The Two-Phase Hele-Shaw Problem with a Nonregular Initial Interface and Without Surface Tension

Bazaliy, Borys V.; Vasylyeva, Nataliya

doi:10.15407/mag10.01.003

Cited by 15 publications

(18 citation statements)

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“…[22,29] (see also the equations (2.1a)-(2.1b)). The first local existence result has been established in [53] by using Newton's iteration method, the analysis in [5,10,15,[18][19][20][21][22][23]36] is based on energy estimates and the energy method, the authors of [49] use methods from complex analysis and a version of the Cauchy-Kowalewski theorem, a fixed point argument is employed in [8] for nonregular initial data, and the approach in [27][28][29] relies on the formulation of the problem as a nonlinear and nonlocal parabolic equation together with an abstract well-posedness result from [24] based on continuous maximal regularity. Other papers study the qualitative aspects of solutions to the Muskat problem for fluids with equal viscosities, such as: global existence of strong and weak solutions [17,18,37], existence of initial data for which solutions turn over [12][13][14], the absence of squirt or splash singularities [22,34].…”

Section: Introduction and The Main Resultsmentioning

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: Introduction and The Main Resultsmentioning

confidence: 99%

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

Matioc

2019

Analysis & PDE

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“…To prove the solvability of system (4.12), (4.13), we reduce one similarly [5] to the nonlocal equation:…”

Section: Proof Ofmentioning

confidence: 99%

“…3. The operator R enables to construct an inverse operator to L by the methods from Section 4 [25] or Section 4 [5]. First of all, we need the following result.…”

Section: The Solvability Of the Linear Problemmentioning

confidence: 99%

“…Note that conditions of Lemma 4.1 mean that f 1 ≡f 1 . Since the proof of Lemma 4.1 is standard (see Section 4 [5]), although with extended calculations and based on the results of Theorem 3.1, we omit these arguments here.…”

Section: The Solvability Of the Linear Problemmentioning

confidence: 99%

“…Thus, Lσ is given by the left-hand side of (4.13).A nonlocal character of system (4.15) causes difficulties requiring technical tricks related to a suitable localization. As suggested in either Chapter 4 from[25] or Section 4 from[5], we introduce the two collections of open sets: {ω m } and {Ω m }, such that…”

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On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study

Vasylyeva

Vynnytska

2014

Nonlinear Differ. Equ. Appl.

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We study the anomalous diffusion version of the quasistationary Stefan problem (the fractional quasistationary Stefan problem) in the multidimensional case Ω(t) ⊂ R n , n ≥ 2. This free boundary problem is a mathematical model of a solute drug released from a polymer matrix (n = 1, 3). We prove the existence and uniqueness of the classical solution for this moving boundary problem locally in time. A numerical solution is constructed in the two-dimensional case. MathematicsSubject Classification. Primary 35R35, 35C15; Secondary 35B65, 35R11. Keywords. Quasistationary Stefan problem, Anomalous diffusion, Caputo derivative, Coercive estimates. 544 N. Vasylyeva and L. Vynnytska NoDEA tional velocity of the boundary Γ(t) in the direction of the normal n t and is represented by (see, e.g., [44]):(1.5) D ν t denotes the Caputo fractional derivative with respect to t and is defined by (see (2.4.6) where Γ(ν) is the Gamma function, < ·, · > denotes the scalar product. We recall that the anomalous diffusion can not be modelled as a standard Brownian motion [8,35], and the mean square displacement of the diffusing species (Δy) 2 scales as a nonlinear power law in time, i.e. (Δy) 2 ∼ t ν for some real number ν. In the case ν ∈ (0, 1), this is referred to as a subdiffusion.If n = 1, 3 and Γ 0 is in the exterior of Γ(t), problem (1.1)-(1.4) describes a solute released from an undissolved monolithic polymeric matrix in the quasistationary case [28,29]. We recall that each matrix consists of two regions: the surface zone (or liquid phase) x ∈ Ω(t) and the core B(t) within the interior of Γ(t), and B(t) ∩ Ω(t) = ∅. The whole solute is dissolved in the surface zone, and the core contains undissolved solute. The two zones are separated by the diffusion front (liquid-solid interface) Γ(t) which moves inward as time progresses. In other words, the domain Ω(t) expends, that is,Concentration of drug in the liquid phase is given by function p(y, t), and function ϕ(y) describes the drug concentration in the bulk liquid phase. Under conditions of practical significance, the amount of drug loaded is usually much bigger than the solubility of drug, and the time scale for the release process is long. Therefore, the rate of movement of the solid-liquid interface is relatively slow compared to that of drug diffusion.Note that Liu and Xu [29] were the first who introduced the timefractional diffusion equation to the drug release process. They obtained the well-known, semi-empirical formula in the controlled drug release system given by Ritger and Peppas.Problem (1.1)-(1.4) can be called the fractional quasistationary Stefan problem which arises under consideration of the materials with memory [44]. If Eq. (1.1) in problem (1.1)-(1.4) is replaced by the subdiffusion equation, we will get the fractional Stefan problem formulated and studied by Atkinson [1] for the motion of planar, cylindrical and spherical boundaries.If ν = 1, free boundary problem (1.1)-(1.4) is interpreted as the mathematical model of the Hele-Shaw problem, which has a hydrod...

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Well-Posedness and Stability Results for Some Periodic Muskat Problems

Matioc

2020

J. Math. Fluid Mech.

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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 $$H^r({\mathbb {S}})$$ H r ( S ) for each $$r\in (2,3)$$ 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 $$H^2({\mathbb {S}})$$ 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.

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The Two-Phase Hele-Shaw Problem with a Nonregular Initial Interface and Without Surface Tension

Cited by 15 publications

References 17 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

On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study

Well-Posedness and Stability Results for Some Periodic Muskat Problems

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