Convergence Rates of the Allen--Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies

Fischer, Julian; Laux, Tim; Simon, Theresa M.

doi:10.1137/20m1322182

Cited by 26 publications

(42 citation statements)

References 11 publications

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“…Due to the concentration of ∇ Q ε near the interface I t , this estimate is not sufficient to derive the convergence of Q ε . Following a recent work of Fisher et al [12] we shall develop in this section a calibrated inequality, which modulates the surface energy.…”

Section: The Modulated Energy Inequalitymentioning

confidence: 99%

“…The following result was first proved in [12] in the case of the Allen-Cahn equation, and can be generalized to the vectorial case:…”

Section: The Modulated Energy Inequalitymentioning

confidence: 99%

“…The proof consists of two key steps: (i) an adaptation of the modulated energy inequality in [12] to the vector-valued case to control the leading-order energy contribution, which is of order O(1) and comes from the phase transition across ∂ + (t). (ii) A version of Chen-Shatah's wedge-product trick in the sense that (1.13) implies…”

Section: Introductionmentioning

confidence: 99%

“…where [•, •] denotes the commutator. In (i) we basically follow [12] but need to carefully regularize the metric d F on Q induced by the conformal structure F(Q) in order to exploit the fine properties of its derivative ∇ q d F ε . In particular, we will use the crucial commutator relation ∇ q d F ε (Q ε ), Q ε = 0 for a.e.…”

Section: Introductionmentioning

confidence: 99%

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Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Laux

Liu

2021

Arch Rational Mech Anal

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In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

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Section: The Modulated Energy Inequalitymentioning

confidence: 99%

“…The following result was first proved in [12] in the case of the Allen-Cahn equation, and can be generalized to the vectorial case:…”

Section: The Modulated Energy Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Laux

Liu

2021

Arch Rational Mech Anal

Self Cite

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“…To list a few, we refer the readers to [17] for the scaling limit of particle system, to [13,14] for the semiclassical limit of Gross-Pitaevskii equation, and to [3,10] for the hydrodynamic limit of Boltzmann equation. For the applications of such a method to interface problems, we mention [12,8,9,11].…”

Section: Introductionmentioning

confidence: 99%

The sharp interface limit of a Navier--Stokes/Allen--Cahn system with constant mobility: Convergence rates by a relative energy approach

Hensel¹,

Liu²

2022

Preprint

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We investigate the sharp interface limit of a diffusive interface system that couples the Allen-Cahn equation with the instationary Stokes system in a bounded domain in R 2 . This model is used to describe a propagating front in a viscous incompressible flow with the width of the transition layer being characterized by a small parameter ε > 0. For well-prepared initial data, we show that the solution converges to a limit system that couples the curve-shortening flow and the Stokes system.

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Sharp Interface Limit for a Stokes/Allen–Cahn System

Abels

Liu

2018

Arch Rational Mech Anal

111

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We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter ε > 0 that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen-Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen-Cahn operator. Moreover, a careful treatment of the coupling terms is needed. Mathematics Subject Classification (2000): Primary: 76T99; Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45 Key words: Two-phase flow, diffuse interface model, sharp interface limit, Allen-Cahn equation, Free boundary problems 1 INTRODUCTION, MAIN RESULT, AND OVERVIEW is the jump of a function u : Ω × [0, T0] → R 2 at Γt in direction of nΓ t , HΓ t and VΓ t are the curvature and the normal velocity of Γt, both with respect to nΓ t . Furthermore, Dv = 1 2 (∇v + (∇v) T ) and σ = R θ ′ 0 (ρ) 2 dρ, where θ0 is the so-called optimal profile that is the unique solution ofIf the material derivative ∂tvε +vε ·∇vε is added to the right-hand side of (1.5) (i.e., the Navier-Stokes equations are considered), the system (1.5)-(1.9) was already suggested by Liu and Shen in [31] as an alternative approximation of a classical sharp interface model for a two-phase flow of viscous, incompressible, Newtonian fluids, which has advantages for numerical simulations since the Allen-Cahn equation is of second order and not of fourth order as the Cahn-Hilliard equation. On the other hand, for solutions of (1.5)-(1.9) the total mass Ω cε(x, t)dx is in general not preserved in time, in contrast to solutions of (1.1)-(1.4), which is a disadvantage if the model is used to approximate a two-phase flow without phase transitions. However, (1.5)-(1.9) can be considered as a simplified model for a two-phase flow with phase transitions. Such models can yield systems of Navier-Stokes/Allen-Cahn type, cf. e.g. Blesgen [11]. Finally, let us mention that in [31] a rigorous result on the sharp interface limes of (1.5)-(1.9) was announced, which was not published so far to the best of the author's knowledge.The limit system (1.11)-(1.16) was also studied by Liu, Sato, and Tonegawa in [30] if the Stokes equation on the right-hand side is replaced by a modified Navier-Stokes equation for a shear thickening non-Newtonian fluid of power-law type. They constructed weak solutions for this system using a Galerkin approximation by a corresponding Navier-Stokes/Allen-Cahn system. In the proof they pass to the limit in the Ga...

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Convergence Rates of the Allen--Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies

Cited by 26 publications

References 11 publications

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

The sharp interface limit of a Navier--Stokes/Allen--Cahn system with constant mobility: Convergence rates by a relative energy approach

Sharp Interface Limit for a Stokes/Allen–Cahn System

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