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

Abstract

We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space H r (S) for each r ∈ (2, 3). When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of H 2 (S) defined by the Rayleigh-Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.2010 Mathematics Subject Classification. 35B35; 35B65; 35K55; 35Q35; 42B20.

Well-Posedness and Stability Results for Some Periodic Muskat Problems

“…A first major difference to the case σ > 0 is due to the fact that the quasilinear character of the problem, which is mainly due to the curvature term, is lost (except for the very special case when μ − = μ + , cf. [47]), and the problem (1.1) is now fully nonlinear. The second important difference, is that the problem is of parabolic type only when the Rayleigh-Taylor condition holds.…”

“…Furthermore, given τ ∈ (1/2, 1), classical (but lengthy) arguments (see [47, where similar integral operators are discussed) show that…”

“…We study first the leading order part Φ σ,1 (f ). In order to establish (4.2) we follow a direct and self-contained approach pursued previously in [29,33,35] and generalized more recently in [32,[47][48][49] in the context of the Muskat problem. The proof of (4.2) uses a localization procedure which necessitates the introduction of certain partitions of unity for the unit circle.…”

“…∀h ∈ H 3 (S) and Reλ ≥ 1. We now come to the proof our first main result which uses on the one hand the abstract theory for quasilinear parabolic problems outlined in [1][2][3][4][5] (see also [50, Theorem 1.1]), and on the other hand a parameter trick which has been employed in various versions in [8,34,[47][48][49]57] in the context of improving the regularity of solutions to certain parabolic evolution equations. We point out that the parameter trick can only be used because the uniqueness claim of Theorem 1.1 holds in the setting of classical solution (the solutions in Theorem 1.1 possess though additional Hölder regularity properties, see the proof of Theorem 1.1).…”

Growth in the Muskat problem

“…Matioc [75], by rewitting the Muskat problem as an abstract evolution equation in an appropriate functional setting, was able to prove the local existence for arbitrary H s , 3/2 < s < 2 initial data (see also [74,76,77]).…”