On nonlinear problems of parabolic type with implicit constitutive equations involving flux

TL;DR

This paper develops a comprehensive theory for nonlinear parabolic PDEs with implicit constitutive relations involving flux, establishing existence, uniqueness, and numerical tractability for broad applications.

Contribution

It introduces four conditions defining a maximal monotone p-coercive graph and generalizes the Minty method to prove solution existence and uniqueness.

Findings

Proved global-in-time existence of weak solutions.

Established uniqueness of solutions under broad conditions.

Developed a numerically tractable unified theory.

Abstract

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone $p$-coercive graph. We then establish the global-in-time and large-data existence of (weak) solution and its uniqueness. Towards this goal, we adopt and significantly generalize the Minty method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable numerically.