Global self-similar solutions for the 3D Muskat equation

TL;DR

This paper proves the existence of global self-similar solutions for the 3D Muskat equation with specific initial data, providing new estimates and insights into the behavior of solutions with different densities and viscosities.

Contribution

It establishes the existence of global self-similar solutions for the 3D Muskat equation with equal viscosities and different densities, using novel estimates in critical and weighted Sobolev spaces.

Findings

Existence of global self-similar solutions with cone initial data.

Quantitative estimates of the difference between solutions and linearized solutions.

Development of new estimates in mixed Sobolev spaces for the Muskat problem.

Abstract

In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted $\dot{W}^{k,\infty}(\mathbb{R}^2)$ spaces for $k=1,2$. The main ingredients of the proof are new estimates in the sense of $\dot{H}^{s_1}(\mathbb{R}^2) \cap \dot{H}^{s_2}(\mathbb{R}^2)$ with $3/2<s_1<2<s_2<3$, which is continuously embedded in critical spaces for the 3D Muskat problem: $\dot{H}^2(\mathbb{R}^2)$ and $\dot{W}^{1,\infty}(\mathbb{R}^2)$.