Zakharov-Kuznetsov (ZK) equation is the long-wave small-amplitude limit of the Euler-Poisson system for cold plasma uniformly magnetized along one space direction. It is also a multi-dimensional extension of the Korteweg-de Vries (KdV) equation and a special case of the partially hyperbolic equations. 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, we obtain the global existence of strong solutions in 3D, which, for similar equations in fluid dynamics, is still open. 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, an analogy to the results of the weak solutions (in the PDE sense) in the deterministic case.
In terms of methodology, the focus is on the handling of the mixed features consisting of the partial hyperbolicity, nonlinearity, anisotropicity and stochasticity of the system, which, sitting at the interface among probability and analysis of the parabolic and hyperbolic PDEs, provides interesting and challenging mathematical complications.