Well-Posedness and Stability Results for Some Periodic Muskat Problems
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Journal of Mathematical Fluid Mechanics
Well-Posedness and Stability Results for Some Periodic Muskat Problems Bogdan-Vasile Matioc Communicated by M. Hieber
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. Mathematics Subject Classification. 35B35, 35B65, 35K55, 35Q35, 42B20. Keywords. Muskat problem, Singular integral, Well-posedness, Parabolic smoothing, Stability.
Contents 1. Introduction and the Main Results 2. The Equations of Motion and the Equivalence of the Formulations 3. The Double Layer Potential and Its Adjoint 4. The Muskat Problem with Surface Tension Effects 5. The Muskat Problem Without Surface Tension Effects 6. Stability Analysis Appendix A. Some Technical Results References
1. Introduction and the Main Results In this paper we study the coupled system of equations ⎧ π f (t, x)(1 + t2 )(T 2 [x,s] f (t)) + t[s] [1 − (T[x,s] f (t)) ] 1 ⎪ [s] ⎪ ⎪ ∂ PV f (t, x) = ω(t, x − s)ds, t ⎪ ⎪ 4π t2[s] + (T[x,s] f (t))2 ⎪ −π ⎪ ⎨ 2k ω(t, x) = (σκ(f (t)) − Θf (t)) (x) ⎪ μ + μ − + ⎪ π f (t, x)t [1 − (T ⎪ 2 2 ⎪ ⎪ [s] [x,s] f (t)) ] − (1 + t[s] )T[x,s] f (t) aμ ⎪ ⎪ ω(t, x − s)ds − PV ⎩ 2π t2[s] + (T[x,s] f (t))2 −π 0123456789().: V,-vol
(1.1a)
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JMFM
for t > 01 and x ∈ R, which is supplemented by the initial condition f (0) = f0 .
(1.1b)
The evolution problem (1.1) describes the motion of the boundary [y = f (t, x) + tV ] separating two immiscible fluid layers with unbounded heights located in a homogeneous porous medium with permeability k ∈ (0, ∞) or in a vertical/horizontal Hele–Shaw cell. It is assumed that the fluid system moves with constant velocity (0, V ), V ∈ R, that the motion is periodic with respect to the horizontal variable x (with period 2π), and that the fluid velocities are asymptotically equal to (0, V ) far away from the interface. The unknowns of the evolution problem (1.1) are the functions (f, ω) = (f, ω)(t, x). We denote by S := R/2πZ the unit circle, functions that depend on x ∈ S being 2π-periodic with respect to the real variable x. To be concise, we have set s
δ[x,s] f , T[x,s] f = tan h δ[x,s] f := f (x) − f (x − s), , t[s] = tan 2 2 and ( · ) denotes the spatial derivative ∂x . We further de
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