Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy

TL;DR

This paper investigates the formation of singularities in a reduced 1D model of the generalized SQG equation, demonstrating finite-time self-similar blow-up solutions with supporting numerical simulations.

Contribution

It introduces a one-dimensional reduction capturing the main singular behavior of the 2D gSQG system and proves the existence of self-similar blow-up solutions.

Findings

Existence of finite-time self-similar blow-up solutions

Derivation of a 1D model capturing 2D singular behavior

Numerical simulations confirming theoretical results

Abstract

We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space $\mathbb{R}^2$ and on the upper half-plane $\mathbb{R}^2_+$, allowing infinite energy. In each case, we derive a one-dimensional reduction that captures the leading-order singular behavior of the original 2D system, and use a fixed-point argument to show the existence of finite-time self-similar blow-up solutions for the 1D systems. We also perform numerical simulations for verification and visualization.