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=Λ^γ f+g in (0,∞)ᵈ where Λ=(-Δ)1/2, 0<γ≤ s, Lₛ is the Pesudo-differential operator defined by {equation} L_su(t,x)=(2π)-d/2∫R^dA(t,x,ξ) u(t,ξ)eix·ξdξ, A(t,x,ξ)~ |ξ|^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.