On well-posedness of the Muskat problem with surface tension

Abstract

We consider the Muskat problem with surface tension for one fluid or two fluids, with or without viscosity jump, with infinite depth or Lipschitz rigid boundaries, and in arbitrary dimension d of the interface. The problem is nonlocal, quasilinear, and to leading order, is scaling invariant in the Sobolev space Hsc(Rd) with sc=1+d2. We prove local well-posedness for large data in all subcritical Sobolev spacesHs(Rd), s>sc, allowing for initial interfaces whose curvatures are unbounded and, furthermore when d=1, not locally square integrable. To the best of our knowledge, this is the first large-data well-posedness result that covers all subcritical Sobolev spaces for the Muskat problem with surface tension. We reformulate the problem in terms of the Dirichlet-Neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation, which is of independent interest.