A Stochastic Lagrangian particle system for the Navier-Stokes equations

TL;DR

This paper introduces a resetting procedure for a stochastic Lagrangian particle system modeling the Navier-Stokes equations, ensuring energy dissipation over time, aligning the particle system's behavior with that of true viscous fluids.

Contribution

It proposes a novel resetting method that guarantees energy dissipation in a stochastic particle system for Navier-Stokes, improving its physical realism.

Findings

The resetting procedure ensures energy dissipation when repeated sufficiently often.

The particle system's behavior aligns with the true Navier-Stokes equations after applying the resetting.

The method provides a new way to simulate viscous fluid dynamics with stochastic particles.

Abstract

This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \to \infty$. In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \to \infty$. The objective of this short note is to describe a resetting procedure that removes this deficiency. We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy.