Global self-similar solutions for the 3D Muskat equation

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 ${\dot{W}}^{k,\mathrm{\infty }}({\mathbb{R}}^2)$ 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 $3\mathrm{/}2<s_1<2<s_2<3$, which is continuously embedded in critical spaces for the 3D Muskat problem: ${\dot{H}}^2({\mathbb{R}}^2)$ and ${\dot{W}}^{1,\mathrm{\infty }}({\mathbb{R}}^2)$.