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Weak-Strong Uniqueness and the d'Alembert Paradox

Hao Quan, Gregory L. Eyink

TL;DR

The paper addresses weak-strong uniqueness for inviscid limits of exterior flows and its relation to the d'Alembert paradox. It combines a velocity decomposition ${\mathbf{u}}={\mathbf{u}}_{\phi}+{\mathbf{u}}_{\omega}$ with the Josephson–Anderson relation to obtain a conditional weak-strong uniqueness result in exterior domains under the vanishing integrated skin-friction condition ${\langle {\mathbf{u}}_{\phi}\cdot {\boldsymbol{\tau}}_w,1\rangle=0}$, showing the rotational part disappears and the flow reduces to the potential solution when these criteria hold. The paper also proves that the Drivas–Nguyen uniform wall-continuity condition implies weak-strong uniqueness for admissible Euler solutions in bounded domains, thereby connecting boundary regularity to energy dissipation and uniqueness. Collectively, the results highlight boundary-layer behavior and dissipation as crucial in determining the relevance of generalized Euler solutions and illuminate why the potential (drag-free) flow may fail to attract inviscid limits in real, bounded-wall scenarios.

Abstract

We prove conditional weak-strong uniqueness of the potential Euler solution for external flow around a smooth body in three space dimensions, within the class of viscosity weak solutions with the same initial data. Our sufficient condition is the vanishing of the streamwise component of the skin friction in the inviscid limit, somewhat weaker than the condition of Bardos-Titi in bounded domains. Because global-in-time existence of the smooth potential solution leads back to the d'Alembert paradox, we argue that weak-strong uniqueness is not a valid criterion for "relevant" notions of generalized Euler solution and that our condition is likely to be violated in the inviscid limit. We prove also that the Drivas-Nguyen condition on uniform continuity at the wall of the normal velocity component implies weak-strong uniqueness within the general class of admissible weak Euler solutions in bounded domains.

Weak-Strong Uniqueness and the d'Alembert Paradox

TL;DR

The paper addresses weak-strong uniqueness for inviscid limits of exterior flows and its relation to the d'Alembert paradox. It combines a velocity decomposition with the Josephson–Anderson relation to obtain a conditional weak-strong uniqueness result in exterior domains under the vanishing integrated skin-friction condition , showing the rotational part disappears and the flow reduces to the potential solution when these criteria hold. The paper also proves that the Drivas–Nguyen uniform wall-continuity condition implies weak-strong uniqueness for admissible Euler solutions in bounded domains, thereby connecting boundary regularity to energy dissipation and uniqueness. Collectively, the results highlight boundary-layer behavior and dissipation as crucial in determining the relevance of generalized Euler solutions and illuminate why the potential (drag-free) flow may fail to attract inviscid limits in real, bounded-wall scenarios.

Abstract

We prove conditional weak-strong uniqueness of the potential Euler solution for external flow around a smooth body in three space dimensions, within the class of viscosity weak solutions with the same initial data. Our sufficient condition is the vanishing of the streamwise component of the skin friction in the inviscid limit, somewhat weaker than the condition of Bardos-Titi in bounded domains. Because global-in-time existence of the smooth potential solution leads back to the d'Alembert paradox, we argue that weak-strong uniqueness is not a valid criterion for "relevant" notions of generalized Euler solution and that our condition is likely to be violated in the inviscid limit. We prove also that the Drivas-Nguyen condition on uniform continuity at the wall of the normal velocity component implies weak-strong uniqueness within the general class of admissible weak Euler solutions in bounded domains.

Paper Structure

This paper contains 7 sections, 7 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the Main Results
  3. Prior work
  4. Main theorem statement
  5. Proof of Theorem \ref{['thm:rot-energy']}
  6. Proof of Theorem \ref{['thm:external']}
  7. Proof of Theorem \ref{['thm:internal']}

Key Result

Proposition 1

Consider a solid body $B\subset\mathbb{R}^3,$ represented by a simply connected $C^{\infty}$ manifold with vanishing genus/first Betti number and a compact boundary. Let ${ \mathbf{u}}_\phi$ be the unique potential flow solution of the incompressible Euler equations eq:euler in $\Omega=\mathbb{R}^3\ Furthermore, the power dissipated by drag ${\mathcal{W}}_\phi(t):=\mathbf{F}_\phi(t){\hbox{\boldmat

Figures (1)

  • Figure 1: Flow around a finite body $B$ in an unbounded region $\Omega$ filled with an incompressible fluid moving at a velocity ${ \mathbf{V}}(t)$ at far distances.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • Definition 1
  • ...and 6 more