March  2008, 9(2): 309-320. doi: 10.3934/dcdsb.2008.9.309

Quasi-static evolution of polyhedral crystals

1. 

Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland

Received  February 2007 Revised  October 2007 Published  December 2007

We examine quasi-static evolution of crystals in three dimensions. We assume that the Wulff shape is a prism with a hexagonal base. We include the Gibbs-Thomson law on the crystal surface and the so-called Stefan condition. We show local in time existence of solutions assuming that the initial crystal has admissible shape.
Citation: Przemysław Górka. Quasi-static evolution of polyhedral crystals. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 309-320. doi: 10.3934/dcdsb.2008.9.309
[1]

Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213

[2]

Weiqing Xie. A free boundary problem arising from the process of Czochralski crystal growth. Conference Publications, 2001, 2001 (Special) : 380-385. doi: 10.3934/proc.2001.2001.380

[3]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357

[4]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293

[5]

Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997

[6]

Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129

[7]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140

[8]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[9]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625

[10]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

[11]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281

[12]

Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure & Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845

[13]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741

[14]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[15]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[16]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[17]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655

[18]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337

[19]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44

[20]

Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]