Anisotropic Inviscid Limit for the Navier-Stokes Equations with Transport Noise Between Two Plates

TL;DR

This paper studies the anisotropic vanishing viscosity limit of 3D stochastic Navier-Stokes equations between two plates, showing convergence to the Euler equations under specific viscosity ratios and addressing challenges posed by anisotropic noise scaling.

Contribution

It introduces a novel analysis of anisotropic viscosity and noise scaling in stochastic Navier-Stokes equations, establishing convergence to Euler solutions when vertical viscosity diminishes faster.

Findings

Weak martingale solutions converge to Euler solutions under viscosity ratio conditions.

Anisotropic noise scaling complicates divergence-free properties, addressed in the analysis.

Results apply to Navier-Stokes equations with boundary conditions between plates.

Abstract

We investigate an anisotropic vanishing viscosity limit of the 3D stochastic Navier-Stokes equations posed between two horizontal plates, with Dirichlet no-slip boundary condition. The turbulent viscosity is split into horizontal and vertical directions, each of which approaches zero at a different rate. The underlying Cylindrical Brownian Motion driving our transport-stretching noise is decomposed into horizontal and vertical components, which are scaled by the square root of the respective directional viscosities. We prove that if the ratio of the vertical to horizontal viscosities approaches zero, then there exists a sequence of weak martingale solutions convergent to the strong solution of the deterministic Euler equation on its lifetime of existence. A particular challenge is that the anisotropic scaling ruins the divergence-free property for the spatial correlation functions of the noise.