On the effect of stochastic Lie transport noise on fluid dynamic equations

Motivated by a recent geometric approach for adding stochastic noise of “Lie transport” type to PDEs developed by Darryl Holm, we investigate the effect of stochastic “Lie transport” noise in deterministic fluid equations. More concretely, first we develop a tool which facilitates the rigorous treatment of these equations and we will need in our research. This tool comprises an ex- tension of the Itˆo-Wentzell formula to allow for advection of k-forms as well as tensors. Afterwards, we proceed to ask whether addition of this type of noise can improve the solution properties of some well-known deterministic fluid equations of interest. In particular, (A) we study the solution properties of the inviscid Burgers’ equation with transport noise, establishing the local well-posedness of strong solutions, and the global well-posedness in the vis- cous case. We also present some shock formation results. Inspired by recent advances in the problem of well-posedness of PDEs by addition of stochas- tic noise, (B) we demonstrate a general well-posedness by noise result for spatially weak, strong in the probabilistic sense solutions to the equation of linear advection of k-forms with Lie transport noise, which is ill-posed in the deterministic case. Finally, (C) we consider a novel Lagrangian-averaged approach for including stochastic noise in fluid equations and prove that it permits establishing the global well-posedness of the Boussinesq equations, where global well-posedness in the deterministic case is a fundamental open problem in mathematics.Open Acces