h-principle for the 2-dimensional incompressible porous media equation with viscosity jump

Abstract

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. $(4)$ 45:3 (2012), 491–509) on the 2-dimensional incompressible porous media system with constant viscosity (Atwood number $A_{\mu }=0$) to the case of viscosity jump (). We prove an h-principle whereby (infinitely many) weak solutions in $C_t{L}_{w^{\ast }}^{\infty }$ 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 $A_{\mu }=0$, 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.