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N. Burq

Publications and source records attributed to N. Burq.

17 recordsLinked to original sources
arXiv2021

$H^1$ scattering for mass-subcritical NLS with short-range nonlinearity and initial data in $Σ$

We consider short-range mass-subcritical nonlinear Schrödinger equations and we show that the corresponding solutions with initial data in $Σ$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and Tsutsumifrom $L^2$ to $H^1$.We also provide some partial results concerning the scattering of the first order moments, as well as a short proof via lens transform of a classical result due to Tsutsumi and Cazenave-Weissler on the scattering in $Σ$.

math.AP↗
arXiv2015

Long time dynamics for damped Klein-Gordon equations

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

math.AP↗
arXiv2007

Global existence for energy critical waves in 3-d domains : Neumann boundary conditions

We prove that the defocusing quintic wave equation, with Neumann boundary conditions, is globally wellposed on $H^1_N(Ω) \times L^2(Ω)$ for any smooth (compact) domain $Ω\subset \mathbb{R}^3$. The proof relies on one hand on $L^p$ estimates for the spectral projector by Smith and Sogge, and on the other hand on a precise analysis of the boundary value problem, which turns out to be much more delicate than in the case of Dirichlet boundary conditions.

math.AP↗
arXiv2007

Invariant measure for a three dimensional nonlinear wave equation

We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of radial initial data in $\cap_{s<1/2} H^s(B(0,1))$ the equation is (globally in time) well posed and we construct an invariant measure.

math.AP↗
arXiv2007

Random data Cauchy theory for supercritical wave equations II : A global existence result

We prove that the subquartic wave equation on the three dimensional ball $Θ$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(Θ)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.

math.AP↗
arXiv2007

Random data Cauchy theory for supercritical wave equations I: Local theory

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in $H^s(M)$, where $s\geq 1/4$ in the case of a boundary less manifold and $s\geq 8/21$ in the case of a manifold with boundary.

math.AP↗
arXiv2006

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.

math.SP↗
arXiv2005

On well-posedness for the Benjamin-Ono equation

We prove existence of solutions for the Benjamin-Ono equation with data in $H^s(\R)$, $s>0$. Thanks to conservation laws, this yields global solutions for $H^\frac 1 2(\R)$ data, which is the natural ``finite energy'' class. Moreover, inconditional uniqueness is obtained in $L^\infty_t(H^\frac 1 2(\R))$, which includes weak solutions, while for $s>\frac 3 {20}$, uniqueness holds in a natural space which includes the obtained solutions.

math.AP↗
arXiv2005

Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear SchrÖdinger Equations

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space $\R^3$ and the flat torus $\T^3$ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.

math.AP↗
arXiv2004

Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications

We prove smoothing estimates for Schrödinger equations $i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.

math.AP↗
arXiv2003

Eigenfunctions for partially rectangular billiards

In this note we further develop the idea of using a ``black box'' point of view (see our previous work) to study eigenfunctions for billiards which have rectangular components: they include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards

math.SP↗
arXiv2003

Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces

We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère $\mathbb{S}^2$ et plus généralement sur toute surface de Zoll, on démontre que pour $s>1/4$, NLS est uniformément bien posée dans $H^s$, ce qui est optimalsur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

math.AP↗
arXiv2003

Bouncing ball modes and quantum chaos

Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only elementary calculus. In the case of the famous Bunimovich billiard table we prove that the wave functions have to spread into any neighbourhood of the wings.

math.AP↗