Partial regularity for a surface growth model

TL;DR

This paper establishes partial regularity results for a surface growth model, showing the singular set has limited size, and draws parallels with the Navier--Stokes equations, advancing understanding of their mathematical similarities.

Contribution

It proves the first partial regularity results for the surface growth model, a lower-dimensional analogue of Navier--Stokes, including novel nonlinear parabolic Poincaré inequalities.

Findings

Singular set has upper box-counting dimension ≤ 7/6

Singular set has parabolic Hausdorff measure zero

Results highlight similarities between surface growth model and Navier--Stokes equations

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 (1998) and Ladyzhenskaya & Seregin (1999), we develop certain nonlinear parabolic Poincar\'e inequality, which is a concept of independent interest. We believe that similar inequalities could be applicable in other parabolic equations.