Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

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 Lévy 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 Lévy processes with time dependent and discontinuous drifts is proved.

19 pages

References in corpus (3)

• A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations
• A stochastic-Lagrangian particle system for the Navier-Stokes equations
• A stochastic Lagrangian proof of global existence of the Navier-Stokes equations for flows with small Reynolds number

Cited by in corpus (5)

• Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
• Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations
• Forward self-similar solutions of the fractional Navier-Stokes Equations
• Local regularity of weak solutions of the hypodissipative Navier-Stokes equations
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation