Global self-similar solutions for the 3D Muskat equation

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 $\dot{W}^{1,\infty}(\mathbb{R}^2)$.