Strong solutions of semilinear matched microstructure models

TL;DR

This paper investigates the mathematical well-posedness and long-term behavior of solutions for a coupled system of parabolic equations modeling Newtonian fluid flows in fractured porous media with microstructure.

Contribution

It establishes local strong solutions for semi-linear matched microstructure models and analyzes their long-time behavior, including global existence and exponential decay.

Findings

Proved local well-posedness of the model

Established global existence for certain cases

Showed solutions decay exponentially over time

Abstract

The subject of this article is a matched microstructure model for Newtonian fluid flows in fractured porous media. This is a homogenized model which takes the form of two coupled parabolic differential equations with boundary conditions in a given (two-scale) domain in Euclidean space. The main objective is to establish the local well-posedness in the strong sense of the flow. Two main settings are investigated: semi-linear systems with linear boundary conditions and semi-linear systems with nonlinear boundary conditions. With the help of analytic semigoups we establish local well-posedness and investigate the long-time behaviour of the solutions in the first case: we establish global existence and show that solutions converge to zero at an exponential rate.