Global well-posedness for the one-phase Muskat problem with or without surface tension

We study the free boundary problem for 2D and 3D incompressible flow in porous media, known as the one-phase Muskat problem. In the absence of surface tension, we prove that when the initial interface is given by a Lipschitz graph, there exists a unique global Lipschitz strong solution. When surface tension is included, we establish small-data global well-posedness results in both the whole-space and periodic settings.