Piecewise constant subsolutions for the Muskat problem

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.

28 pages

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