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

We establish a Schauder-type estimate for general local and non-local linear parabolic system $${\mathrm{\partial }}_{t}u+{\mathbf{L}}_{s}u={\mathrm{\Lambda }}^{\gamma }f+g$$ in $(0,\mathrm{\infty })\times {\mathbb{R}}^d$ where $\mathrm{\Lambda }=(-\mathrm{\Delta }{)}^{\frac{1}{2}}$, $0<\gamma \le s$, ${\mathbf{L}}_s$ is the Pesudo-differential operator defined by \begin{equation} $\mathbf{L}$su(t,x)=(2π)^{-\frac{d}{2}}\int{$\mathbb{R}$^d}$\mathsf{A}$(t,x,ξ)\hat u(t,ξ)e^{ix\cdotξ}dξ,\quad\quad $\mathsf{A}$(t,x,ξ)\sim |ξ|^s. \end{equation} To prove this, we develop a new freezing coefficient method for kernel, where we freeze the coefficient at $x_0$, then derive a representation formula of the solution, and finally we take $x_0=x$ when estimating the solution. 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.