Existence for coupled pseudomonotone-strongly monotone systems and application to a Cahn-Hilliard model with elasticity

TL;DR

This paper establishes conditions for the existence of solutions in coupled systems involving pseudomonotone and strongly monotone operators, and applies these results to a complex Cahn-Hilliard model with elasticity.

Contribution

It provides a novel reduction of coupled operator systems to single operator equations, enabling analysis of existence for complex PDE models.

Findings

Existence of solutions under certain coupling restrictions

Reduction of coupled systems to single operator equations

Application to a Cahn-Hilliard model with elasticity

Abstract

A system of two operator equations is considered - one of pseudomonotone type and the other of strongly monotone type - both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single operator equation for a pseudomonotone mapping. This result is applied to a coupled system consisting of a parabolic equation of forth order in space of Cahn-Hilliard type and a nonlinear elliptic equation of second order to a quasi-steady mechanical equilibrium. Using an appropriate notation of weak solutions and a framework for evolution equations developed by Gr\"oger, the system is reduced to a single parabolic operator equation and the existence of solutions are shown under restrictions on the strength of the coupling.