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Faceted Crystals Grown from Solution A Stefan Type Problem with a Singular Interfacial Energy

Faceted Crystals Grown from Solution A Stefan Type Problem with a Singular Interfacial Energy. Yoshikazu Giga University of Tokyo and Hokkaido University COE. Joint work with Piotr Rybka December , 2005 Lyon.

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Faceted Crystals Grown from Solution A Stefan Type Problem with a Singular Interfacial Energy

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  1. Faceted Crystals Grown from Solution A Stefan Type Problem with a Singular Interfacial Energy Yoshikazu Giga University of Tokyo and Hokkaido University COE Joint work with Piotr Rybka December , 2005 Lyon

  2. A basic problem from pattern formation in the theory of crystal growth.In what situation a flat portion (a FACET) of crystal surface breaks or not ? Goal : We shall prove : ‘All facets are stable near equilibrium for a cylindrical crystal by analysizing a Stefan type problem’

  3. Contents 1 Model 2 Problem 3 Main mathematical results 4 Three ingredients － ODE analysis － Berg’s effect － Facet splitting criteria－ 5 Open problems

  4. 1 Model Crystals grown from vapor (snow crystal) from solution (NaCl crystal) <driving force : supersaturation> (density of atoms outside crystal is small)

  5. Stefan like Model

  6. unnormalized version :

  7. We shall consider (1)－(5) for given quasi-stationary One phase Stefan problem with Gibbs-Thomson + kinetic effect

  8. Solvability (smooth ) K. Deckelnik - C. Elliott ’99 ( Hele Shaw type ) No …Friedman –Hu ’92 Liu – Yuan ’94

  9. Others (No ) Kuroda-Irisawa-Ookawa ‘77 Stability of facets Experiment e.g. Gonda-Gomi ’85 (No ) : Fingering : Saffman-Taylor R.Almgrem ’95

  10. 2.Problem (specific to ours)

  11. 3. Main Math Results

  12. Th (Rybka-G ‘04) If is close to the Equilibrium then the solution solves the original problem (1),(2),(3), (4),(5),(6),(7) Near equilibrium Facet does not break.

  13. Reduction to ODE

  14. Near equilibrium : close to zero / bounded away from zero

  15. 5. Open problems • Existence of solution of the • Original • problem is widly open if • is not near • equilibrium • (Even if is given • M.-H. Giga – Y. Giga ’98 • graphs) • ( : constant • M.-H. Giga – Y. Giga ‘01 • level set approach : unique • existence of generalized sol • (2-D)) • Uniqueness of the solution of • the original problem • (Sol is unique for • Reduced problems)

  16. All my preprints are in Hokkaido University Preprint Series on Math. http:coe.math.sci.hokudai.ac.jp

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