Decay Rates of Solutions to the Surface Growth Equation and the Navier–Stokes System

By means of Guo and Wang’s pure energy method, we present the optimal decay rates of higher-order spatial derivatives of solutions to the surface growth model arising in epitaxial growth of monocrystals, under the smallness condition in scaling-invariant (critical) Sobolev space. We also obtain the optimal decay estimates of higher-order spatial derivatives of the 3D Navier–Stokes system with small initial data in critical Lebesgue space, which is larger than the critical Sobolev space in previous works. The proof relies on our new observation that the solutions of 3D Navier–Stokes equations keep the smallness in critical Lebesgue space as well as the critical Sobolev space.