We classify in this article the structure and its transitions/evolution of the Taylor vortices with perturbations in one of the following categories: a) the Hamiltonian vector fields, b) the divergence-free vector fields, and c). the solutions of the Navier-Stokes equations on the two-dimensional torus.This is part of a project oriented toward to developing a geometric theory of incompressible fluid flows in the physical spaces.

In this article we introduce an adaptive multi-levelmethod in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of differenttime-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a correspondingadaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments,in particular for a convection-dominated problem.

We solve an optimal cost problem for a stochasticNavier-Stokes equation in spacedimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.

We first prove an abstract result for a class of nonlocalproblems using fixed point method. We apply this result toequations revelant from plasma physic problems. These equationscontain terms like monotone or relative rearrangement of functions.So, we start the approximation study by using finite element todiscretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

We give local and global well-posedness results for a system of twoKadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhuto model the oblique interaction of weakly nonlinear, two dimensional,long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves.We use the Fourier transform restriction norms introduced by J. Bourgain and the Strichartz estimates for the linear KP group. Finallywe extend the result of [3] for lower order perturbationof the system in the absence of transverse effects.