Piecewise constant subsolutions for the Muskat problem

TL;DR

This paper demonstrates the existence of infinitely many weak solutions to the Muskat problem using piecewise constant subsolutions, highlighting non-uniqueness due to initial data discontinuities.

Contribution

It introduces a novel approach to construct admissible subsolutions with piecewise constant densities for the Muskat problem, simplifying the proof of non-uniqueness.

Findings

Existence of infinitely many weak solutions for all Muskat initial data with regular interfaces.

Non-uniqueness is instantaneous and related to initial data discontinuities.

Method applies to both stable and unstable regimes.

Abstract

We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with $C^{3,\alpha}$-regularity of the interface in the unstable regime and for all non-horizontal data with $C^{3,\alpha}$-regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero - thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.