Well-posedness for local and nonlocal quasilinear evolution equations in fluids and geometry. Lecture 1

In this talk, I will present a Schauder-type estimate for general local and non-local linear parabolic system
$$\partial_tu+\mathcal{L}_su=\Lambda^\gamma f+g$$
in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathcal{L}_s$ is the Pesudo-differential operator of the order $s$. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier–Stokes equations, the surface quasi- geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.