The Second Iterate of the Muskat Equation in Supercritical Spaces

TL;DR

This paper investigates the ill-posedness of the Muskat problem in supercritical spaces, showing that the second Picard iterate becomes discontinuous near the origin in these spaces, indicating instability.

Contribution

It demonstrates the discontinuity of the second Picard iterate for the Muskat problem in supercritical spaces, highlighting ill-posedness in these regimes.

Findings

Second Picard iterate is discontinuous near the origin in supercritical spaces.

Ill-posedness is established for the Muskat problem in certain supercritical spaces.

Approaching the critical space, the instability persists, indicating fundamental challenges in these regimes.

Abstract

The ill-posedness for the Muskat problem in spaces that are supercritical with respect to the scaling is studied. The main result of the paper establishes that for a sequence of approximations of the Muskat equation obtained via Taylor expansion, their corresponding second Picard's iterate is discontinuous around the origin in a certain family of supercritical spaces approaching a critical space.