Martingale solution of stochastic hybrid Korteweg - de Vries - Burgers equation

TL;DR

This paper establishes the existence of a martingale solution for a stochastic hybrid Korteweg-de Vries-Burgers equation with multiplicative noise, using approximation and compactness methods.

Contribution

It introduces a novel approach to prove the existence of solutions for a complex stochastic PDE with hybrid features and multiplicative noise.

Findings

Existence of a martingale solution is proven.

The solution is obtained via approximation and convergence techniques.

The method can be applied to similar stochastic PDEs.

Abstract

In the paper, we consider a stochastic hybrid Korteweg - de Vries - Burgers type equation with multiplicative noise in the form of cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. The proof of the existence of the solution is based on two approximations of the considered problem and the compactness method. First, we introduce an auxiliary problem corresponding to the equation studied. Then, we prove the existence of a martingale solution to this problem. Finally, we show that the solution of the auxiliary problem converges, in some sense, to the solution of the equation under consideration.