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

Abstract

We study the rigidity problem for $(-\hat{I}\pm )$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\hat{I}{\text{c◯}}_{a,b,\hat{I}{¸}_0}:=\{(r,\hat{I}¸):a<r<b,0<\hat{I}¸<\hat{I}{¸}_0\}$, where $\hat{I}\pm \in \mathbb{R}$, $0⩽a<b⩽+\infty$ and $0<\hat{I}{¸}_0⩽2\ddot{I}€$. 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 $\hat{I}{\text{c◯}}_{a,b,\hat{I}{¸}_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{\hat{I}{\text{c◯}}_{a,b,\hat{I}{¸}_0}}\setminus \{0\}$, then it must be homogeneous throughout $\overline{\hat{I}{\text{c◯}}_{a,b,\hat{I}{¸}_0}}\setminus \{0\}$.

39 pages, 1 figures