Related papers: Strichartz estimates for Dirichlet-wave equations …
In this paper, we prove new multilinear Strichartz estimates, which are obtained by using techniques of Bourgain. These estimates lead to new critical well-posedness results for the nonlinear Schr\"odinger equation on irrational tori in two…
The Cauchy problem for the Zakharov system in the energy-critical dimension $d=4$ is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground…
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.…
We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the…
This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an…
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,…
In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz…
We prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions four and higher. This generalizes a result of Shibata and Tsutsumi in spatial…
We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and $N$. This allows one to extend the local existence results of the second and third author globally in…
We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application we consider the critical wave equation and prove the…
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 second-order stochastic parabolic equations in a cylindrical domain with homogeneous Dirichlet boundary conditions. Under a natural compatibility condition on the gradient-type noise, we establish global Schauder estimates in…
The optimal $L^4$-Strichartz estimate for the Schr{\"o}dinger equation on the two-dimensional rational torus $\mathbb{T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach…
The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional…
We obtain local energy decay as well as global Strichartz estimates for the solutions $u$ of the wave equation $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,\ t\in{\R},\ x\in{\R}^n,$ with time-periodic non-trapping metric $a(t,x)$ equal to $1$…
We prove certain weighted Strichartz estimates and use these to prove a sharp theorem for global existence of small amplitude solutions of $\square u= |u|^p$, thus verifying the so-called "Strauss conjecture".
We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as…
We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy…
We study the inverse problem of determining a real-valued potential in the two-dimensional Schr\"odinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of…
We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…