On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension

Pruess, Jan; Shao, Yuanzhen; Simonett, Gieri

doi:10.4171/ifb/354

Cited by 17 publications

(20 citation statements)

References 69 publications

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“…We are now in a position to prove Theorem 1.3 when σ = 0, where we use a parameter trick which appears, in other forms, also in [7,31,44]. We present a new idea which uses only the abstract result Theorem 1.5 in the context of an evolution problem related to (1.1), and not explicitly the maximal regularity property as in [7,31,44]. The proof when σ > 0 is almost identical and is also discussed below, but it relies on some properties established in Section 6.…”

Section: The Conclusion Is Now Obviousmentioning

confidence: 99%

The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

Matioc

2019

Analysis & PDE

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In this paper we consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class H s , s ∈ (3/2, 2), when neglecting surface tension, respectively in H s , s ∈ (2, 3), when surface tension effects are included. Besides, we provide new criteria for the global existence of solutions.

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Section: The Conclusion Is Now Obviousmentioning

confidence: 99%

The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

Matioc

2019

Analysis & PDE

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“…We now come to the proof of our first main result, which is mainly based on the abstract theory for quasilinear parabolic problems due to H. Amann, cf. [3,Section 12], and a recent idea from the proof of [40,Theorem 1.3] where a parameter trick, also used in [7,28,44], is employed in a different manner.…”

Section: Proofmentioning

confidence: 99%

Viscous displacement in porous media: the Muskat problem in 2D

Matioc¹

2018

Trans. Amer. Math. Soc.

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Abstract. We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh-Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local wellposedness for arbitrary large initial data in H s , s > 2, if surface tension is taken into account, respectively for arbitrary large initial data in H 2 that additionally satisfy the Rayleigh-Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.

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“…. , 2 p+1 } and τ ∈ [0, 1] operators A c j,τ ∈ L(H 3 (S), L 2 (S)) (A c j,τ is the complexification of A j,τ defined in (4.10)) such that 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 [8,35,[47][48][49]55] 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.…”

Section: Noticing Thatmentioning

confidence: 99%

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

Matioc

2019

Journal of Differential Equations

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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.

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On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension

Abstract: Abstract. We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, Lp-maximal regularity theory, and the implicit function theorem.

Cited by 17 publications

References 69 publications

The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

Viscous displacement in porous media: the Muskat problem in 2D

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

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