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Jean-Marc Bouclet

Publications and source records attributed to Jean-Marc Bouclet.

At least 19 recordsLinked to original sources
arXiv2026

Short-time parametrix for the Fokker--Planck semigroup and applications

We construct a short-time parametrix for the Fokker--Planck semigroup in Euclidean space. Among possible applications, we obtain smoothing and localization properties of the semigroup, the derivation of an approximate short-time polar decomposition for the semigroup, the construction of a parametrix of the resolvent, pseudospectral estimates, and estimates of the asymptotics of the number of eigenvalues of the Fokker--Planck operator. As a step in the proofs, we introduce a class of operators, which we call subsectorial operators, to which the Fokker--Planck operator belongs, and describe some of their functional analytic and spectral properties.

math.AP↗
arXiv2018

Sharp resolvent and time decay estimates for dispersive equations on asymptotically Euclidean backgrounds

The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{ö}dinger equations, on curved geometries, showing under very general assumptions the exact same decay as for the Euclidean case. Then we also extend these decay properties to the case of boundary value problems.

math.AP↗
arXiv2016

Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities

This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey-Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the full set of global-in-time Strichartz estimates including the endpoint case is derived. In the proof of Strichartz estimates, we develop a general criterion on perturbations ensuring that both homogeneous and inhomogeneous endpoint estimates can be recovered form resolvent estimates. Finally, we also investigate uniform resolvent estimates for long range repulsive potentials with critical singularities by using an elementary version of the Mourre theory.

math.AP↗
arXiv2016

Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping

We prove global Strichartz inequalities for the Schrödinger equation on a large class of asymptotically conical manifolds. Letting $ P $ be the nonnegative Laplace operator and $ f_0 \in C_0^{\infty}({\mathbb R}) $ be a smooth cutoff equal to $1$ near zero, we show first that the low frequency part of any solution $ e^{-itP} u_0 $, i.e. $ f_0 (P) e^{-itP} u_0 $, enjoys the same global Strichartz estimates as on $ {\mathbb R}^n $ in dimension $ n \geq 3 $. We also show that the high energy part $ (1-f_0)(P) e^{-itP} u_0$ also satisfies global Strichartz estimates without loss of derivatives outside a compact set, even if the manifold has trapped geodesics but in a temperate sense. We then show that the full solution $ e^{-itP}u_0 $ satisfies global space-time Strichartz estimates if the trapped set is empty or sufficiently filamentary, and we derive a scattering theory for the $ L^2 $ critical nonlinear Schrödinger equation in this geometric framework.

math.AP↗
arXiv2014

Strichartz inequalities on surfaces with cusps

We prove Strichartz inequalities for the wave and Schrödinger equations on noncompact surfaces with ends of finite area, i.e. with ends isometric to $ \big( (r_0,\infty) \times {\mathbb S}^1 , dr^2 + e^{- 2 ϕ(r)}d θ^2 \big) $ with $ e^{-ϕ} $ integrable. We prove first that all Strichartz estimates, with any derivative loss, fail to be true in such ends. We next show for the wave equation that, by projecting off the zero mode of $ {\mathbb S}^1 $, we recover the same inequalities as on $ {\mathbb R}^2 $. On the other hand, for the Schrödinger equation, we prove that even by projecting off the zero angular modes we have to consider additional losses of derivatives compared to the case of closed surfaces; in particular, we show that the semiclassical estimates of Burq-Gérard-Tzvetkov do not hold in such geometries. Moreover our semiclassical estimates with loss are sharp.

math.AP↗
arXiv2014

Sharp low frequency resolvent estimates on asymptotically conical manifolds

On a class of asymptotically conical manifolds, we prove two types of low frequency estimates for the resolvent of the Laplace-Beltrami operator. The first result is a uniform $ L^2 \rightarrow L^2 $ bound for $ \langle r \rangle^{-1} (- Δ_G - z)^{-1} \langle r \rangle^{-1} $ when $ \mbox{Re}(z) $ is small, with the optimal weight $ \langle r \rangle^{-1} $. The second one is about powers of the resolvent. For any integer $N$, we prove uniform $ L^2 \rightarrow L^2 $ bounds for $ \langle εr \rangle^{-N} (-ε^{-2} Δ_G - Z)^{-N} \langle εr \rangle^{-N} $ when $ \mbox{Re}(Z) $ belongs to a compact subset of $ (0,+\infty) $ and $ 0 < ε\ll 1 $. These results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity.

math.AP↗
arXiv2013

Local Energy Decay for the Damped Wave Equation

We prove local energy decay for the damped wave equation on R^d. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding "resolvent", both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre's commutators method.

math-ph↗
arXiv2013

Refined Sobolev inequalities on manifolds with ends

By considering a suitable Besov type norm, we obtain refined Sobolev inequalities on a family of Riemannian manifolds with (possibly exponentially large) ends. The interest is twofold: on one hand, these inequalities are stable by multiplication by rapidly oscillating functions, much as the original ones \cite{GMO}, and on the other hand our Besov space is stable by spectral localization associated to the Laplace-Beltrami operator (while $ L^p $ spaces, with $ p \ne 2 $, are in general not preserved by such localizations on manifolds with exponentially large ends). We also prove an abstract version of refined Sobolev inequalities for any selfadjoint operator on a measure space (Proposition \ref{general}).

math.CA↗
arXiv2013

Normal form of the metric for a class of Riemannian manifolds with ends

In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which contains asymptotically conical and hyperbolic manifolds. We study the decay rate to the metric at infinity associated to radial coordinates and also show that the latter metric is always conformally equivalent to the metric at infinity associated to the original coordinate system. We finally give several examples illustrating the sharpness of our results.

math.DG↗
arXiv2008

Semiclassical Resonances of Schrödinger operators as zeroes of regularized determinants

We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \prod_{w = {\rm resonances}}(z-w) \exp (φ_p(z,h)) $ and give semiclassical bounds on $ \partial_z φ_p $ as well as a representation of Koplienko's regularized spectral shift function. Here the index $ p \geq 1 $ depends on the decay rate at infinity of the perturbation.

math.SP↗
arXiv2006

Strichartz estimates for long range perturbations

We study local in time Strichartz estimates for the Schroedinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside of a large ball centered at the origin, the solutions of the Schroedinger equation enjoy the same Strichartz estimates as in the non perturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is non trapping, we prove that the Strichartz estimates hold in the whole space.

math.AP↗