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Inviscid limit on $L^p$-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions

Mustafa Sencer Aydın

TL;DR

This work addresses the inviscid limit for the 3D Navier–Stokes equations in the half-space under Navier boundary conditions, unifying $L^p$-based Sobolev conormal spaces with conormal regularity. The authors develop a comprehensive a priori framework combining conormal derivative estimates, normal derivative control via the auxiliary field $\eta$, pressure bounds, and maximal parabolic regularity to obtain uniform-in-$\nu$ estimates and a convergence rate to Euler for $p \in [2,\infty]$. They prove existence, uniqueness, and uniform bounds for Navier–Stokes solutions in these anisotropic spaces, establish the inviscid limit with an explicit rate, and show Euler well-posedness in the same conormal setting, with the friction coefficient $\mu$ influencing boundary behavior. The results extend prior vanishing-viscosity analyses by relaxing regularity requirements, introduce a robust maximal-regularity approach for the normal derivatives, and offer a versatile Euler limit theory applicable to Lipschitz data and beyond, contributing a new rigorous framework for boundary-layer-like phenomena in anisotropic function spaces.

Abstract

We establish uniform bounds and the inviscid limit in $L^p$-based Sobolev conormal spaces for the solutions of the Navier-Stokes equations with the Navier boundary conditions in the half-space. We extend the vanishing viscosity results of~\cite{BdVC1} and~\cite{AK1} by weakening the normal and the conormal regularity assumptions, respectively. We require the initial data to be Lipschitz with three integrable conormal derivatives. We also assume that the initial normal derivative has one or two integrable conormal derivative depending on the sign of the friction coefficient. Finally, we establish the existence and uniqueness of the Euler equations with a bounded normal derivate, two bounded conormal derivatives, and three integrable conormal derivatives.

Inviscid limit on $L^p$-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions

TL;DR

This work addresses the inviscid limit for the 3D Navier–Stokes equations in the half-space under Navier boundary conditions, unifying -based Sobolev conormal spaces with conormal regularity. The authors develop a comprehensive a priori framework combining conormal derivative estimates, normal derivative control via the auxiliary field , pressure bounds, and maximal parabolic regularity to obtain uniform-in- estimates and a convergence rate to Euler for . They prove existence, uniqueness, and uniform bounds for Navier–Stokes solutions in these anisotropic spaces, establish the inviscid limit with an explicit rate, and show Euler well-posedness in the same conormal setting, with the friction coefficient influencing boundary behavior. The results extend prior vanishing-viscosity analyses by relaxing regularity requirements, introduce a robust maximal-regularity approach for the normal derivatives, and offer a versatile Euler limit theory applicable to Lipschitz data and beyond, contributing a new rigorous framework for boundary-layer-like phenomena in anisotropic function spaces.

Abstract

We establish uniform bounds and the inviscid limit in -based Sobolev conormal spaces for the solutions of the Navier-Stokes equations with the Navier boundary conditions in the half-space. We extend the vanishing viscosity results of~\cite{BdVC1} and~\cite{AK1} by weakening the normal and the conormal regularity assumptions, respectively. We require the initial data to be Lipschitz with three integrable conormal derivatives. We also assume that the initial normal derivative has one or two integrable conormal derivative depending on the sign of the friction coefficient. Finally, we establish the existence and uniqueness of the Euler equations with a bounded normal derivate, two bounded conormal derivatives, and three integrable conormal derivatives.

Paper Structure

This paper contains 10 sections, 17 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the Main Results
  3. Conormal Derivative Estimates
  4. Normal Derivative Estimates
  5. Pressure Estimates
  6. $L^\infty$ estimates
  7. Maximal parabolic regularity estimates
  8. Trace estimates
  9. Concluding the a priori estimates
  10. Proofs of Theorems \ref{['T01']}, \ref{['T03']}, and \ref{['T04']}

Key Result

Theorem 2.1

Assume that $\nu \in (0,\bar{\nu}]$ and $\mu \in \mathbb{R}$. Let $u_0 \in L^2(\Omega)\cap W^{3,3+\delta}_\text{co}(\Omega) \cap W^{3,6+2\delta}_\text{co}(\Omega)$, with $\nabla u_0 \in W^{2,3+\delta}_\text{co}(\Omega) \times L^\infty(\Omega)$ be such that $\mathop{\rm div}\nolimits u_0 = 0$ on $\Om

Theorems & Definitions (24)

  • Theorem 2.1: existence, uniqueness, and inviscid limit
  • Theorem 2.2: A sharper result for non-negative friction
  • Theorem 2.3: Well-posedness of the Euler equations in Sobolev conormal spaces
  • Lemma 2.1
  • Lemma 2.2: An interpolation inequality for Sobolev conormal spaces
  • Proposition 2.3: A priori estimates
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['P.Con']}
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['P.Nor']}
  • ...and 14 more