscosity is equal to the
heat diffusivity , turning the diffusion operator into the classical
Laplacian. Obtaining the same results without this assumption is much more
difficult as it involves a non-local diffusion operator. The key to the main
result is a family of a priori estimates for the 3D-QG system that we obtained
in a companion paper
Local in time results for local and non-local capillary Navier-Stokes systems with large data
In this article we study three capillary compressible models (the classical
local Navier-Stokes-Korteweg system and two non-local models) for large initial
data, bounded away from zero, and with a reference pressure state
which is not necessarily stable ( can be non-positive). We
prove that these systems have a unique local in time solution and we study the
convergence rate of the solutions of the non-local models towards the local
Korteweg model. The results are given for constant viscous coefficients and we
explain how to extend them for density dependant coefficients.Comment: 39 page
Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data
In this article we prove highly improved and flexible Strichartz-type
estimates allowing us to generalize the asymptotics we obtained for a
stratified and rotating incompressible Navier-Stokes system: for large (and
less regular) initial data, we obtain global well-posedness, asymptotics (as
the Rossby number goes to zero) and convergence rates as a power of
the small parameter . Our approach is lead by the special structure
of the limit system: the 3D quasi-geostrophic system
Lagrangian methods for a general inhomogeneous incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients
International audienceWe study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p is different from 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier-Stokes system with constant coefficients
Convergence of capillary fluid models: from the non-local to the local Korteweg model
In this paper we are interested in the barotropic compressible Navier-Stokes
system endowed with a non-local capillarity tensor depending on a small
parameter such that it heuristically tends to the local Korteweg
system. After giving some physical motivations related to the theory of
non-classical shocks (see [28]) we prove global well-posedness (in the whole
space with ) for the non-local model and we also prove the
convergence, as goes to zero, to the solution of the local Korteweg
system
Existence of strong solutions in a larger space for the shallow-water system
This paper is dedicated to the study of both viscous compressible barotropic
fluids and Navier-Stokes equation with dependent density, when the viscosity
coefficients are variable, in dimension . We aim at proving the local
and global well-posedness for respectively {\it large} and \textit{small}
initial data having critical Besov regularity and more precisely we are
interested in extending the class of initial data velocity when we consider the
shallow water system, improving the results in \cite{CMZ1,H2} and \cite{arma}.
Our result relies on the fact that the velocity can be written as the sum
of the solution of the associated linear system and a remainder
velocity term ; then in the specific case of the shallow-water system
the remainder term is more regular than by taking into
account the regularizing effects induced on the bilinear convection term. In
particular we are able to deal with initial velocity in as
Fujita and Kato for the incompressible Navier-Stokes equations (see \cite{FK})
with an additional condition of type . We would
like to point out that this type of result is of particular interest when we
want to deal with the problem of the convergence of the solution of
compressible system to the incompressible system when the Mach number goes to
0
Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity
We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas & Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids
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