The lamination convex hull of stationary IPM

TL;DR

This paper calculates the lamination convex hull of stationary IPM equations and explores the properties of subsolutions, revealing that velocity vanishes and density depends solely on height in bounded domains, with implications for the infinite time limit.

Contribution

It introduces the computation of the lamination convex hull for stationary IPM and characterizes subsolutions in bounded domains, linking stationary and non-stationary IPM behaviors.

Findings

Velocity vanishes for subsolutions in bounded domains.

Density depends only on height for subsolutions.

Results connect stationary IPM to its infinite time limit.

Abstract

We compute the lamination convex hull of the stationary IPM equations. We also show in bounded domains that for subsolutions of stationary IPM taking values in the lamination convex hull, velocity vanishes identically and density depends only on height. We relate the results to the infinite time limit of non-stationary IPM.