Existence of Homogeneous Euler Flows of Degree \(-\alpha \notin [-2,0]\)

Abstract

We consider (\(-\alpha \))-homogeneous solutions to the stationary incompressible Euler equations in \({\mathbb {R}}^{3}\backslash \{0\}\) for \(\alpha \geqq 0\) and in \({\mathbb {R}}^{3}\) for \(\alpha <0\). Shvydkoy (2018) demonstrated the nonexistence of (\(-1\))-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) and (\(-\alpha \))-homogeneous solutions in the range \(0\leqq \alpha \leqq 2\) for the Beltrami and axisymmetric flow; namely, that no (\(-\alpha \))-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) for \(1\leqq \alpha \leqq 2\) and \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) for \(0\leqq \alpha < 1\) exist among these particular classes of flows other than irrotational solutions for integers \(\alpha \). The nonexistence result of the Beltrami (\(-\alpha \))-homogeneous solutions \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) holds for all \(\alpha <1\). We show the nonexistence of axisymmetric (\(-\alpha \))-homogeneous solutions without swirls \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) for \(-2\leqq \alpha <0\). The main result of this study is the existence of axisymmetric (\(-\alpha \))-homogeneous solutions in the complementary range \(\alpha \in {\mathbb {R}}\backslash [0,2]\). More specifically, we show the existence of axisymmetric Beltrami (\(-\alpha \))-homogeneous solutions \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) for \(\alpha >2\) and \((u,p)\in C({\mathbb {R}}^{3})\) for \(\alpha <0\) and axisymmetric (\(-\alpha \))-homogeneous solutions with a nonconstant Bernoulli function \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\) for \(\alpha >2\) and \((u,p)\in C({\mathbb {R}}^{3})\) for \(\alpha <-2\), including axisymmetric (\(-\alpha \))-homogeneous solutions without swirls \((u,p)\in C^{2}({\mathbb {R}}^{3}\backslash \{0\})\) for \(\alpha >2\) and \((u,p)\in C^{1}({\mathbb {R}}^{3}\backslash \{0\})\cap C({\mathbb {R}}^{3})\) for \(\alpha <-2\). This is the first existence result on (\(-\alpha \))-homogeneous solutions with no explicit forms. The level sets of the axisymmetric stream function of the irrotational (\(-\alpha \))-homogeneous solutions in the cross-section are the Jordan curves for \(\alpha =3\). For \(2<\alpha <3\), we show the existence of axisymmetric (\(-\alpha \))-homogeneous solutions whose stream function level sets are the Jordan curves. They 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 \(``\infty ''\).