Global well-posedness and regularity of 3D Burgers equation with multiplicative noise

TL;DR

This paper introduces a novel low regularity theory for the 3D Burgers equation with multiplicative noise, establishing a random maximum principle that aids in analyzing moment estimates and long-term behavior of stochastic hydrodynamic systems.

Contribution

It develops a new approach for stochastic PDEs, including a random maximum principle, applicable to 3D Burgers and other stochastic hydrodynamic equations.

Findings

Established a random maximum principle for stochastic 3D Burgers equations.

Derived moment estimates crucial for understanding long-term behavior.

Demonstrated the method's applicability to other stochastic PDEs.

Abstract

In this paper, we develop low regularity theory for 3D Burgers equation perturbed by a linear multiplicative stochastic force. This method is new and essentially different from the deterministic partial differential equations(PDEs). Our results and method can be widely applied to other stochastic hydrodynamic equations and the deterministic PDEs. As a further study, we establish a random version of maximum principle for random 3D Burgers equations, which will be an important tool for the study of 3D stochastic Burgers equations. As we know establishing moment estimates for highly nonlinear stochastic hydrodynamic equations is difficult. But moment estimates are very important for us to study the probabilistic properties and long-time behavior for the stochastic systems. Here, the random maximum principle helps us to achieve some important moment estimates for 3D stochastic Burgers equations and lays a solid foundation for the further study of 3D stochastic Burgers equations.