Nonlinear stochastic transport partial differential equations: well-posedness and applications to data assimilation

Abstract

In this thesis I study analytical properties and applications to data assimilation for nonlinear stochastic transport partial differential equations which originate in fluid dynamics. The thesis has two parts. In the first part (Chapters 3-5) I focus on theoretical results for three two-dimensional stochastic transport models. I prove analytical properties (existence, uniqueness, continuity with respect to initial conditions) for the corresponding classes of stochastic partial differential equations (SPDEs) driven by transport noise: the stochastic Euler equation (SE), the stochastic great lake equation (SGLE), and the stochastic rotating shallow water (SRSW) model. The well-posedness strategy is based on constructing an approximating sequence of solutions which is proven to be relatively compact and to converge to the solution of the original equation in a suitable sense. The solution is global for the 2D SE equation and local for the SGLE and for the SRSW model. In the second part (Chapter 6) I prove the applicability of one of these models (the SRSW model) in a data assimilation setting. I implement a data assimilation methodology based on a bootstrap particle filter combined with two additional procedures (tempering and jittering). The methodology is tested first on the Lorenz ’63 model and then applied to the SRSW model.