Mohamed Majdoub

Scattering for the two-dimensional energy-critical wave equation

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain’s induction argument. We show the same result for the “subcritical” nonlinear Schrödinger equation.

Scattering for the two-dimensional NLS with exponential nonlinearity

We investigate existence and asymptotic completeness of the wave operators for nonlinear Schrodinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger–Moser type inequality. We prove that if the Hamiltonian is below the critical value, then the solution approaches a free Schrodinger solution at the time infinity.

EXISTENCE ET UNICITÉ DE SOLUTIONS POUR LE SYSTÈME DE NAVIER-STOKES AXISYMÉTRIQUE

We study the Navier-Stokes equations, written in a domain of R 3 which is invariant under rotation around the vertical axis, or in the whole space R 3; the solutions seeked are also invariant by that rotation, and we look for conditions on the initial data which are close to the natural assumptions in the case of two space dimensions.

Global well posedness for a 2D drift–diffusion–Maxwell system

Abstract We prove global existence of regular solutions to the drift–diffusion–Maxwell system in two space dimension. We also provide an exponential growth estimate for the norm of the solution when the time goes to infinity.

Weak solutions for generalized stationary Oldroyd-B fluid with a diffusive stress

Abstract This paper focuses on the analysis of the stationary case of incompressible viscoelastic generalized Oldroyd-B fluids derived in [2] by Bejaoui and Majdoub. The studied model is different from the classical Oldroyd-B fluid model in having a viscosity function which is shear-rate depending, and a diffusive stress added to the equation of the elastic part of the stress tensor. Under some conditions on the viscosity stress tensor and for a large class of models, we prove the existence of weak solutions in both two-dimensional and three-dimensional bounded domains for shear-thickening flows.