Related papers: Strichartz inequalities with white noise potential…
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and…
The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*} \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) +…
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…
In this paper we establish the strong existence, pathwise uniqueness and a comparison theorem to a stochastic partial differential equation driven by Gaussian colored noise with non-Lipschitz drift, H\"older continuous diffusion…
We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven…
We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity,…
We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schr\"odinger equation $$i\partial_tu+\Delta u +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and…
We prove spacetime weighted-L^2 estimates for the Schrodinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.
In this article, we study a $d$-dimensional stochastic quadratic nonlinear Schr\"{o}dinger equation (SNLS), driven by a fractional derivative (of order $-\alpha<0$) of a space-time white noise: $$\left\{ \begin{array}{l}i\partial_t u-\Delta…
We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao, which obtains the existence of maximizers in the context of the Schr\"odinger…
We prove Strichartz estimates without loss for the Schr\"odinger equation and the wave equation outside finitely many strictly convex obstacles verifying Ikawa's condition, extending the approach we introduced previously for the two convex…
This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schr\"{o}dinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing…
Some analogues of the Schr\"odinger refined Strichartz inequalities (Du, Guth, Li and Zhang) are obtained for the wave equation. These are used to improve the best known $L^2$ fractal Strichartz inequalities for the wave equation in…
We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schr\"odinger equation $iu_t + {1/2}…
We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which…
In this undergraduate thesis, we present an analytical proof of the Morse inequalities for closed smooth $n$-manifolds following Witten's approach. Using techniques from PDE theory, the proof is reduced to study the eigenspaces and…
In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with…
This work deals with Schr\"odinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such…
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…