Some results on regularity of weak solutions to the Navier-Stokes equations published recently in [3] follow easily from a classical theorem on compact operators. Further, weak solutions of the Navier-Stokes equations in the space $L^2(0,T,W^{1,3}{\left(𝛺\right)}^3)$ are regular.

Nous démontrons dans cet article que le système MHD tridimensionnel à densité et viscosité variables est localement bien posé lorsque $({{\rho }_0}^{-1}-1,u_0,B_0)\in {\dot{B}}_{p1}^{\frac{3}{p}}\left({ℝ}^3\right)\times {\dot{B}}_{p1}^{\frac{3}{p}-1}\left({ℝ}^3\right)\times {\dot{B}}_{p1}^{\frac{3}{p}-1}\left({ℝ}^3\right),$ pour $p\in ]1,3]$ et la densité initiale est proche d’une constante strictement positive. Nous démontrons également un résultat d’existence et d’unicité dans l’espace de Sobolev $H^{\frac{3}{2}+\alpha }\left({ℝ}^3\right)\times H^{\frac{3}{2}-1+\alpha }\left({ℝ}^3\right)\times H^{\frac{3}{2}-1+\alpha }\left({ℝ}^3\right)$ pour $\alpha \>0,$ sans aucune condition de petitesse sur la densité.

La compréhension du passage des équations de la mécanique des fluides compressibles aux équations incompressibles a fait de grands progrès ces vingt dernières années. L’objectif de cet exposé est de présenter l’évolution des méthodes mathématiques mises en œuvre pour étudier ce passage à la limite, depuis les travaux de S. Klainerman et A. Majda dans les années quatre–vingts, jusqu’à ceux récents de G. Métivier et S. Schochet (pour les équations non isentropiques). Suivant les conditions initiales...

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for...

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for...

The shape and velocity of a sliding droplet are computed by solving the Navier-Stokes equation with free interface boundary conditions. The Galerkin finite element method is implemented in a 2D computation domain discretized using an unstructured mesh with triangular elements. The mesh is refined recursively at the corners (contact points). The stationary sliding velocity is found to be strongly dependent on grid refinement, which is a consequence of the contact line singularity resolved through...

The most important result stated in this paper is a theorem on the existence of global solutions for the Navier-Stokes equations in Rn when the initial velocity belongs to the space weak Ln(Rn) with a sufficiently small norm. Furthermore, this fact leads us to obtain self-similar solutions if the initial velocity is, besides, an homogeneous function of degree -1. Partial uniqueness is also discussed.

We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Hölder space $C^{1,r}$. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic...