On the dynamics of stochastic nonlinear dispersive partial differential equations

Abstract

This thesis contributes towards the well-posedness theory of stochastic dispersive partial differential equations. Our investigation focuses on initial value problems as sociated with the stochastic nonlinear Schro¨dinger (SNLS) and stochastic Korteweg de Vries (SKdV) equations. We divide this thesis into four main topics, which are the contents of Chapters 2–5. Chapter 2 is concerned with the SNLS posed on the d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, we prove local-in time well-posedness for initial data and noise at subcritical regularities. We are also able to extend this to global-in-time well-posedness at energy subcritical regularity for certain cases. In the next two chapters, we focus on SNLS posed on the d dimensional Euclidean space with additive noise. In Chapter 3, we prove local well posedness with the noise at supercritical regularity while the initial data stays at critical regularity. In Chapter 4, we restrict our attention to dimension 4 and study SNLS with non-vanishing boundary conditions. In particular, we use perturbative techniques to prove global well-posedness with data in H1(R4) + 1. In Chapter 5, we move on from SNLS to SKdV, where we prove L2(T)-global well-posedness of SKdV with multiplicative noise on the circle. We also verify that a result on the stabilisation of noise by Tsutsumi [84] continues to hold in our low regularity setting.