Showing papers by "Hajer Bahouri published in 2018"

Tempered distributions and Fourier transform on the Heisenberg group

Abstract: The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$ is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable function to be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$ endowed with a suitable distance $\wh d$. This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.

On the stability of global solutions to the three-dimensional Navier-Stokes equations

Abstract: We prove a weak stability result for the three-dimensional homogeneous incom-pressible 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 u0 which generates a global smooth solution, does u0,n generate a global smooth solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples u_{0,n} = nϕ0(n·) or u_{0,n}) = ϕ0(· − x_n) with |x_n| → ∞. 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.