Sharp local well-posedness of $C^1$ vortex patches

It is well known that the boundary dynamics of vortex patches is globally well-posed in the H\"older space $C^{1,\alpha}$ for $0<\alpha<1$, whereas the well-posedness in $C^1$ remains an open problem, even locally. In this paper, we establish the local well-posedness for vortex patches in the space $C^{1,\varphi}$ defined via a modulus of continuity $\varphi$ that satisfies certain structural assumptions. Our class includes curves that are strictly rougher than the H\"older-continuous ones, with prototypical examples being $\varphi(r) = (-\log r)^{-s}$ for $s>3$. Motivated by the fact that the velocity operator in the contour dynamics equation is a nonlinear variant of the Hilbert transform, we study the system of equations satisfied by the curve parametrization $\gamma \in C^{1,\varphi}$ and its Hilbert transform. In doing so, we derive several properties of the Hilbert transform and its variants in critical spaces, which are essential for controlling the velocity operator and its Hilbert transform.