Stochastic inviscid Leray-$\alpha$ model with transport noise: convergence rates and CLT

TL;DR

This paper proves that transport noise regularizes the stochastic inviscid Leray-$\

Contribution

It demonstrates convergence of solutions to the viscous Leray-$\alpha$ model under noise scaling and establishes a CLT with explicit rates, revealing noise-induced regularization effects.

Findings

Weak solutions converge to the viscous model in negative Sobolev spaces

Transport noise regularizes the inviscid Leray-$\alpha$ model

Explicit convergence rates and CLT are established

Abstract

We consider the stochastic inviscid Leray-$\alpha$ model on the torus driven by transport noise. Under a suitable scaling of the noise, we prove that the weak solutions converge, in some negative Sobolev spaces, to the unique solution of the deterministic viscous Leray-$\alpha$ model. This implies that transport noise regularizes the inviscid Leray-$\alpha$ model so that it enjoys approximate weak uniqueness. Interpreting such limit result as a law of large numbers, we study the underlying central limit theorem and provide an explicit convergence rate.