Title

Regularity and Monotonicity for Solutions to a Continuum Model of Epitaxial Growth with Nonlocal Elastic Effects

Document Type

Article

Publication Date

4-16-2021

Abstract

We study the following parabolic nonlocal 4-th order degenerate equation: ut= – [2πH (ux) + ln(uxx + α) + 3/2(uxx + α)2]xx, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and α > 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 + α approaches zero. Thus we show that, if the initial datum u0 is such that (u0)xx + α 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.

Identifier

85104785488 (Scopus)

DOI

10.1515/acv-2020-0114

Publisher

De Gruyter

ISSN

18648258

Publication Information

Advances in Calculus of Variations

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