Homogeneous steady states for the generalized surface quasi-geostrophic equations

Abstract

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 $ψ$ and advected scalar $ω$ are of the form \begin{align*} ψ=\frac{w(θ)}{r^β},\quad ω=\frac{g(θ)}{r^{β+2s}}, \end{align*} in polar coordinates $(r,θ)$ with parameter $β\in \mathbb{R}$. 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 $-m_0-2s<β<-2s$ and $0<β<m_0+2$ ($1/2-s<β< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq β\leq 0$. 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.