Existence of homogeneous Euler flows of degree $-α\notin [-2,0]$

Abstract

We consider ($-α$)-homogeneous solutions to the stationary incompressible Euler equations in $\mathbb{R}^{3}\backslash\{0\}$ for $α\geq 0$ and in $\mathbb{R}^{3}$ for $α<0$. Shvydkoy (2018) demonstrated the nonexistence of ($-1$)-homogeneous solutions and ($-α$)-homogeneous solutions in the range $0\leq α\leq 2$ for the Beltrami and axisymmetric flows. The nonexistence result of the Beltrami ($-α$)-homogeneous solutions holds for all $α<1$. We show the nonexistence of axisymmetric ($-α$)-homogeneous solutions without swirls for $-2\leq α<0$. The main result of this study is the existence of axisymmetric ($-α$)-homogeneous solutions in the complementary range $α\in \mathbb{R}\backslash [0,2]$. More specifically, we show the existence of axisymmetric Beltrami ($-α$)-homogeneous solutions for $α\in \mathbb{R}\backslash [0,2]$ and axisymmetric ($-α$)-homogeneous solutions with a nonconstant Bernoulli function for $α\in \mathbb{R}\backslash [-2,2]$. This is the first existence result on ($-α$)-homogeneous solutions with no explicit forms. For $2<α<3$, constructed ($-α$)-homogeneous solutions provide new examples of the Beltrami/Euler flows in $\mathbb{R}^{3}\backslash\{0\}$ whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign $``\infty"$.