Slim Ibrahim

Trudinger–Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to

Global Small Solutions for the Navier–Stokes–Maxwell System

L^\infty

Well-posedness of the Navier—Stokes—Maxwell equations

. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrodinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

Scattering for the two-dimensional energy-critical wave equation

We consider a full system of magnetohydrodynamic equations. The system does not enjoy any property of scaling invariance and, at least formally, has an energy estimate. Nevertheless, the existence of a global weak solution seems to remain an interesting open problem in both two and three space dimensions. In three dimensions, we show the existence of global small mild solutions. In two dimensions, we prove the same result in a space “close” to the energy space.

Exponential Energy Decay for Damped Klein–Gordon Equation with Nonlinearities of Arbitrary Growth

We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equa- tions through Ohms law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions two and three but also refine those in (10). The main simplification comes from an a priori L 2(L ∞ ) estimate for solutions of the forced Navier-Stokes equations.

Threshold solutions in the case of mass-shift for the critical Klein-Gordon 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 derive a uniform exponential decay of the total energy for the nonlinear Klein–Gordon equation with a damping around spatial infinity in ℝ N or in the exterior of a star-shaped obstacle. Such a result was first proved by Zuazua [37, 38] for defocusing nonlinearity with moderate growth, and later extended to the energy subcritical case by Dehman et al. [7], using linear approximation and unique continuation arguments. We propose a different approach based solely on Morawetz-type a priori estimates, which applies to defocusing nonlinearity of arbitrary growth, including the energy critical case, the supercritical case and exponential nonlinearities in any dimensions. One advantage of our proof, even in the case of moderate growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than the one of the ground state, once we get control of the nonlinear part in Morawetz-type estimates. In particular this can be achieved when we have the scattering for the undamped equation.

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

We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation.

GEOMETRIC-OPTICS FOR NONLINEAR CONCENTRATING WAVES IN FOCUSING AND NON-FOCUSING TWO GEOMETRIES

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.

A Contraction Argument for Two-Dimensional Spiking Neuron Models

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.