Homogeneous steady states for the generalized surface quasi-geostrophic equations

We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $\psi$ and advected scalar $\omega$ are of the form \begin{align*} \psi=\frac{w(\theta)}{r^{\beta}},\quad \omega=\frac{g(\theta)}{r^{\beta+2s}}, \end{align*} in polar coordinates $(r,\theta)$ with parameter $\beta\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<\beta<-2s$ and $0<\beta<m_0+2$ ($1/2-s<\beta< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq \beta\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions.