  We establish a Schauder-type estimate for general local and non-local linear
parabolic system $$\partial_tu+\mathbf{L}_su=\Lambda^\gamma f+g$$ in
$(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$,
$0<\gamma\leq s$, $\mathbf{L}_s$ is the Pesudo-differential operator defined by
\begin{equation}
  \mathbf{L}_su(t,x)=(2\pi)^{-\frac{d}{2}}\int_{\mathbb{R}^d}\mathsf{A}(t,x,\xi)\hat
u(t,\xi)e^{ix\cdot\xi}d\xi,\quad\quad \mathsf{A}(t,x,\xi)\sim |\xi|^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.
