Accessible Requires Authentication Pre-published online by De Gruyter April 16, 2021

Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

Yuan Gao ORCID logo, Xin Yang Lu and Chong Wang


We study the following parabolic nonlocal 4-th order degenerate equation:


arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a>0 is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when uxx+a approaches zero. Thus we show that, if the initial datum u0 is such that (u0)xx+a is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.

MSC 2010: 35K65; 35R06; 49J40

Communicated by Irene Fonseca

Funding statement: The second author is very grateful for the support of NSERC Discovery Grants and acknowledges the support of his Lakehead University internal funding.


[1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2008. Search in Google Scholar

[2] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. Search in Google Scholar

[3] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philos. Trans. Roy. Soc. London Ser. A 243 (1951), 299–358. Search in Google Scholar

[4] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Volume I: One Dimensional Theory, Academic Press, New York, 1971. Search in Google Scholar

[5] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma and Q.-K. Xue, Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator, Science 340 (2013), no. 6129, 167–170. Search in Google Scholar

[6] G. Dal Maso, I. Fonseca and G. Leoni, Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces, Arch. Ration. Mech. Anal. 212 (2014), no. 3, 1037–1064. Search in Google Scholar

[7] C. Duport, P. Politi and J. Villain, Growth instabilities induced by elasticity in a vicinal surface, J. Phys. I 5 (1995), no. 10, 1317–1350. Search in Google Scholar

[8] I. Fonseca, G. Leoni and X. Y. Lu, Regularity in time for weak solutions of a continuum model for epitaxial growth with elasticity on vicinal surfaces, Comm. Partial Differential Equations 40 (2015), no. 10, 1942–1957. Search in Google Scholar

[9] Y. Gao, Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity, J. Differential Equations 267 (2019), no. 7, 4429–4447. Search in Google Scholar

[10] Y. Gao, J.-G. Liu and J. Lu, Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, J. Nonlinear Sci. 27 (2017), no. 3, 873–926. Search in Google Scholar

[11] Y. Gao, J.-G. Liu and X. Y. Lu, Gradient flow approach to an exponential thin film equation: Global existence and latent singularity, ESAIM Control Optim. Calc. Var. 25 (2019), Paper No. 49. Search in Google Scholar

[12] Y. Gao, J.-G. Liu, X. Y. Lu and X. Xu, Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 55. Search in Google Scholar

[13] M.-H. Giga and Y. Giga, Very singular diffusion equations: Second and fourth order problems, Jpn. J. Ind. Appl. Math. 27 (2010), no. 3, 323–345. Search in Google Scholar

[14] Y. Giga and R. V. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst. 30 (2011), no. 2, 509–535. Search in Google Scholar

[15] J.-G. Liu and X. Xu, Existence theorems for a multidimensional crystal surface model, SIAM J. Math. Anal. 48 (2016), no. 6, 3667–3687. Search in Google Scholar

[16] J.-G. Liu and X. Xu, Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions, SIAM J. Math. Anal. 49 (2017), no. 3, 2220–2245. Search in Google Scholar

[17] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Phys. Rev. B 42 (1990), no. 8, Article ID 5013. Search in Google Scholar

[18] A. Pimpinelli and J. Villain, Physics of Crystal Growth. Vol. 19, Cambridge university, Cambridge, 1998. Search in Google Scholar

[19] L.-H. Tang, Flattening of grooves: Ffrom step dynamics to continuum theory, Dynamics of Crystal Surfaces and Interfaces, Springer, Boston (2002), 169–184. Search in Google Scholar

[20] J. Tersoff, Y. H. Phang, Z. Zhang and M. G. Lagally, Step-bunching instability of vicinal surfaces under stress, Phys. Rev. Lett. 75 (1995), no. 14, Article ID 2730. Search in Google Scholar

[21] E. Weinan and N. K. Yip, Continuum theory of expitaxial crystal growth. I, J. Statist. Phys. 104 (2001), no. 1–2, 211–253. Search in Google Scholar

[22] Y. Xiang, Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface, SIAM J. Appl. Math. 63 (2002), no. 1, 241–258. Search in Google Scholar

[23] Y. Xiang and E. Weinan, Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces, Phys. Rev. B. 69 (2004), no. 3, Article ID 035409. Search in Google Scholar

[24] X. Xu, Existence theorems for a crystal surface model involving the p-Laplace operator, SIAM J. Math. Anal. 50 (2018), no. 4, 4261–4281. Search in Google Scholar

Received: 2020-04-06
Revised: 2021-02-28
Accepted: 2021-03-02
Published Online: 2021-04-16

© 2021 Walter de Gruyter GmbH, Berlin/Boston