The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

TL;DR

This paper studies the lifespan of classical solutions to the inviscid Surface Quasi-geostrophic equation, showing extended existence times for small perturbations and constructing global rotating solutions through bifurcation.

Contribution

It introduces a modified energy method to significantly extend the lifespan of solutions and constructs global rotating solutions via bifurcation, advancing understanding of solution behavior.

Findings

Existence time extends from 1/ε to 1/ε^4 for small perturbations.

Global smooth solutions that rotate uniformly are constructed.

The method applies bifurcation techniques to generate new solutions.

Abstract

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $\epsilon$ from a radial stationary solution $\theta=|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{\epsilon}$ to a time scale of $\frac{1}{\epsilon^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.