Problème de la surface libre de l'équation de Navier-Stokes —Cas stationnaire et cas périodique—
On s’intéresse à la résolution du système de Navier-Stokes incompressible à densité variable dans le demi-espace en dimension On considère des données initiales à régularité critique. On établit que si la densité initiale est proche d’une constante strictement positive dans et si la vitesse initiale est petite par rapport à la viscosité dans l’espace de Besov homogène alors le système de Navier-Stokes admet une unique solution globale. La démonstration repose sur de nouvelles estimations...
We consider sequences of solutions of the Navier-Stokes equations in , associated with sequences of initial data bounded in . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in , up to a remainder term small in ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If is an “admissible” space (in particular ...
We consider a stochastic system of particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly...
We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality with a large class of non-autonomous external forces. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal...
The work is devoted to reaction-diffusion-convection problems in unbounded cylinders. We study the Fredholm property and properness of the corresponding elliptic operators and define the topological degree. Together with analysis of the spectrum of the linearized operators it allows us to study bifurcations of solutions, to prove existence of convective waves, and to make some conclusions about their stability.
In this short note we give a link between the regularity of the solution to the 3D Navier-Stokes equation and the behavior of the direction of the velocity . It is shown that the control of in a suitable norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very...
In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.
Existence of a global attractor for the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has been shown already. In this paper we prove the higher regularity of the attractor.