Global Well-Posedness and Regularity of Stochastic 3D Burgers Equation with Multiplicative Noise

Abstract

By utilizing the so-called Doss–Sussman transformation, we link our stochastic 3D Burgers equation with linear multiplicative noise to a random 3D Burger equation. With the help of techniques from partial differential equations (PDEs) and probability, we establish the global well-posedness of stochastic 3D Burgers with the diffusion coefficient being constant. Next, by developing a solution which is orthogonal with the gradient of coefficient of the noise, we extend the global well-posedness to a more general case in which the diffusion coefficient is spatial dependent, i.e., it is a function of the spatial variable. Our results and methodology pave a way to extend some regularity results of stochastic 1D Burgers equation to stochastic 3D Burgers equations.