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A Two-Sink Solution to the Self-Similar Euler Equations

Hyungjun Choi, Matei P. Coiculescu

Abstract

We construct the first known example of a self-similar solution to the two-dimensional incompressible Euler equations whose pseudo-velocity has more than one stagnation point. The solution is also a homogeneous steady state of the Euler equations. In contrast, any homogeneous steady state with bounded vorticity necessarily admits only a single stagnation point at the origin. Our construction develops cusps in the velocity along two lines passing through the origin, thereby allowing stagnation points other than the origin.

A Two-Sink Solution to the Self-Similar Euler Equations

Abstract

We construct the first known example of a self-similar solution to the two-dimensional incompressible Euler equations whose pseudo-velocity has more than one stagnation point. The solution is also a homogeneous steady state of the Euler equations. In contrast, any homogeneous steady state with bounded vorticity necessarily admits only a single stagnation point at the origin. Our construction develops cusps in the velocity along two lines passing through the origin, thereby allowing stagnation points other than the origin.
Paper Structure (11 sections, 5 theorems, 62 equations, 5 figures)

This paper contains 11 sections, 5 theorems, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Homogeneous Solutions of the Euler Equations and a Hamiltonian System
  3. Construction of the Two-Sink Solution
  4. Local solution
  5. Gluing Local Solutions
  6. Asymptotics as $P \to 0^-$ of the Periods $T_\pm$
  7. Relation to the Homogeneous Shear Flow
  8. Convergence to the Shear Flow as Pressure Goes to Zero
  9. The Mellin Transform and its Analytic Continuation
  10. Asymptotic Expansions for Integrals
  11. Numerical Data

Key Result

Theorem 1

For any value of the scaling parameter $\alpha\in (0,1)$, there exists a self-similar profile $\Omega$ that is $(-\alpha)$-homogeneous and whose corresponding pseudo-velocity field $U-\frac{\xi}{\alpha}$ has two sink stagnation points away from the origin and a saddle stagnation point at the origin.

Figures (5)

  • Figure 1: Vector Plot of the Pseudo-Velocity of the Two-Sink Solution when $\alpha = \frac{3}{4}$ on $[-1,1]^2$
  • Figure 2: Vector Plot of the Pseudo-Velocity of the Two-Sink Solution when $\alpha = \frac{3}{4}$ on $[0, 0.3] \times [-0.06, 0]$. The red dot is the location of a stagnation point.
  • Figure 3: Graph of $T-\pi$ near $P=0$ when $\lambda = 1.9$
  • Figure 4: Log-log plot of $P^*$ by $\alpha$: data for this graph are located at Appendix C.
  • Figure 5: Numerical investigation of $P^*$ for several values of $\lambda$ approaching $2$

Theorems & Definitions (14)

  • Theorem
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Proposition 3.7
  • ...and 4 more