We extend the results of Córdoba, Faraco and Gancedo (Arch. Ration. Mech. Anal.200:3 (2011), 725–746) and Székelyhidi (Ann. Sci. Éc. Norm. Supér.45:3 (2012), 491–509) on the 2-dimensional incompressible
porous media system with constant viscosity (Atwood number
) to the case of
viscosity jump ().
We prove an h-principle whereby (infinitely many) weak solutions in
are
recovered via convex integration whenever a subsolution is provided. As a first example,
nontrivial weak solutions with compact support in time are obtained. Secondly, we
construct mixing solutions to the unstable Muskat problem with initial flat interface.
As a byproduct, we check that the connection, established by Székelyhidi (2012) for
,
between the subsolution and the Lagrangian relaxed solution of
Otto (Comm. Pure Appl. Math. 52:7 (1999), 873–915) holds for
too.
For different viscosities, we show how a pinch singularity in the relaxation prevents
the two fluids from mixing wherever there is neither Rayleigh–Taylor nor vorticity at
the interface.
