Symmetry Breaking and Uniqueness for the Incompressible Navier-Stokes Equations

TL;DR

This paper explores the relationship between the structure of deterministic Navier-Stokes equations and stochastic cascades, offering insights into symmetry breaking and potential approaches to the longstanding uniqueness problem.

Contribution

It establishes connections between Navier-Stokes equations and stochastic cascades, introduces new branching Markov chains, and suggests novel methods for investigating symmetry breaking and uniqueness.

Findings

Explosion criteria for stochastic cascades coincide

New branching Markov chains are introduced

Connections suggest new approaches to symmetry breaking

Abstract

The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group $(0,\infty)$, naturally arise as a result of this investigation.