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

Daniel Goodair

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.

Anisotropic Inviscid Limit for the Navier-Stokes Equations with Transport Noise Between Two 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.
Paper Structure (12 sections, 8 theorems, 159 equations)

This paper contains 12 sections, 8 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Deterministic Theory
  3. Structure of the Noise
  4. Aspects of the Proof
  5. Preliminaries
  6. Functional Analytic Preliminaries
  7. Stochastic Preliminaries
  8. Transport-Stretching Noise
  9. Anisotropic Inviscid Limit
  10. Definitions and Main Result
  11. Selection of Martingale Weak Solutions
  12. Proof of the Main Result

Key Result

Lemma 2.1

Let $0 < \theta$ be an arbitrary parameter and fix $0 < \nu_z$. There exists a function $\mathscr{B} \in C\left([0,T]; H^{\gamma} \cap L^2_{\sigma} \right)$ such that $w_t + B_t \in W^{1,2}_{\sigma}$ for every $t \in [0,T]$, of the form where $\mathscr{M}(z) \in \mathbb{R}^{3 \times 3}$ and $\mathscr{M}$ is only supported on $\left[0, \frac{1}{4}\right] \cup \left[\frac{3}{4}, 1\right]$. Moreover

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 4 more