Related papers: About the wave equation outside two strictly conve…

Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how…

Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive…

The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small…

We prove Strichartz inequalities for the wave and Schr\"odinger 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 \phi (r)}d \theta^2 \big) $…

We consider the problem of large data scattering for the defocusing cubic nonlinear Schr\"odinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$. This equation is critical both at the level of energy and mass. The key ingredients…

We establish a mixed-norm Strichartz type estimate for the wave equation on Riemannian manifolds $(\Omega,g)$, for the case that $\Omega$ is the exterior of a smooth, normally hyperbolic trapped obstacle in $n$ dimensional Euclidean space,…

In this paper, we study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a non eclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic…

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…

We establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there…

In the present paper we consider Schr\"odinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity.…

Scattering for the energy critical non-linear wave equation for domains exterior to non trapping obstacles in 3+1 dimension is known for the star-shaped case. In this paper, we extend the scattering for a class of non star-shaped obstacles…

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 scattering for a massless wave equation which is critical in two space dimensions. Our method combines conformal inversion with decay estimates from Struwe's previous work on global existence of a similar equation.

Schultz \cite{S98} proved dispersive estimates for the wave equation on lattice graphs $\mathbb{Z}^d$ for $d=2,3,$ which was extended to $d=4$ in \cite{BCH23}. By Newton polyhedra and the algorithm introduced by Karpushkin \cite{K83}, we…

We study local-in-time and global-in-time bilinear Strichartz estimates for the Schr\"odinger equation on waveguides. As applications, we apply those estimates to study global well-posedness of nonlinear Schr\"odinger equations on these…

This note is concerned with Strichartz estimates for the wave equation and orthonormal families of initial data. We provide a survey of the known results and present what seems to be a reasonable conjecture regarding the cases which have…

In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$…

A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two finite-energy solutions to the wave equation are…

We establish Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds $(\Omega,\g)$ with boundary, for both the compact case and the case that $\Omega$ is the exterior of a smooth, non-trapping obstacle in Euclidean…

We prove the (local in time) Strichartz estimates (for the full range of parameters given by the scaling unless the end point) for asymptotically flat and non trapping perturbations of the flat Laplacian in $\R^n$, $n\geq 2$. The main point…