On the elasto-plastic filtration equation

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of viscosity solutions to a nonlinear heat equation modeling elastic fluid flow in porous media, including asymptotic analysis and game-theoretic characterizations.

Contribution

It provides new results on the well-posedness and asymptotic properties of solutions to a fully nonlinear elasto-plastic filtration equation, linking solutions to a minimization dynamic game.

Findings

Existence and uniqueness of viscosity solutions established.

Asymptotic behavior characterized as time approaches infinity.

Solutions linked to limits of a minimization dynamic game.

Abstract

We study the fully nonlinear heat equation $b(\partial_tu)\partial_tu=\Delta u$ posed in a bounded domain with Dirichlet boundary conditions. Here $b(s)=b^-$ if $s<0$, $b(s)=b^+$ if $s>0$, $b^-\neq b^+$ being two positive constants. This equation models the flow of an elastic fluid in an elasto-plastic porous medium. We are interested in the existence and uniqueness of viscosity solutions and in their asymptotic behaviour as $t\to\infty$ and when $b^-\to 0^+$ or $b^+\to +\infty$. We also characterize solutions of the problem as limits of a minimization dynamic game.