Fast singular oscillating limits ofthe three-dimensional "primitive" equations ofgeophysical fluid flows are analyzed.We prove existence on infinite time intervals of regular solutions to the3D "primitive" Navier-Stokes equations for strongstratification (large stratification parameter N).This uniform existence is proven forperiodic or stress-free boundary conditionsfor all domain aspect ratios,including the case of three wave resonances which yield nonlinear " $2\frac{1}{2}$ dimensional" limit equations for N → +∞;smoothness assumptions are the same as for localexistence theorems, that is initial data in Hα , α ≥ 3/4.The global existence is proven using techniques ofthe Littlewood-Paley dyadic decomposition.Infinite time regularity for solutions of the3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonantequations and convergence theorems.

In this article we consider local solutions for stochastic Navier Stokesequations, based on the approach of Von Wahl, for the deterministic case. Wepresent several approaches of the concept, depending on the smoothnessavailable. When smoothness is available, we can in someway reduce thestochastic equation to a deterministic one with a random parameter. In thegeneral case, we mimic the concept of local solution for stochasticdifferential equations.

Evolution equations featuring nonlinearity, dispersion anddissipation are considered here. For classes of such equationsthat include the Korteweg-de Vries-Burgers equation and the BBM-Burgers equation, the zero dissipation limit is studied.Uniform bounds independent of the dissipation coefficient are derived and zero dissipation limit results with optimal convergence rates are established.

Non reflecting boundary conditions on artificial frontiersof the domain are proposed for bothincompressible and compressible Navier-Stokes equations.For incompressible flows, the boundary conditions lead to a well-posedproblem, convey properly the vortices without any reflections on theartificial limits and allow to compute turbulent flows at high Reynoldsnumbers.For compressible flows, the boundary conditions convey properly thevortices without any reflections on the artificial limits and alsoavoid acoustic waves that go back into the flow and change itsbehaviour.Numerical tests illustrate the efficiency of the various boundaryconditions.

Motivated by rotating fluids, we study incompressible fluidswith anisotropic viscosity.We use anisotropic spaces that enable us to prove existencetheoremsfor less regular initial data than usual. In the case of rotatingfluids, in the whole space, we prove Strichartz-type anisotropic, dispersive estimateswhich allow us to prove global wellposedness for fast enoughrotation.

We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stableprovided the bifurcation parameter $u_*$ does not exceed a critical value $u_{*}^{c}$ . The latter isthe limit of a decreasing sequence $(u_{*}^{k})$ of bifurcation points. The paper deals with the study of the 2D bifurcated branchesfrom the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we canapply the Crandall-Rabinowitz bifurcation theorem for a local study.We point out that the fully nonlinear reformulation of the FBP can also serve to developefficient numerical schemes in view of global information, such as techniques based on arc length continuation.

In this paper we present an analysis of the partial differential equations thatdescribe the Chemical Vapor Infiltration (CVI) processes. The mathematical modelrequires at least two partial differential equations, one describing thegas phase and one corresponding to the solid phase.A key difficulty in the process is the long processing times that are typicallyrequired. We address here the issue of optimization and show that we can chooseappropriate pressure and temperature to minimize these processing times.

The behavior of an ordinary differential equation for the low wave number velocitymode is analyzed. This equation was derived in [5]by an iterative process on the two-dimensional Navier-Stokes equations (NSE). Itresembles the NSE in form, exceptthat the kinematic viscosity is replaced by an iterated viscositywhich is a partial sum, dependent on the low-mode velocity. The convergence of this sum as the number of iterations is taken to be arbitrarily large is explored.This leads to a limiting dynamical system which displaysseveral unusual mathematical features.

Bhattacharya and Kohn have used small-strain (geometrically linear)elasticity to analyze the recoverable strains of shape-memory polycrystals.The adequacy of small-strain theory is open to question, however, since someshape-memory materials recover as much as 10 percent strain. This paperprovides the first progress toward an analogous geometrically nonlineartheory. We consider a model problem, involving polycrystals madefrom a two-variant elastic material in two space dimensions. The lineartheory predicts that a polycrystal with sufficient symmetry can have norecoverable strain. The nonlinear theory corrects this to the statement thata polycrystal with sufficient symmetry can have recoverable strain nolarger than the 3/2 power of the transformation strain. This result isin a certain sense optimal. Our analysis makes use of Fritz John'stheory of deformations with uniformly small strain.

By means of a result on the semi-global C 1 solution, we establish the exact boundary controllability for the reducible quasilinearhyperbolic system if the C 1 norm of initial data and final state issmall enough.

"Least regret control" consists in trying tofind a control which "optimizes the situation" with the constraint of not making things tooworse with respect to a known reference control,in presence of more or less significantperturbations. This notion was introduced in [7].It is recalled on a simple example (an ellipticsystem, with distributed control and boundary perturbation) in Section 2. We show that the problem reduces to a standard optimal control problem for augmented state equations.On another hand, we have introduced in recentnotes [9-12] the method ofvirtual control, aimed at the"decomposition of everything" (decomposition ofthe domain, of the operator, etc). Anintroduction to this method is presented, withouta priori knowledge needed, in Sections 3 and 4,directly on the augmented state equations.For problems without control, or with "standard" control, numerical applications of the virtualcontrol ideas have been given in the notes[9-12] and in the note[5].One of the first systematic paper devoted to allkind of decomposition methods, including multicriteria, is a jointpaper with A. Bensoussan and R. Temam, towhom this paper is dedicated, cf. [1].

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.