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.