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Hajer Bahouri

Publications and source records attributed to Hajer Bahouri.

At least 19 recordsLinked to original sources
arXiv2026

Restriction problems on the three-dimensional Heisenberg nilmanifold

In this paper, we prove a spectral restriction theorem on the three-dimensional Heisenberg nilmanifold. Since this manifold is an $\mathbb S^1$-bundle over the flat torus $\mathbb T^2$, the result provides a sub-elliptic counterpart of Zygmund's restriction theorem on $\mathbb T^2$ \cite{zygmund}. We also establish its sharpness by means of the discrete short-time Fourier transform.

math.CA↗
arXiv2025

Some geometric and spectral aspects of restriction problems

This texts commemorates the memory of Haim Brezis and explores some aspects of the restriction problem, particularly its connections to spectral and geometric analysis. Our choice of subject is motivated by Brezis' significant contributions to various domains related to this problem, including harmonic analysis, partial differential equations, spectral theory, representation theory, number theory, and many others.

math.CA↗
arXiv2022

Spectral summability for the quartic oscillator with applications to the Engel group

In this article, we investigate spectral properties of the sublaplacian $-Δ_{G}$ on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying $F(-Δ_{G})u=u\star k_{F}$, for suitable scalar functions $F$, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.

math.AP↗
arXiv2021

Strichartz estimates and Fourier restriction theorems on the Heisenberg group

This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group $\mathbb{H}^d$ for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on $\mathbb{H}^d$ is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated by Tomas and Stein, is based on Fourier restriction theorems on $\mathbb{H}^d$, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

math.AP↗
arXiv2020

Local dispersive and Strichartz estimates for the Schr{ö}dinger operator on the Heisenberg group

It was proved by H. Bahouri, P. G{é}rard and C.-J. Xu in [9] that the Schr{ö}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{ö}dinger equation, by a refined study of the Schr{ö}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d$.

math.AP↗
arXiv2020

Global well-posedness for the derivative nonlinear Schrödinger equation

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac12}$ with mass strictly less than $4π$ or general initial conditions in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $H^{\frac12}$ and that furthermore the $H^{\frac12}$ norm of the solutions remains globally bounded in time. One should recall that for $H^s$, with $s < 1 / 2 $, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.

math.AP↗
arXiv2019

Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to Simons cone

In this article, we establish the existence of a family of hypersurfaces $(Γ(t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive approach consists in proving the existence of finite time blow up solutions of this hyperbolic equation under the form $u(t,x) \sim t^ {ν+1} Q\Big(\frac {x} {t^ {ν+1}} \Big) $, where~$Q$ is a stationary solution and $ν$ an arbitrary large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru. However contrary to these works, the equation to be handled in this article is quasilinear. This induces a number of difficulties to face.

math.AP↗
arXiv2017

Tempered distributions and Fourier transform on the Heisenberg group

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 functionto 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.

math.FA↗
arXiv2016

A Frequency Space for the Heisenberg Group

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by taking advantage of basic properties of Hermite functions) the Fourier transform f\_H of f to be a uniformly continuous mapping on the set N^d x N^d xR \ {0} endowed with a suitable distance. This enables us to extend f\_H to the completion of that space, and to get an explicit asymptotic description of the Fourier transform when the 'vertical' frequency tends to 0. We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the Euclidean case that are based on Fourier analysis. As an example, we here establish an explicit extension of the Fourier transform for smooth functions on H^d that are independent of the vertical variable.

math.CA↗
arXiv2016

Dispersive estimates for the Schrödinger operator on step 2 stratified Lie groups

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schrödinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay with exponant -(k+p-1)/2. In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples.

math.AP↗
arXiv2013

Stability by rescaled weak convergence for the Navier-Stokes equations

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.

math.AP↗
arXiv2013

Phase-space analysis and pseudodifferential calculus on the Heisenberg group

A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microlocal directions and completes, with the Littlewood-Paley theory developed in \cite{bgx} and \cite{bg}, a microlocal analysis of the Heisenberg group.

math.AP↗
arXiv2013

Phase-space analysis and pseudodifferential calculus on the Heisenberg group

This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microlocal directions and completes, with the Littlewood-Paley theory developed in \cite{bgx} and \cite{bg}, a microlocal analysis of the Heisenberg group.

math.AP↗
arXiv2013

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

Let $X$ be a suitable function space and let $\cG \subset X$ 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 $\cG$ belongs to $\cG$ if $n$ is large enough, provided the convergence holds "anisotropically" in frequency space. Typically that 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 does not belong to $\cG$ (hence which produces a solution blowing up in finite time) cannot have a strong anisotropy in its frequency support.

math.AP↗
arXiv2013

Scattering for the critical 2-D NLS with exponential growth

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger 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.

math.AP↗