Well-posedness of a needle crystal growth problem with anisotropic surface tension

References

• Caffarelli , L.A. and Evans , L.C. 1983 . Continuity of the temperature in the two-phase Stefan problem . Archive for Rational Mechanics and Analysis. , 81 : 199 – 220 .
   Web of Science ®Google Scholar
• Friedman , A. and Reitich , F. 1991 . The Stefan problem with small surface tension . Transactions of the American Mathematical Society. , 328 : 465 – 515 .
   Web of Science ®Google Scholar
• Luckhaus , S. 1990 . Solutions for the two-phase Stefan problem with the Gibbs-Thompson Law for the melting temperature . European Journal of Applied Mathematics. , 1 : 101 – 111 .
   Google Scholar
• Escher , J. , Pruss , J. and Simonett , G. 2003 . Analytic solutions for a stefan problem with Gibbs-Thompson correction . Journal fur die reine und angewandte Mathematik. , 563 : 1 – 52 .
   Web of Science ®Google Scholar
• Hadzic , M. and Guo , Y. 2010 . Stability in the Stefan problem with surface tension (I) . Communications in Partial Differential Equations. , 35 ( 2 ) : 201 – 244 .
   Web of Science ®Google Scholar
• Mullins , W.W. and Sekerka , R.F. 1963 . Morphological stability of a particle growing by diffusion of heat flow . Journal of Applied Physics. , 34 : 323 – 328 .
   Web of Science ®Google Scholar
• Langer , J.S. 1987 . In: Souletie J, editor , Chance and matter. Amsterdam : North-Holland .
   Google Scholar
• Kessler , D. , Koplik , J. and Levine , H. 1988 . Patterned selection in Fingered Growth Phenomena . Advances in Physics. , 37 : 255
   Web of Science ®Google Scholar
• 1988 . Dynamics of curved fronts , New York : Academic .
   Google Scholar
• Brener , E.A. and Mel’nikov , V.I. 1991 . Pattern selection in two-dimensional dendritic growth . Advances in Physics. , 40 ( 1 ) : 53 – 97 .
   Web of Science ®Google Scholar
• Ball , J.M. , Kinderlehrer , D. , Podio-Guidugli , P. and Slemrod , M. 1999 . Evolving phase interfaces in solids , New York (NY) : Springer .
   Google Scholar
• Davis SH. Theory of solidification. Cambridge University Press; 2000.
   Google Scholar
• Xu JJ. Dynamical theory of dendritic growth in convective flow. Kluwer Academic; 2003.
   Google Scholar
• Kunka , M.D. , Foster , M.R. and Tanveer , S. 1996 . Dendritic crystal growth for weak undercooling . Physical Review E. , 56 : 3068 – 3100 .
   Web of Science ®Google Scholar
• Kunka MD, Foster MR, Tanveer S. Dendritic crystal growth for weak undercooling. II. Surface energy effects on nonlinear evolution. Physical Review E. 1999;59:673–710.
   Google Scholar
• Howison , S.D. 1992 . Complex variable methods in a Hele-Shaw moving boundary problems . European Journal of Applied Mathematics. , 3 : 209 – 224 .
   Google Scholar
• Tanveer , S. and Xie , X. 2003 . Analyticity and nonexistence of classical steady Hele-Shaw Fingers . Communications on Pure and Applied Mathematics. , 56 : 353 – 402 .
   Web of Science ®Google Scholar
• Xie , X. and Tanveer , S. 2003 . Rigorous results in steady finger selection in viscous fingering . Archive for Rational Mechanics and Analysis. , 166 : 219 – 286 .
   Web of Science ®Google Scholar
• Xie , X. 2004 . Rigorous results in selection of steady needle crystals . Journal of Differential Equations. , 197 : 349 – 426 .
   Web of Science ®Google Scholar
• Xie , X. 2005 . Analyticity of classical steady needle crystal . Journal of Differential Equations. , 216 : 1 – 31 .
   Web of Science ®Google Scholar
• Xie , X. 2007 . Existence and uniqueness of analytic solution for unsteady crystals with zero surface tension . Journal of Mathematical Analysis and Applications. , 327 : 1307 – 1319 .
   Web of Science ®Google Scholar
• Xie X. On linear stability of steady needle crystals. New York (NY): Nova; 2008. Chapter 25. Discrete and computational mathematics; p. 355.
   Google Scholar
• Xie , X. 2007 . A note on the linear stability of steady needle crystals with no surface tension . Applicable Analysis. , 86 : 405 – 410 .
   Web of Science ®Google Scholar
• Chen , X. , Hong , J. and Yi , F. 1996 . Existence, uniqueness and regularity of classical solutions of the Mullin-Sekerka problem . Communications in Partial Differential Equations. , 21 ( 11–12 ) : 1705 – 1727 .
   Web of Science ®Google Scholar
• Chen , X. 1993 . The Hele-Shaw problem and area-preserving curve-shortening motions . Archive for Rational Mechanics and Analysis. , 123 : 117 – 151 .
   Web of Science ®Google Scholar
• Kim , I. 2003 . Uniqueness and existence results on the Hele-Shaw and the Stefan problems . Archive for Rational Mechanics and Analysis. , 168 : 299 – 328 .
   Web of Science ®Google Scholar
• Kim , I. and Pozar , N. 2011 . Viscosity solutions for the two-phase Stefan problem . Communications in Partial Differential Equations. , 36 ( 1 ) : 42 – 66 .
   Web of Science ®Google Scholar
• Almgren , F. and Wang , L. 2000 . Mathematical existence of crystal growth with Gibbs-Thomson curvature effects . Journal of Geometric Analysis. , 10 : 1 – 100 .
   Google Scholar
• Duchon , J. and Robert , R. 1984 . Evolution d’une interface par capillarite’ t difusion de volume . Annales de l’Institute Henri Poincare. , 1 : 361 – 378 .
   Web of Science ®Google Scholar
• Constantin , P. and Pugh , M. 1993 . Global solutions for small data to Hele-Shaw problem . Nonlinearity. , 6 : 393
   Web of Science ®Google Scholar
• Escher , J. and Simonett , G. 1997 . Classical solutions for Hele-Shaw models with surface tension . Advances in Differential Equations. , 2 : 619 – 642 .
   Google Scholar
• Prokert , G. 1998 . Existence result for Hele-Shaw flow driven by surface tension . European Journal of Applied Mathematics. , 9 : 195 – 221 .
   Web of Science ®Google Scholar
• Gunther , M. and Prokert , G. 2005 . On a Hele-Shaw-Type domain evolution with convected surface energy density . SIAM Journal on Mathematical Analysis. , 37 : 372 – 410 .
   Web of Science ®Google Scholar
• Xie X. Local smoothing effect and existence for the one-phase Hele-Shaw problem with surface tension. Journal of Nonlinear Systems and Applications.
   Google Scholar
• Caginalp G. Length scales in phase transition models: phase field, Cahn-Hilliard and blow-up problems. Pattern formation: symmetry methods and applications; 1993; Waterloo, ON. Fields Institute Communications 5. Providence (RI): American Mathematical Society; 1996. p. 67–83.
   Google Scholar
• McFadden , G. , Wheeler , A. , Braun , R. and Coriel , S. 1993 . Phase-field models for anisotropic interfaces . Physical Review E. , 48 : 2016
   PubMed Web of Science ®Google Scholar
• Carrier , G.F. and Krook , M. 1966 . Pearson CE , Functions of a complex variable : McGraw-Hill .
   Google Scholar
• Taylor , M.E. 1996 . Partial differential equations I, II, III , New York (NY) : Springer-Verlag .
   Google Scholar
• Wu S, Wu S. Mathematical analysis of Vortex Sheets. Communications on Pure and Applied Mathematics. 2006;LIX:1–142.
   Google Scholar
• Ambrose D. Well-posedness of vortex sheets with surface tension. SIAM Journal on Mathematical Analysis. 35(1):211–44.
   Google Scholar
• Kepler J. De Nive Sexangula. Frankfurt am Main; 1611.
   Google Scholar