Homogeneous steady states for the generalized surface quasi-geostrophic equations

TL;DR

This work classifies homogeneous stationary self-similar solutions of the generalized surface quasi-geostrophic equations across the full parameter range 0< s <1. It builds a linear-nonlinear framework on high-frequency angular spaces, delivering a sharp polar-mode formula for the fractional Laplacian and introducing the non-local 2s-th order operator $\mathcal{L}(s,β)$. Through rigorous L^p and Hölder estimates and a PS-based variational scheme, the authors prove existence of odd-symmetric solutions for β in $(-m_0-2s, -2s)$ and $(0, m_0+2)$ and nonexistence for $-2s\le β\le 0$, while identifying five distinct regimes relating to scale-invariance and local well-posedness. Numerical experiments corroborate the analytical results and illustrate how the angular profiles encode high-frequency content as β varies, including the approach to Bahouri–Chemin-type patches. The results illuminate the landscape of self-similar structures in gSQG and provide a robust methodology for analyzing nonlocal, scale-invariant PDEs in fluid dynamics.

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.

Figures (19)

• Figure 1: $1/2< s< 1$ (Case $s=3/4$)
• Figure 2: $s=1/2$
• Figure 3: $0<s< 1/2$ (Case $s=1/4$)
• Figure 6: Five self-similar regimes (i)-(v) in critical and supercritical regimes $\beta\geq -2$ for local well-posedness of gSQG (Case $s=3/4$ and $m_0=2$)
• Figure 7: The ranges of $\beta$ and variational principles. Case $s=1/4$ and $m_0=2$

Theorems & Definitions (82)

• Theorem 1.2
• Lemma 2.1
• Proposition 2.3: Poles and zeros for $|m|$
• proof
• Proposition 2.4: Monotonicity
• proof
• Proposition 2.5: Asymptotics
• proof
• Proposition 2.6: Poles and zeros for $\beta$
• proof