Well-posedness and stability results for a quasilinear periodic Muskat problem

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.