Global Strichartz Estimates for nontrapping Geometries: About an Article by H. Smith and C. Sogge

The purpose of this note is to present an alternative proof of a result by H. Smith and C. Sogge showing that in odd dimension of space, local (in time) Strichartz estimates and exponential decay of the local energy for solutions to wave equations imply global Strichartz estimates. Our proof allows to extend the result to the case of even dimensions of spaceComment: 9 page

Semi-classical measures for inhomogeneous Schr\"odinger equations on tori

The purpose of this note is to investigate the high frequency behaviour of solutions to linear Schr\"odinger equations. More precisely, Bourgain and Anantharaman-Macia proved that any weak-* limit of the square density of solutions to the time dependent homogeneous Schr\"odinger equation is absolutely continuous with respect to the Lebesgue measure on $R\times T^d$. Our contribution is that the same result automatically holds for non homogeneous Schr\"odinger equations, which allows for abstract potential type perturbations of the Laplace operator.Comment: Corrected a few typos, added, as an illustration, an appplication to a nonlinear equatio

Decays for Kelvin-Voigt damped wave equations I : the black box perturbative method

We show in this article how perturbative approaches~from our work with Hitrik (see also the work by Anantharaman-Macia) and the {\em black box} strategy from~ our work with Zworski allow to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates : Carleman estimates or geometric control estimates for Helmoltz equationCarleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin Voigt damping, such approach is unlikely to allow for the optimal results when additional geometric assumptions are considered (see \cite{BuCh, Bu19}), it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori).Comment: 13 pages, to appear SIAM Journal on Control and Optimization (SICON

Smoothing effect for Schr\"odinger boundary value problems

We show the necessity of the non trapping condition for the plain smoothing effect ($H^{1/2}$) for Schr\"odinger equation with Dirichlet boundary conditions in exterior problems. We also give a class of trapped obstacles (Ikawa's example) for which we can prove a weak ($H^{1/2 - \epsilon}$) smoothing effect.Comment: 19 pages, various modifications: slight simplifications, typos fixe

Stabilisation of wave equations on the torus with rough dampings

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings)

Imperfect geometric control and overdamping for the damped wave equation

We consider the damped wave equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in \cite{Chr-NC-erratum} is optimal in the case of one hyperbolic periodic geodesic. We show if the equation is overdamped, then the energy decays exponentially. Finally we show if the equation is overdamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially

Probabilistic well-posedness for the cubic wave equation

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs measures), and second, we show that the flow that we are able to construct enjoys very nice dynamical properties, including a new notion of probabilistic continuity

Strichartz estimates and the Cauchy problem for the gravity water waves equations

This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and H\"older estimates. We first prove tame estimates in Sobolev spaces depending linearly on H\"older norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these H\"older norms.Comment: 118 pages. This preprint expands and supersedes our previous submission arXiv:1308.1423. It contains in addition some results that have been withdrawn from our previous preprint arXiv:1212.062

On the Cauchy problem for gravity water waves

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+\epsilon}$-class for some $\epsilon>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.Comment: This is a shortened version (of our previous arXiv submission 1212.0626) which appeared in Inventiones 201

Cauchy theory for the gravity water waves system with non localized initial data

In this article, we develop the local Cauchy theory for the gravity water waves system, for rough initial data which do not decay at infinity. We work in the context of $L^2$-based uniformly local Sobolev spaces introduced by Kato. We prove a classical well-posedness result (without loss of derivatives). Our result implies also a local well-posedness result in H\"older spaces (with loss of $d/2$ derivatives). As an illustration, we solve a question raised by Boussinesq on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested by Boussinesq does indeed apply to this setting.Comment: 60 pages. This new version contains in addition an application to water-waves in a canal which have been withdrawn from our previous submission arXiv:1212.062