Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes

TL;DR

This paper develops a probabilistic framework for constructing weak solutions to the Navier-Stokes equations by linking them to diffusion processes, offering a new perspective on solving these complex fluid dynamics problems.

Contribution

It introduces a novel method that reduces the Navier-Stokes Cauchy problem to a stochastic problem using diffusion processes for probabilistic representation.

Findings

Constructed diffusion processes for Navier-Stokes solutions

Provided a probabilistic representation of weak solutions

Established a new approach linking stochastic processes to fluid dynamics

Abstract

We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for the Navier- Stokes system.