Related papers: Generalized Strichartz estimates for wave and Dira…
We study the dispersive properties of the Schr\"odinger equation. Precisely, we look for estimates which give a control of the local regularity and decay at infinity {\it separately}. The Banach spaces that allow such a treatment are the…
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…
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…
We present direct and elementary commutator techniques for the Dirac equation with long-range electric and mass perturbations. The main results are absence of generalized eigenfunctions and locally uniform resolvent estimates, both in terms…
We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1(\mathbb{R}^3)$. The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation.…
We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the case of wave, Klein-Gordon and fractional Schr\"odinger equations. Our results generalize the classical (single-function) Strichartz…
This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach,…
We study the deacy and Strichartz estimates for the massive Dirac Hamiltonian in a constant magnetic fields in $\mathbb{R}_t\times\mathbb{R}^2_x$: \begin{equation*} \begin{cases} i\partial_tu(t,x)-\mathcal{D}_Au(t,x)=0, u(0,x)=f,…
We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global…
We derive new solutions of the Schr\"odinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of…
We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
We study the $L^p-L^q$-type uniform resolvent estimates for 2D-Schr\"odinger operators in scaling-critical magnetic fields, involving the Aharonov-Bohm model as a main example. As an application, we prove localization estimates for the…
We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schr\"odinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both…
We study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are…
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,…
Applying the spectral measure estimates obtained in the author's joint work with A. Hassell, we establish global-in-time Strichartz estimates without loss via truncated / microlocalized dispersive estimates as well as energy estimates.
We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its…
We establish new Strichartz estimates for orthonormal systems on compact Riemannian manifolds in the non-sharp admissible region of exponents, covering wave, Klein-Gordon, and fractional Schr\"odinger equations. Our approach combines the…
We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time…