Inviscid Limit for Stochastic Second-Grade Fluid Equations

TL;DR

This paper proves that solutions of stochastic second-grade fluid equations with transport noise converge uniformly in time to Euler solutions in the inviscid limit, under certain regularity and parameter conditions, without boundary layer dissipation.

Contribution

It establishes the inviscid limit for stochastic second-grade fluids with boundary conditions, extending previous results to stochastic and boundary layer contexts.

Findings

Uniform convergence in $L^2$ norm over time

Inviscid limit holds without boundary layer energy dissipation

Convergence depends on initial regularity and parameter behavior

Abstract

We consider in a smooth and bounded two dimensional domain the convergence in the $L^2$ norm, uniformly in time, of the solution of the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, that assuming proper regularity of the initial conditions of the Euler equations and a proper behavior of the parameters $\nu$ and $\alpha$, then the inviscid limit holds without requiring a particular dissipation of the energy of the solutions of the second-grade fluid equations in the boundary layer.