Blow-up of strong solutions of the Thermal Quasi-Geostrophic equation

TL;DR

This paper establishes a criterion for finite-time blow-up of solutions to the Thermal Quasi-Geostrophic equation, extending classical blow-up criteria from fluid dynamics to this coupled system.

Contribution

It introduces a blow-up criterion for the TQG equation, providing conditions under which solutions become singular in finite time.

Findings

Derived a Beale-Kato-Majda type blow-up criterion for TQG

Identified conditions leading to solution singularity

Extended classical fluid dynamics blow-up results to TQG

Abstract

The Thermal Quasi-Geostrophic (TQG) equation is a coupled system of equations that governs the evolution of the buoyancy and the potential vorticity of a fluid. It has a local in time solution as proved in [4]. In this paper, we give a criterion for the blow-up of solutions to the Thermal Quasi-Geostrophic equation, in the spirit of the classical Beale-Kato-Majda blow-up criterion (cf. [3]) for the solution of the Euler equation.