Desingularization of small moving corners for the Muskat equation

TL;DR

This paper studies the evolution of interfaces with small corners in the Muskat equation, showing how these corners smooth out and move over time through a renormalization approach.

Contribution

It introduces a novel renormalization method to precisely describe the desingularization and motion of small corners in the Muskat equation.

Findings

Corners desingularize and move during evolution

Renormalization captures corner dynamics accurately

Provides a detailed description of interface behavior

Abstract

In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution.