Partial Regularity for a Surface Growth Model

Abstract

We prove two partial regularity results for the scalar equation ut+uxxxx+∂xxu 2 x = 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 (1998) and Ladyzhenskaya & Seregin (1999), 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.

A sufficient integral condition for local regularity of solutions to the surface growth model

“…Such an extension can be proved in the same way as (1.4) (see Theorem 3.1 in Ożański & Robinson (2017)). Thus if the condition (1.6) is satisfied with 1/q + 4/q ′ = 1 − ε, then an application of Hölder's inequality to (1.7) together with the Campanato lemma give ε-Hölder continuity of u in U × (t 1 , t 2 ).…”

“…The 2012 paper proves local existence in a critical space of a similar type to that occurring in the paper by Koch & Tataru (2001) for the NSE. Very recently Ożański & Robinson (2017) developed a partial regularity theory for the surface growth model, an analogue of the celebrated Caffarelli et al (1982) theorem for the NSE. The central result of this theory is that there exists ε 0 > 0 such any "suitable" weak solution of the surface growth model (see Definition 2.5) satisfying…”

“…is Hölder continuous (with any exponent α ∈ (0, 1)) in Q(z, r/2) (see Theorem 4.5 in Ożański & Robinson (2017)). Here Q(z, r) denotes a space-time cylinder of radius r centred at z and "suitable" refers to weak solutions satisfying a local form of an energy inequality, Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK w.s.ozanski@warwick.ac.uk see Ożański & Robinson (2017) for precise definitions. A straightforward consequence of this result is that one can estimate the dimension of the singular set S := {(x, t) ∈ T × [0, ∞) : u is not space-time Hölder continuous in any neighbourhood of (x, t)}.…”

“…Interestingly, it seems that the proof of the partial regularity theory cannot be obtained by the method of Caffarelli et al (1982); instead Ożański & Robinson (2017) used the approach of Lin (1998) and Ladyzhenskaya & Seregin (1999); the analysis there also required a novel tool, namely a nonlinear parabolic Poincaré inequality, and u z,r/2 denotes the mean of u over Q(z, r/2).…”

“…This is related to the fact that the lowest spatial derivative of u involved in the nonlinear term in the surface growth model (1.1) is u x (rather than the 0-th derivative, which is the case in the NSE). Note that the condition for local regularity from Ożański & Robinson (2017) is also stated in terms of u x (see (1.2)). Therefore, our main result is a next step towards understanding the remarkable similarity between the SGM and the NSE.…”