Models of Phase Transitions

Visintin, Augusto

doi:10.1007/978-1-4612-4078-5

Cited by 250 publications

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“…We recall here this well-known result (see [1,31]) and present the proof for the simplest case, namely for strong solutions to the system with non-degenerate coefficient functions. The L 1 -contraction property derived in Theorem 2.3 and Remark 2.4 implies the stability of (1.1)-(1.2) in the sense of Definition 2.2 with the choice ρ = ε/2.…”

Section: Stability Resultsmentioning

confidence: 87%

Instability of gravity wetting fronts for Richards equations with hysteresis

Schweizer¹

2012

Interfaces Free Bound.

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Abstract:We study the evolution of saturation profiles in a porous medium. When there is a more saturated medium on top of a less saturated medium, the effect of gravity is a downward motion of the liquid. While in experiments the effect of fingering can be observed, i.e. an instability of the planar front solution, it has been verified in different settings that the Richards equation with gravity has stable planar fronts.In the present work we analyze the Richards equation coupled to a play-type hysteresis model in the capillary pressure relation. Our result is that, in an appropriate geometry and with adequate initial and boundary conditions, the planar front solution is unstable. In particular, we find that the Richards equation with gravity and hysteresis does not define an L 1 -contraction.

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Section: Stability Resultsmentioning

confidence: 87%

Instability of gravity wetting fronts for Richards equations with hysteresis

Schweizer¹

2012

Interfaces Free Bound.

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“…The Stefan problem has been studied in the mathematical literature for over a century, see [47], [39] and [53], pp. 117-120, for a historic account, and has attracted the attention of many prominent mathematicians.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Analytic solutions for a Stefan problem with Gibbs-Thomson correction

Gruyter¹

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

133

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Abstract. We provide existence of a unique smooth solution for a class of one-and two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal L pregularity for the linearized problem, which is fully developed in this paper as well.

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“…This will be referred to as mean curvature flow with a forcing term. For instance, let us consider phase transition in a solid-liquid system [8,14,25]. Let us denote by θ the temperature, by M t the interface between phases, by v its normal velocity, by H the sum of the principal curvatures.…”

Section: Introductionmentioning

confidence: 99%

Mean curvature flow with a constant forcing

Liu

2012

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, the motion of surface by its mean curvature and a forcing term θ was studied. We show that to each uniformly convex bounded initial surface M0, there exists a unique θ * such that the surface shrinks or expands depending on whether θ < θ * or θ > θ * . Also, we show that the surface with θ = θ * converges to a limiting surface.Mathematics Subject Classification. 35K57, 58E15.

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Models of Phase Transitions

Abstract: Models of phase transitions / Augusto Visintin. p.cm. --(Progress in nonlinear differential equations and their applications: v. 28)Includes bibliographical references and indes.

Cited by 250 publications

References 0 publications

Instability of gravity wetting fronts for Richards equations with hysteresis

Instability of gravity wetting fronts for Richards equations with hysteresis

Analytic solutions for a Stefan problem with Gibbs-Thomson correction

Mean curvature flow with a constant forcing

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