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 ₜu+Lₛu=^ f+g in (0, ) ᵈ where = (-) ^1{2}, 0< s, Lₛ is the Pesudo-differential operator defined by equation Lₛu (t, x) = (2) ^-d{2}ₑ㵧A (t, x, ) u (t, ) e^ixd, A (t, x, ) ||ˢ. equation To prove this, we develop a new freezing coefficient method for kernel, where we freeze the coefficient at x₀, then derive a representation formula of the solution, and finally we take x₀=x when estimating the solution. By applying our Schauder-type estimate to suitably chosen differential operators Lₛ, 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.