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

Abstract

We establish a Schauder-type estimate for general local and non-local linear parabolic system $${\partial }_{t}u+{\mathbf{L}}_{s}u={\Lambda }^{\gamma }f+g$$ in $(0,\infty )\times {\mathbb{R}}^d$ where $\Lambda =(-\Delta {)}^{\frac{1}{2}}$, $0<\gamma \le s$, ${\mathbf{L}}_s$ 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.

184 pages, 1 figure