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The purpose in this paper is to prove end point Strichartz estimates for the Schr\"odinger equation in the exterior domain of a generic non-trapping obstacle in the case $n \geq 3.$ In the case $n=2$ we have the same range of Strichartz…

We prove a $\langle t\rangle^\varepsilon$ bound on the growth of Sobolev norms for unbounded time dependent perturbations of the Laplacian on flat tori.

Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and…

We analyze long range wave propagation in three-dimensional random waveguides. The waves are trapped by top and bottom boundaries, but the medium is unbounded in the two remaining directions. We consider scalar waves, and motivated by…

In this paper, we study the endpoint reversed Strichartz estimates along general time-like trajectories for wave equations in $\mathbb{R}^{3}$. We also discuss some applications of the reversed Strichartz estimates and the structure of wave…

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are…

We study a quantum and classical correspondence related to the Strichartz estimates. First we consider the orthonormal Strichartz estimates on manifolds with ends. Under the nontrapping condition we prove the global-in-time estimates on…

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial\_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where…

For the one dimensional nonlinear Schr\"odinger equation with triple power nonlinearity and general exponents, we study analytically and numerically the existence and stability of standing waves. Special attention is paid to the curves of…

We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside…

This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity…

We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a…

We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R 3 and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary,…

We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix…

We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition,…

We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…

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,…

We prove global weighted Strichartz estimates for radial solutions of linear Schr\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields…

In this paper, we studied the space-time estimates for the solution to the Schr\"odinger equation. By polynomial partitioning, induction arguments, bilinear to linear arguments and broad norm estimates, we set up several maximal estimates…

We prove sharper Strichartz estimates than expected from theoptimal dispersion bounds.