Global well-posedness of arbitrarily large Lipschitz solutions for the Muskat problem with surface tension

Abstract

We prove a global well-posedness result for the 2D Muskat problem with surface tension. Given any regular enough initial data which is small in some critical space but possibly large in Lipschitz, we prove that the associated Cauchy problem has a unique global solution. Our result allows for the slope of the interface between the two fluids to be arbitrarily large.