# Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations

## TL;DR

This paper establishes the existence of stochastic Lagrangian trajectories for Leray solutions of the 3D Navier-Stokes equations, linking probabilistic methods with classical fluid dynamics solutions.

## Contribution

It introduces a novel stochastic framework for Leray solutions, proving existence and convergence of associated stochastic particle paths.

## Key findings

- Existence of weak solutions to the SDE with Leray velocity fields.
- Density of solutions belongs to specific Sobolev spaces under certain conditions.
- Weak convergence of mollified solutions to the original stochastic trajectories.

## Abstract

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $\rho_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2\nu}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.