Navier-Stokes Equations with Navier Boundary Conditions and Stochastic Lie Transport: Well-Posedness and Inviscid Limit

TL;DR

This paper establishes the well-posedness of 2D stochastic Navier-Stokes equations with Navier boundary conditions and demonstrates the inviscid limit leading to solutions of stochastic Euler equations, highlighting the role of SALT noise.

Contribution

It proves existence and uniqueness of solutions under Navier boundary conditions for a broad class of stochastic forces, including SALT noise, and analyzes the inviscid limit.

Findings

Existence and uniqueness of solutions for 2D stochastic Navier-Stokes with Navier boundary conditions.

Inviscid limit results leading to stochastic Euler equations under specific conditions.

Energy estimates enabled by Navier boundary conditions with transport-type noise.

Abstract

We prove the existence and uniqueness of global, probabilistically strong, analytically strong solutions of the 2D Stochastic Navier-Stokes Equation under Navier boundary conditions. The choice of noise includes a large class of additive, multiplicative and transport models. We emphasise that with a transport type noise, the Navier boundary conditions enable direct energy estimates which appear to be prohibited for the usual no-slip condition. The importance of the Stochastic Advection by Lie Transport (SALT) structure, in comparison to a purely transport Stratonovich noise, is also highlighted in these estimates. In the particular cases of SALT noise, the free boundary condition and a domain of non-negative curvature, the inviscid limit exists and is a global, probabilistically weak, analytically weak solution of the corresponding Stochastic Euler Equation.