Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

TL;DR

This paper introduces a stochastic Lagrangian particle method to analyze fractal Navier-Stokes equations, providing new existence results for solutions using Lévy processes and gradient estimates.

Contribution

It offers a novel stochastic representation for fractal Navier-Stokes equations and proves local and global existence of solutions under certain conditions.

Findings

Existence of local unique solutions for fractal Navier-Stokes equations.

Global solutions in two dimensions or with large viscosity.

Gradient estimates for Lévy processes with time-dependent drifts.

Abstract

In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by L\'evy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for L\'evy processes with time dependent and discontinuous drifts is proved.