Faceted Crystals Grown from Solution A Stefan Type

A basic problem from pattern formation in theory of crystal growth. In what situation

Contents 1 Model 2 Problem 3 Main mathematical results 4 Three ingredients － ODE

1 Model Crystals grown from vapor (snow crystal) from solution (Na. Cl crystal) <driving

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

Solvability (smooth ) K. Deckelnik - C. Elliott ’ 99 ( Hele Shaw type

Others (No ) Kuroda-Irisawa. Ookawa ‘ 77 Stability of facets Experiment e. g. Gonda-Gomi

Th (Rybka-G ‘ 04) If is close to the Equilibrium then the solution solves

Near equilibrium : close to zero / bounded away from zero 27

5. Open problems • Existence of solution of the Original problem is widly open

All my preprints are in Hokkaido University Preprint Series on Math. http: coe. math.

Slides: 30

Download presentation

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 1

A basic problem from pattern formation in 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’ 2

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

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

Stefan like Model 5

unnormalized version : 7

We shall consider (1)－(5) for given quasi-stationary One phase Stefan problem with Gibbs. Thomson + kinetic effect 10

Solvability (smooth ) K. Deckelnik - C. Elliott ’ 99 ( Hele Shaw type ) No 　　… Friedman –Hu ’ 92 　　Liu – Yuan ’ 94 11

Others (No ) Kuroda-Irisawa. Ookawa ‘ 77 Stability of facets Experiment e. g. Gonda-Gomi ’ 85 (No ) : Fingering : Saffman-Taylor R. Almgrem ’ 95 12

2. Problem (specific to ours) 13

3. Main Math Results 19

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. 20

Reduction to ODE 21

Near equilibrium : close to zero / bounded away from zero 27

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) 28

All my preprints are in Hokkaido University Preprint Series on Math. http: coe. math. sci. hokudai. ac. jp 30