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Nicolas Burq

Publications and source records attributed to Nicolas Burq.

68 records · Page 4Linked to original sources
arXiv2011

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.

math.AP↗
arXiv2011

Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schrödinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{itΔ}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

math.AP↗
arXiv2010

Minoration de la resolvante dans le cas captif

In this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schroedinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The proof rely on an idea of X. P. Wang, and on propagation of coherent states for Ehrenfest times.

math.AP↗
arXiv2010

Long time dynamics for the one dimensional non linear Schrödinger equation

In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).

math.AP↗
arXiv2010

Strichartz Estimates for Water Waves

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [2]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at (? = 0, ? = 0)).

math.AP↗
arXiv2009

On the Water Waves Equations with Surface Tension

The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Métivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.

math.AP↗
arXiv2006

Global existence for energy critical waves in 3-D domains

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(Ω) \times L^2(Ω)$ for any smooth (compact) domain $Ω\subset \mathbb{R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

math.AP↗
arXiv2005

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions $u_λ$ of the Bunimovich stadium $S$, satisfying $(Δ- λ^2) u_λ= 0$. Write $S = R \cup W$ where $R$ is the central rectangle and $W$ denotes the ``wings,'' i.e. the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in $R$ as $λ\to \infty$. We obtain a lower bound $C λ^{-2}$ on the $L^2$ mass of $u_λ$ in $W$, assuming that $u_λ$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_λ$ is controlled by $λ^2$ times the $L^2$ norm in $W$. Moreover, if $u_λ$ is a $o(λ^{-2})$ quasimode, the same result holds, while for a $o(1)$ quasimode we prove that $L^2$ norm of $u_λ$ is controlled by $λ^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_λ$ may be controlled by the integral of $w \abs{\partial_N u}^2$ along $\partial S \cap W$, where $w$ is a smooth factor on $W$ vanishing at $R \cap W$. These results complement recent work of Burq-Zworski which shows that the $L^2$ norm of $u_λ$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.

math.AP↗
arXiv2003

Smoothing effect for Schrödinger boundary value problems

We show the necessity of the non trapping condition for the plain smoothing effect ($H^{1/2}$) for Schrödinger 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 ($H^{1/2 - ε}$) smoothing effect.

math.AP↗
arXiv2003

Quantum ergodicity of boundary values of eigenfunctions: A control theory approach

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.

math.AP↗