Zusammenfassung
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 ...
Zusammenfassung
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 (S), with s is an element of (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. (C) 2018 Elsevier Inc. All rights reserved.