Self-similar solutions for the Muskat equation

TL;DR

This paper demonstrates the existence of self-similar solutions to the Muskat equation, which start as corner-shaped profiles with small slope and evolve into smooth solutions over time.

Contribution

It introduces a new class of self-similar solutions for the Muskat equation, highlighting their formation from initial corners with small slope.

Findings

Existence of self-similar solutions parameterized by small slope s

Solutions start as corners and become smooth over time

Exact corner solutions at initial time

Abstract

We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0<s \ll 1$; they are exact corners of slope $s$ at $t=0$ and become smooth in $x$ for $t>0$.