THE MUSKAT PROBLEM IN TWO DIMENSIONS: EQUIVALENCE OF FORMULATIONS, WELL-POSEDNESS, AND REGULARITY RESULTS

We consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional-analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class H-s, s is an element of (3/2, 2), when neglecting surface tension, respectively in H-s, s is an element of (2, 3), when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.