Martingale and Pathwise Solutions to the Stochastic Zakharov-Kuznetsov Equation with Multiplicative Noise

TL;DR

This paper proves the existence of martingale solutions and, in some cases, pathwise solutions for the stochastic Zakharov-Kuznetsov equation with multiplicative noise, using novel methods to handle boundary conditions and regularity issues.

Contribution

It introduces new techniques for establishing solutions to the stochastic Zakharov-Kuznetsov equation, especially addressing boundary conditions and solution uniqueness under low regularity.

Findings

Global existence of martingale solutions in 2D and 3D

Pathwise uniqueness and solutions in 2D

Novel methods for boundary and regularity challenges

Abstract

We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise. We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions. New methods are employed in the passage to the limit on a special type of boundary conditions and in the verification of the pathwise uniqueness of martingale solutions with a lack of regularity, where both difficulties arise due to the partly hyperbolic feature of the model.