PARTIAL REGULARITY FOR A SURFACE GROWTH MODEL

We prove two partial regularity results for the scalar equation u(t+)u(xxxx)+ &PARTIAL(xx)u(x)(2) =0, 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(1991), pp. 356-387], we develop certain nonlinear parabolic Poincare inequality, which is a concept of independent interest. We believe that similar inequalities could be applicable in other parabolic equations.