Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains

TL;DR

This paper investigates the rigidity of homogeneous solutions to the 2D stationary Euler equations in sector domains, showing under certain conditions that solutions must be globally homogeneous.

Contribution

It establishes new rigidity results for homogeneous solutions in sector-type domains, extending understanding of solution structure in fluid dynamics.

Findings

Homogeneous solutions are necessarily globally homogeneous under specified boundary conditions.

Results depend on domain type and boundary homogeneity assumptions.

Radial or angular velocity components non-vanishing implies global homogeneity.

Abstract

We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where $\alpha\in\mathbb{R}$, $0\leqslant a < b \leqslant +\infty$ and $0< \theta_0 \leqslant 2\pi$. For each type of domains, depending on whether $a = 0$ or $a > 0$, and $b = +\infty$ or $b < +\infty$, we show that if a solution satisfies some homogeneity assumptions on the boundary of $\Omega_{a, b, \theta_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$.