Well posedness of nonlinear parabolic systems beyond duality

TL;DR

This paper introduces a novel methodology to establish well-posedness for nonlinear parabolic systems in optimal regularity spaces, especially addressing cases where traditional duality methods are inapplicable, covering the full range of integrability exponents.

Contribution

It develops a new approach using weighted spaces to prove existence, uniqueness, and regularity for nonlinear parabolic systems beyond the scope of standard duality theory.

Findings

Established well-posedness for all q in (1, ∞)

Extended regularity results to previously inaccessible cases

Provided a framework applicable to a broad class of nonlinear systems

Abstract

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system \[ \partial_tu-\mathrm{div} ( \nu(|\nabla u|) \nabla u )= -\mathrm{div} f \] with a given {strictly} positive bounded function $\nu$, {such that $\lim_{k\to \infty} \nu(k)=\nu_\infty$} and $f \in L^q$ with $q\in (1,\infty)$. The {existence, uniqueness and regularity} results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty)$.