Self-similar spirals for the generalized surface quasi-geostrophic equations

TL;DR

This paper constructs the first rigorous class of non-radial, self-similar spiral solutions for the generalized surface quasi-geostrophic equations, advancing understanding of potential singularity formation in these fluid models.

Contribution

It provides the first rigorous construction of non-trivial self-similar solutions for gSQG equations, specifically of spiral type, which were previously only numerically hypothesized.

Findings

Constructed a large class of non-radial self-similar solutions

Solutions are locally integrable and may have mixed signs

Resemble finite time singularity scenarios proposed in prior numerical studies

Abstract

In this paper we construct a large class of non-trivial (non-radial) self-similar solutions of the generalized surface quasi-geostrophic equation (gSQG). To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. The solutions are of spiral type, locally integrable, and may have mixed sign. Moreover, they bear some resemblance with the finite time singularity scenario numerically proposed by Scott and Dritschel [R. K. Scott, D. G. Dritschel., Journal of Fluid Mechanics, 863:R2, 2019] in the SQG patch setting.