Can probability theory really help tame problems in mathematical hydrodynamics?

TL;DR

This paper explores whether probability theory can address complex issues in mathematical hydrodynamics, especially in relation to non-uniqueness and the impact of stochastic models on fluid equations.

Contribution

It analyzes the effectiveness of stochastic approaches and convex integration techniques in resolving or understanding problems in hydrodynamics.

Findings

Convex integration techniques can overcome stochastic perturbations.

Stochastic models do not necessarily eliminate non-uniqueness in solutions.

Probabilistic methods may have limited impact on certain pathological behaviors.

Abstract

Recent years have seen spectacular progress in the mathematical study of hydrodynamic equations. Novel tools from convex integration in particular prove extremely versatile in establishing non-uniqueness results. Motivated by this 'pathological' behavior of solutions in the deterministic setting, stochastic models of fluid dynamics have enjoyed growing interest from the mathematical community. Inspired by the theory of 'regularization by noise', it is hoped for that stochasticity might help avoid 'pathologies' such as non-uniqueness of weak solutions. Current research however shows that convex integration methods can prevail even in spite of random perturbations.