Showing papers by "Hajer Bahouri published in 2013"

On the stability in weak topology of the set of global solutions to the Navier-Stokes equations

Abstract: Let X be a suitable function space and let \({\mathcal{G} \subset X}\) be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of \({\mathcal{G}}\) belongs to \({\mathcal{G}}\) if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to \({\mathcal{G}}\) (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.

Scattering for the critical 2-D NLS with exponential growth

Abstract: In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrodinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

On the Elements Involved in the Lack of Compactness in Critical Sobolev Embedding

Abstract: The goal of this paper is to emphasize ideas coming from microlocal analysis to refine the study of lack of compactness in critical Sobolev embedding. In particular, we shall highlight that the elements involving in the characterization of the defect of compactness of critical Sobolev embedding in Lebesgue spaces in [13] are of different kinds than the ones intervening in the study of the critical Sobolev embedding of \( H^1(\mathbb{R}^2)\) intoth e Orlicz space \( \mathcal{L}(\mathbb{R}^2)\) in [9]; although in the two frameworks the obstruction to compactness is expressed in the same manner in terms of defect measures. This observation is based on the characterization of these embedding by means of asymptotic decompositions and puts in light the relevance of microlocal analysis behind this approach.

Stability by rescaled weak convergence for the Navier-Stokes equations

Abstract: We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence $(u_{0, n})_{n\in \N}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_0$ which generates a global regular solution, does $u_{0, n}$ generate a global regular solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0,n} = n \vf_0(n\cdot)$ or $u_{0,n} = \vf_0(\cdot-x_n)$ with $|x_n|\to \infty$. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

A Fourier approach to the profile decomposition in Orlicz spaces

Abstract: This paper is devoted to the characterization of the lack of compactness of the Sobolev embedding of $H^N(R^{2N})$ into the Orlicz space using Fourier analysis.