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

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.