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