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

The surface growth model, u(t) + u(xxxx) + partial derivative(xx)u(x)(2) = 0, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition u(x) epsilon L-q' L-q (Q) is satisfied, where q, q' epsilon [1, infinity] are such that either 1/q 4/q' < 1 or 1/q 4/q' = 1, q' < infinity. Crown Copyright (C) 2019 Published by Elsevier Inc. All rights reserved.