Ozanski, Wojciech and Robinson, James C. (2019) Partial regularity for a surface growth model. SIAM Journal of Mathematical Analysis, 51 (1). pp. 228-255. doi:10.1137/18M1166821 ISSN 0036-1410.

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Abstract

We prove two partial regularity results for the scalar equation $u_t+u_{xxxx}+\partial_{xx}u_x^2=0$, a model of surface growth arising from the physical process of molecular epitaxy. We show that the set of space-time singularities has (upper) box-counting dimension no larger than 7/6 and 1-dimensional (parabolic) Hausdorff measure zero. These parallel the results available for the three-dimensional Navier--Stokes equations. In fact the mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier--Stokes equations, and the partial regularity results are the next step towards understanding this remarkable similarity. As far as we know the surface growth model is the only lower-dimensional “mini-model” of the Navier--Stokes equations for which such an analogue of the partial regularity theory has been proved. In the course of our proof, which is inspired by the rescaling analysis of Lin [Comm. Pure Appl. Math., 51(1998), pp. 241--257] and Ladyzhenskaya and Seregin [J. Math. Fluid Mech., 1(1999), pp. 356--387], we develop certain nonlinear parabolic Poincaré inequality, which is a concept of independent interest. We believe that similar inequalities could be applicable in other parabolic equations.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	SIAM Journal of Mathematical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Official Date:	5 February 2019
Dates:	Date Event 5 February 2019 Available 13 November 2018 Accepted
Volume:	51
Number:	1
Page Range:	pp. 228-255
DOI:	10.1137/18M1166821
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	14 November 2018
Date of first compliant Open Access:	24 April 2019
Related URLs:	Publisher
Open Access Version:	ArXiv

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