Paralinearization and extended lifespan for solutions of the $α$-SQG sharp front equation

Abstract: In this paper we paralinearize the contour dynamics equation for sharp-fronts of $\alpha$-SQG, for any $\alpha \in (0,1) \cup (1,2)$, close to a circular vortex. This turns out to be a quasi-linear Hamiltonian PDE. After deriving the asymptotic expansion of the linear frequencies of oscillations at the vortex disk and verifying the absence of three wave interactions, we prove that, in the most singular cases $\alpha \in (1,2)$, any initial vortex patch which is $\varepsilon$-close to the disk exists for a time interval of size at least $\sim \varepsilon^{-2}$. This quadratic lifespan result relies on a paradifferential Birkhoff normal form reduction and exploits cancellations arising from the Hamiltonian nature of the equation. This is the first normal form long time existence result of sharp fronts.