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({mathbb {S}}) for each rin (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({mathbb {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.
Highlights
We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities
In the presence of surface tension effects, that is for σ > 0, (1.1) has been studied previously only in [7] where the author proved well-posedness of the problem in Hr in the more general setting of interfaces which are parameterized by curves, and the zero surface tension limit of the problem has been considered there
The nonperiodic counterpart to (1.1) has been investigated in [48] where it was shown that the problem is well-posed in Hr(S) for each r ∈ (2, 3) by exploiting the fact that the problem is quasilinear parabolic together with the abstract theory outlined in [4,5] for such problems
Summary
The first main result of this paper is the following theorem establishing the well-posedness of the Muskat problem with surface tension in the setting of classical solutions and for general initial data together with other qualitative properties of the solutions. (i) Despite that we deal with a third order problem in the setting of classical solutions, the curvature of the initial data in Theorem 1.1 may be unbounded and/or discontinuous It becomes instantaneously real-analytic under the flow. In Theorem 1.3 below we describe the stability properties of some of the equilibria to (1.1) when σ > 0 Given ω ∈ (0, k(σ + Θ)/(μ− + μ+)), there exist constants δ > 0 and M > 0, with the property that if f0 ∈ Hr(S) satisfies f0 Hr ≤ δ, the solution to (1.1) exists globally and f (t; f0) Hr ≤ M e−ωt f0 Hr for all t ≥ 0
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