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Free boundary problems for the two-dimensional Euler equations in exterior domains

Daomin Cao, Boquan Fan, Weicheng Zhan

Abstract

In this paper we present some classification results for the steady Euler equations in two-dimensional exterior domains with free boundaries. We prove that, in an exterior domain, if a steady Euler flow devoid of interior stagnation points adheres to slip boundary conditions and maintains a constant norm on the boundary, along with certain additional conditions at infinity, then the domain is the complement of a disk, and the flow is circular, namely the streamlines are concentric circles. Additionally, we establish that in the entire plane, if all the stagnation points of a steady Euler flow coincidentally form a disk, then, under certain additional reasonable conditions near the stagnation points and at infinity, the flow must be circular. The proof is based on a refinement of the method of moving planes.

Free boundary problems for the two-dimensional Euler equations in exterior domains

Abstract

In this paper we present some classification results for the steady Euler equations in two-dimensional exterior domains with free boundaries. We prove that, in an exterior domain, if a steady Euler flow devoid of interior stagnation points adheres to slip boundary conditions and maintains a constant norm on the boundary, along with certain additional conditions at infinity, then the domain is the complement of a disk, and the flow is circular, namely the streamlines are concentric circles. Additionally, we establish that in the entire plane, if all the stagnation points of a steady Euler flow coincidentally form a disk, then, under certain additional reasonable conditions near the stagnation points and at infinity, the flow must be circular. The proof is based on a refinement of the method of moving planes.

Paper Structure

This paper contains 6 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction and Main results
  2. Proofs for the main results
  3. Some common lemmas
  4. Proof of Theorem \ref{['th3']}
  5. Proof of Theorem \ref{['th2']}
  6. Proof of Theorem \ref{['th1']}

Key Result

Theorem \oldthetheorem

Let $\mathbf{v}$ be a $C^2({\mathbb{R}^2\backslash\Omega})$ flow solving 1-1. Assume also that: Then $\Omega=B_R$ for some $R>0$. Furthermore, $\mathbf{v}$ is a circular flow, that is, there is a $C^2(R,+\infty)$ function $V: (R, +\infty)\to \mathbb{R}$ with constant strict sign such that for all $|x|>R$.

Theorems & Definitions (16)

  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • Lemma \oldthetheorem: HN
  • Lemma \oldthetheorem: HN
  • Lemma \oldthetheorem
  • proof
  • ...and 6 more