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

Abstract The surface growth model, u t + u x x x x + ∂ x x 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 ∈ L q ′ L q ( Q ) is satisfied, where q , q ′ ∈ [ 1 , ∞ ] are such that either 1 / q + 4 / q ′ 1 or 1 / q + 4 / q ′ = 1 , q ′ ∞ .