Stationary homogeneous solutions for the inviscid SQG equation

TL;DR

The paper addresses the construction of stationary nontrivial homogeneous solutions to the inviscid SQG equation by reducing to a nonlocal 1D problem on the circle via a homogeneous ansatz. The authors adapt a Schauder fixed-point framework, establishing existence of a positive fixed point for a nonlinear integral equation involving the operator $-\mathcal{S}_m$, which yields stationary solutions with $m$-fold symmetry when extended to the torus. They prove regularity of the solutions and verify the necessary hypotheses to apply the fixed-point theorem, culminating in a main theorem that, for every $\alpha>\tfrac{1}{2}$ and $m\ge3$, provides a $C^{\gamma}$-regular, odd, $2\pi/m$-periodic stationary solution with a nonvanishing $m$-th Fourier coefficient solving the folded equation $\alpha\mathcal{S}_m f\partial_x f=\partial_x \mathcal{S}_m f f$. The results also generalize to the generalized De Gregorio equation, highlighting a broad class of stationary homogeneous states and connecting to questions about nonuniqueness and singularity formation in nonlocal transport models.

Abstract

In this paper, we prove existence of stationary nontrivial homogeneous solutions for SQG equation with infinite energy (unbounded at the infinity). Our analysis also covers the existence of stationary solutions for generalized De Gregorio equation $w_t+αuw_x=u_xw,\ u_x=Hw$ with $α>\frac{1}{2}.$

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Lemma 2.8