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. 

　　 

　　Publication: 

　　SIAM Journal on Mathematical Analysis, Volume 55, Issue 3 (2023) 

　　https://doi.org/10.1137/21M1413377 

　　  

　　Author: 

　　Zhao Dong 

　　RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. 

　　Email: dzhao@amt.ac.cn 

　　 

　　Boling Guo 

　　Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China. 

　　 

　　Jiang-Lun Wu 

　　Department of Mathematics, Computational Foundry, Swansea University, Swansea SA1 8EN, UK. 

　　 

　　Guoli Zhou  

　　School of Statistics and Mathematics, Chongqing University, Chongqing 400044, China.