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The interaction between rough vortex patch and boundary layer

Jingchi Huang, Chao Wang, Jingchao Yue, Zhifei Zhang

TL;DR

The paper resolves the vanishing-viscosity limit for the 2D Navier–Stokes equations in the half-plane with no-slip boundary and rough vortex-patch initial data away from the boundary, proving $L^p$ convergence to the Euler solution for $2\le p<\infty$ on a short time $[0,T_0]$ with rate $\nu^{\frac{1}{4p}-C't}$. A key contribution is the development of a quantitative Kato-type criterion tailored to Yudovich data, together with uniform-in-$\nu$ velocity bounds and a carefully designed functional framework that accommodates strong boundary-layer effects and an initial layer. The analysis combines detailed vorticity estimates near the boundary via Maekawa-type kernels, a representation formula for boundary-layer corrections, and weighted energy methods for vorticity away from the boundary, ultimately yielding controlled interaction between rough patches and the boundary layer. The results advance the rigorous understanding of vortex–boundary-layer interactions at high Reynolds numbers and provide a robust basis for inviscid-limit analysis in domains with boundaries and irregular data.

Abstract

In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from the boundary. We establish the $L^p$ ($2\leq p< \infty$) convergence of solutions from the Navier-Stokes equations to those of the Euler equations. One of the main difficulties stems from the limited regularity of the initial data, which hinders the derivation of an asymptotic expansion. To overcome this challenge, we first prove a Kato-type criterion adapted to the Yudovich class setting. We then obtain uniform estimates for the Navier-Stokes equations -- a non-trivial task due to the strong boundary layer effects. A key component of our approach is the introduction of a suitable functional framework, which enables us to control the interaction between the rough vortex patch and the boundary layer.

The interaction between rough vortex patch and boundary layer

TL;DR

The paper resolves the vanishing-viscosity limit for the 2D Navier–Stokes equations in the half-plane with no-slip boundary and rough vortex-patch initial data away from the boundary, proving convergence to the Euler solution for on a short time with rate . A key contribution is the development of a quantitative Kato-type criterion tailored to Yudovich data, together with uniform-in- velocity bounds and a carefully designed functional framework that accommodates strong boundary-layer effects and an initial layer. The analysis combines detailed vorticity estimates near the boundary via Maekawa-type kernels, a representation formula for boundary-layer corrections, and weighted energy methods for vorticity away from the boundary, ultimately yielding controlled interaction between rough patches and the boundary layer. The results advance the rigorous understanding of vortex–boundary-layer interactions at high Reynolds numbers and provide a robust basis for inviscid-limit analysis in domains with boundaries and irregular data.

Abstract

In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from the boundary. We establish the () convergence of solutions from the Navier-Stokes equations to those of the Euler equations. One of the main difficulties stems from the limited regularity of the initial data, which hinders the derivation of an asymptotic expansion. To overcome this challenge, we first prove a Kato-type criterion adapted to the Yudovich class setting. We then obtain uniform estimates for the Navier-Stokes equations -- a non-trivial task due to the strong boundary layer effects. A key component of our approach is the introduction of a suitable functional framework, which enables us to control the interaction between the rough vortex patch and the boundary layer.

Paper Structure

This paper contains 22 sections, 31 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof
  4. Kato type criterion
  5. Uniform boundedness of the velocity
  6. The initial layer
  7. Functional spaces
  8. Energy estimates
  9. Proof of Proposition \ref{['prop: uniform for U omega NS']}
  10. Estimates of the vorticity near the boundary
  11. Representation formula
  12. Some basic estimates
  13. Estimates of $N$ and $B$
  14. Proof of Proposition \ref{['prop: Y(t)']}
  15. Proof of some basic estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume that the initial vorticity $\omega_0\in L_c^{\infty}(\mathbb R^2_+)$ and $\operatorname{supp}\omega_0\subseteq\{20\leq y\leq30\}$. Then there exist a time $T_0>0,$ and two positive constants $C, C'$, (independent of $\nu$) such that for $2\le p<\infty$, where $U^e$ is the solution of eq: Euler with the boundary condition $v^{e}=0$ on $y=0$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • ...and 41 more