Caltech/UCLA Joint Analysis Seminar

Zakharov-Kuznetsov (ZK) equation is a long-wave small-amplitude limit of the Euler-Poisson system of the cold plasma uniformly magnetized along one space direction. It is also a multi-dimensional extension of the Korteweg-de Vries (KdV) equation. The talk will focus on the well-posedness and regularity of both the deterministic and stochastic ZK equation, subjected to a rectangular domain in space dimensions two and three.

Particularly, in the deterministic case, the global existence of strong solutions is established in 3D. For the stochastic ZK equation driven by a white noise, in 3D the existence of martingale solutions, and in 2D the uniqueness and existence of the pathwise solution are established. This is an analogy to the results of the existence and uniqueness of the weak solutions (in the PDE sense) in the deterministic case.

In terms of methodology, the talk focuses on the handling of the mixed features consisting of the partial parabolicity, hyperbolicity, nonlinearity, anisotropicity and stochasticity of the system, which provides interesting and challenging mathematical complications.