On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof

Abstract

We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

30 pages, 6 figures

References in corpus (3)

• Breakdown of smoothness for the Muskat problem
• A maximum principle for the Muskat problem for fluids with different densities
• Finite time singularities for water waves with surface tension

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