# Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids

## TL;DR

This paper proves the existence of steady states in elastic media with fluid diffusion using variational methods and fixed-point theory, addressing complex coupled flow and heat transfer problems in poroelastic solids.

## Contribution

It introduces novel variational approaches combined with Schauder's fixed-point theorem to handle non-potential and non-pseudomonotone operators in coupled flow and elasticity models.

## Key findings

- Existence of steady states in elastic media with diffusion established.
- Extension to electrically-charged multi-component flows and heat transfer.
- Application of variational methods to complex coupled poroelastic problems.

## Abstract

Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed-point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non-potential and non-pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically-charged multi-component flows through an electrically charged elastic solid are treated, employing critical points of the saddle-point type. Eventually, anisothermal variants involving heat-transfer equation are treated, too.