Critical spaces for quasilinear parabolic evolution equations and applications

TL;DR

This paper develops a comprehensive theory of critical spaces for quasilinear parabolic equations using maximal regularity, with applications to various fluid dynamics and electro-chemistry models.

Contribution

It introduces a unified approach to identify critical spaces for a broad class of quasilinear parabolic equations based on maximal $L_p$-regularity.

Findings

Critical spaces coincide with scaling invariant spaces for scale-invariant PDEs.

Application of the theory to Navier-Stokes vorticity equations.

Extension to convection-diffusion, Nernst-Planck-Poisson, chemotaxis, and MHD equations.

Abstract

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.