— Thanks to an approach inspired from Burq-Lebeau [6], we prove stochastic versions of Strichartz estimates for Schrödinger with harmonic potential. As a consequence, we show that the nonlinear Schrödinger equation with quadratic potential and any polynomial nonlinearity is almost surely locally well-posed in L(R) for any d ≥ 2. Then, we show that we can combine this result with the high-low fr...

We investigate the well-posedness for the 2D viscous shallow water equations in the theory of compressible fluid. Making use of the Fourier frequency localization and Bony paraproduct decomposition, closing to a stable equilibrium h0, we obtain the local well-posedness for general initial data u0 and the global well-posedness for small initial data u0 in certain scaling invariant Besov spaces, ...

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev spaceHs for an integer s ≥ 2. Our analysis relies on local well-posedness results of Schäfer & Wayne [12], the correspondence of the short-pulse equation to the sine– Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and...

This thesis is devoted to the study of stochastic nonlinear Schrödinger equations (abbreviated as SNLS) with linear multiplicative noise in two aspects: the wellposedness in L(R), H(R) and the noise effects on blowup phenomena in the non-conservative case. 1. The well-posedness in L(R). The first fundamental question when dealing with SNLS is the well-posedness problem. In the first chapter, we...