Well posedness of nonlinear parabolic systems beyond duality
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 $q\in (1,\infty)$.
- Publication:
-
Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
- Pub Date:
- August 2019
- DOI:
- 10.1016/j.anihpc.2019.01.004
- arXiv:
- arXiv:1810.05061
- Bibcode:
- 2019AIHPC..36.1467B
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35D99;
- 35K51;
- 35K61;
- 35A01;
- 35A02
- E-Print:
- Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 36, No. 5, 1467--1500, 2019