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2024 | OriginalPaper | Chapter

Connecting the Deep Quench Obstacle Problem with Surface Diffusion via Their Steady States

Authors : Eric A. Carlen, Amy Novick-Cohen, Lydia Peres Hari

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

The chapter focuses on the Deep Quench Obstacle Problem (DQOP) and its connection with motion by surface diffusion (SD). It discusses the steady states of both models, emphasizing minimum energy configurations and the conditions under which they exist. The text also explores the dynamics of phase separation and the challenges in mathematically justifying the transition between diffuse and sharp interface descriptions. Additionally, it outlines tools for bridging the two evolutions and highlights the attractor dynamics for both DQOP and SD, suggesting similar stability properties. The chapter concludes with a discussion on minimizing motion evolution formulations and the potential for a rigorous connection between the two evolutions.

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previous chapter Intemperate Lévy Processes and White Noises

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Here E(t) has been scaled so that typically as equilibrium is approached, \(E(t) \propto \frac{{L}(t)}{|\varOmega |}\), where L(t) reflects the length of the interface of between the two phases following phase separation; see [2].

Here \(\text {Ent}(t)\) reflects the physical entropy of the system, while E(t) is a (scaled) free energy.

If \(v_q\) vanishes at some point \(\bar{q} \in (q_-, q_+)\), then Sturm’s Separation Theorem [11] implies that v vanishes at least once in each of the intervals \([q_-, \bar{q}]\), \([\bar{q}, q_+],\) and hence v cannot be monotone.

Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

Baňas, L., Novick-Cohen, A., Nürnberg, R.: The degenerate and non-degenerate deep quench obstacle problem: a numerical comparison. Netw. Heterog. Media 8, 37–64 (2013)MathSciNetCrossRef

Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine. Angew. Math. 384, 153–179 (1988)MathSciNet

Carlen, E.A., Carvalho, M.C., Orlandi, E.: Approximate solutions of the Cahn-Hilliard equation via corrections to the Mullins-Sekerka motion. Arch. Ratio. Mech. Anal. 178, 1–55 (2005)MathSciNetCrossRef

Chen, X.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262–311 (1996)MathSciNetCrossRef

Cahn, J.W., Elliott, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7, 287–301 (1996)

de Figueiredo, D.G., dos Santos, E.M., Miyagaki, O.H.: Sobolev spaces of symmetric functions and applications. J. Funct. Anal. 261, 3735–3770 (2011)MathSciNetCrossRef

Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Part. Differ. 34, 193–231 (2009)MathSciNetCrossRef

Elliott, C.M., Luckhaus, S.: A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy. University of Minnesota, IMA Preprint 887 (1991)

10.

Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ratio. Mech. Anal. 205, 425–466 (2012)MathSciNetCrossRef

11.

Hartman, P.: Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia (2002)

12.

Jerrard, R.L., Sternberg, P.: Critical points via \(\Gamma \)-convergence: general theory and applications. J. Eur. Math. Soc. (JEMS) 11, 705–753 (2009)MathSciNetCrossRef

13.

Kohn, R.V., Otto, F.: Upper bounds on coarsening rates. Comm. Math. Phys. 229, 375–395 (2002)MathSciNetCrossRef

14.

Lisini, S., Matthes, D., Savaré, G.: Cahn-Hilliard and thin film equation with nonlinear mobility as gradient flows in weighted-Wasserstein metrics. J. Differ. Equ. 253, 814–850 (2012)MathSciNetCrossRef

15.

Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957)CrossRef

16.

Novick-Cohen, A.: Wetting and prewetting of anti-phase boundaries. Meccanica 40, 471–483 (2005)MathSciNetCrossRef

17.

Novick-Cohen, A.: Upper bounds for coarsening for the deep quench obstacle problem. J. Stat. Phys. 141, 142–157 (2010)MathSciNetCrossRef

18.

Novick-Cohen, A., Shishkov, A.: Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discret. Contin. Dyn. Syst. 25, 251–272 (2009)MathSciNetCrossRef

19.

Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A. (eds.): NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.1.10 of 2023-06-15. (see also: [1])

20.

Oono, Y., Puri, S.: Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling. Phys. Rev. A 38, 434–453 (1988)

21.

Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ratio. Mech. Anal. 101, 209–260 (1988)MathSciNetCrossRef

22.

Taylor, J.E., Cahn, J.W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77, 183–197 (1994)MathSciNetCrossRef

23.

Wheeler, G.: On the curve diffusion flow of closed plane curves. Ann. Mat. Pura Appl. 4(192), 931–950 (2013)MathSciNetCrossRef

Title: Connecting the Deep Quench Obstacle Problem with Surface Diffusion via Their Steady States
Authors: Eric A. Carlen
Amy Novick-Cohen
Lydia Peres Hari
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_11

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