Initial and Boundary value problems for the Deterministic and Stochastic Zakharov-Kuznetsov Equation in a Bounded Domain

Abstract

We study in this thesis the well-posedness and regularity of the Zakharov-Kuznetsov (ZK) equation in the deterministic and stochastic cases, subjected to a rectangular domain in space dimensions 2 and 3. Mainly we have established the existence, in 3D, and uniqueness, in 2D, of the weak solutions, and the local and global existence of strong solutions in 3D. Then we extend the results to the stochastic case and obtain in 3D the existence of martingale solutions, and in 2D the pathwise uniqueness and existence of pathwise solutions. The main focus is on the mixed features of the partial hyperbolicity, nonlinearity, nonconventional boundary conditions, anisotropicity and stochasticity, which requires methods quite different than those of the classical models of fluid dynamics, such as the Navier-Stokes equation, Primitive Equation and related equations.