Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains
Li Li, Xukai Yan, Zhibo Yang
TL;DR
The paper studies rigidity of $(-\alpha)$-homogeneous solutions to the 2D incompressible stationary Euler equations in sector-type domains. It develops a framework based on the stream function $\psi$, proving that under boundary $(-\alpha)$-homogeneity and a non-vanishing velocity component, $\psi$ satisfies an elliptic equation $\Delta\psi=g(\psi)$, which via a Liouville-type argument forces $\psi$ (and thus $\mathbf{u}$) to be $(-\alpha)$-homogeneous in the interior. The authors treat multiple domain configurations $\Omega_{1,2,\theta_0}$, $\Omega_{1,\infty,\theta_0}$, $\Omega_{0,1,\theta_0}$, and $\Omega_{0,\infty,\theta_0}$, obtaining explicit velocity-pressure structures of the form $\mathbf{u}=\frac{v(\theta)}{r^{\alpha}}\mathbf{e}_{\theta}+\frac{f(\theta)}{r^{\alpha}}\mathbf{e}_{r}$ with $v,f$ solving ODEs and $P=\frac{p}{r^{2\alpha}}+C$. This extends rigidity phenomena for Euler flows to sector-type geometries and complements existing results in strips, annuli, and channels by providing a unified hull of homogeneous solutions under natural boundary data.
Abstract
We study the rigidity problem for $(-α)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $Ω_{a, b, θ_0}:= \{(r,θ): a<r<b, \ 0<θ<θ_0\}$, where $α\in\mathbb{R}$, $0\leqslant a < b \leqslant +\infty$ and $0< θ_0 \leqslant 2π$. 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 $Ω_{a, b, θ_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{Ω_{a, b, θ_0}}\setminus\{\bm{0}\}$.