A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations

TL;DR

This paper introduces a stochastic variational framework for deriving viscous Camassa-Holm and Leray-alpha equations, analyzing their solutions as diffusion processes on the diffeomorphism group of the torus.

Contribution

It presents a novel stochastic variational derivation of viscous fluid equations and investigates the existence of solutions within a probabilistic setting.

Findings

Existence of solutions in H1 space established.

Diffusion processes on the diffeomorphism group characterized.

Stochastic approach provides new insights into viscous fluid equations.

Abstract

We derive the (d-dimensional) periodic incompressible and viscous Camassa-Holm equation as well as the Leray-alpha equations via a stochastic variational principle. We discuss the existence of solution for this equation in the space H1 using the probabilistic characterisation. The underlying Lagrangian flows are diffusion processes living in the group of diffeomorphisms of the torus. We study in detail these diffusions.