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
,
, when neglecting surface
tension, respectively in
,
, when
surface-tension effects are included. Besides, we provide new criteria for the global
existence of solutions.
Keywords
Muskat problem, surface tension, singular integral
