Schauder-type Estimates and Well-posedness for Nonlocal Quasilinear Evolution Equations in Fluid Dynamics

TL;DR

This paper develops Schauder-type estimates for nonlocal parabolic systems with variable coefficients, enabling a robust well-posedness theory for critical nonlocal fluid equations like Muskat and Peskin problems.

Contribution

It introduces a kernel-adapted freezing-coefficient method to derive explicit bounds, advancing the analysis of quasilinear nonlocal evolution equations at critical regularity.

Findings

Established Schauder-type estimates in critical spaces for nonlocal operators.

Proved local and global well-posedness for Muskat and Peskin problems in critical regimes.

Unified framework applicable to various nonlocal fluid dynamics equations.

Abstract

We establish Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order $s>0$. These estimates are formulated in critical time-weighted H\"older/Besov-type spaces and are tailored to quasilinear equations at scaling-critical regularity. A key ingredient is a kernel-adapted freezing-coefficient method. After freezing the coefficients at a reference point, we derive explicit representation formulas through the corresponding fundamental kernels and then evaluate the resulting bounds at the physical point. This avoids treating the coefficient variation as a separate lower-order perturbation and yields robust control of the residual terms within the leading-order dynamics. As an application, we obtain a general well-posedness framework for a class of nonlocal quasilinear parabolic equations in critical spaces. In particular, we prove critical local and, in suitable regimes, global well-posedness for the Muskat equation with surface tension and for the two- and three-dimensional Peskin problems with nonlinear elastic tension. These results provide a unified critical framework for distinct nonlocal evolution equations arising in fluid dynamics and related areas.