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

Abstract

The surface growth model, ut +uxxxx +∂xxu 2 x = 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 ux ∈ 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 ′ < ∞.

On regularity properties of a surface growth model

“…• Ożański and Robinson [16] showed that smallness of I implies Hölder continuity of v (rather than of v x ), which is not sufficient to further bootstrap the regularity of v and exhibits a mismatch between an assumption involving v x and a result for v. Part (ii) fills this gap, as (1.13) guarantees smoothness. Indeed, using for example the regularity condition of [15], one has Corollary 1.7. Under assumptions of theorem…”

“…For more applicational motivations see [11,11,18,19,23], and for certain stochastic aspects see [1][2][3][4]7]. The analytical results for the SGM share striking similarities with the 3D incompressible Navier-Stokes equations, which has been explored in the recent years by the authors of [5,6,15,16,24].…”