Localized mixing zone for Muskat bubbles and turned interfaces

Abstract

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.

Global well-posedness for the one-phase Muskat problem

“…In particular, for the horizontal one-phase Hele-Shaw problem, the authors in [22] proved that if the initial Lipschitz norm of the free boundary is small, then for a short time the unique viscosity solution satisfies the equations pointwise. We also mention that by using the methodology of convex integration, non-unique weak solutions have been constructed in the purely unstable [48,14,62] and partially unstable scenarios [15].…”

On bounded two-dimensional globally dissipative Euler flows

“…Solutions obtained by means of convex integration have not only served as counterexamples, but, due to their highly oscillatory nature, have also been utilized to describe naturally occurring turbulent behaviour in fluids. Examples include turbulence emanating from vortex sheet initial data for the homogeneous Euler equations [31,36], and from the mixture of two different density fluids due to gravity, see [2,3,4,10,21,24,30,32,35] for the incompressible porous media equation as underlying model and [22,23] for the inhomogeneous Euler equations. The construction of these solutions crucially relies on an explicit relaxation of the differential inclusion associated with the considered partial differential equations.…”

“…4 there exists wild initial data w ∈ L 2 (T 2 ) and for each µ ∈ B δ,− an associated subsolution zµ with turbulent zone T 2 × (0, T ]. Furthermore there holdsv(•, 0) − w 2 2 (x, 0) − 1 2 |v(x, 0)| 2 dx.Hence, we may use (3.2) to conclude thatw − w 2 2 ≤ 2 v(•, 0) − w 2 2 + 2 v(•, 0) − w 2 (x, 0) − 1 2 |v(x, 0)| 2 dx + 2δ ≤ 20δ.…”