  We study the Muskat problem describing the spatially periodic motion of two
fluids with equal viscosities under the effect of gravity in a vertical
unbounded two-dimensional geometry. We first prove that the classical
formulation of the problem is equivalent to a nonlocal and nonlinear evolution
equation expressed in terms of singular integrals and having only the interface
between the fluids as unknown. Secondly, we show that this evolution equation
has a quasilinear structure, which is at a formal level not obvious, and we
also disclose the parabolic character of the equation. Exploiting these
aspects, we establish the local well-posedness of the problem for arbitrary
initial data in $H^s(\mathbb{S})$, with $s\in(3/2,2)$, determine a new
criterion for the global existence of solutions, and uncover a parabolic
smoothing property. Besides, we prove that the zero steady-state solution is
exponentially stable.
