Deterministic and Stochastic Nonlinear Partial Differential Equations and Applications

Deterministic and stochastic nonlinear partial differential equations with mixed features (e.g. partial hyperbolicity and anisotropicity) have played an important role in describing the models in physics, chemistry, finance and other real-world phenomena. Moreover, sitting at the interface between probability theory, mathematical analysis and theory of parabolic and hyperbolic partial differential equations, these problems provide interesting and challenging mathematical complications. Motivated by this background, my research interests lie in the study of such models, especially those derived from mathematical physics like plasma physics, fluid dynamics and thermomechanics. The investigation emphasis has been on regularity behavior of solutions and the numerical implementations. Specifically, my projects consist of three interrelated directions. One theme is to establish the basic mathematical theory for the deterministic and stochastic Zakharov-Kuznetsov (ZK) equation, a multi dimensional extension of the celebrated Korteweg-de Vries (KdV) equation. A second topic is to extend the time discrete numerical scheme and Courant-Friedrichs-Levy condition for the evolution equations of geophysical fluid dynamics from the deterministic to the stochastic case. Finally, a third subject is to investigate how to utilise noise perturbations to model the self-organized criticality for the atmospheric equations.