On the higher derivatives estimate of the surface growth equation

In this paper, we are concerned with the higher derivatives estimate of the surface growth equation which has similar features to the 3D Navier-Stokes equations. We extend the well-known apriori estimates for the Navier-Stokes equations due to Foias-Guillope-Temam in C. Foias et al. (1981) to this model. Several new sufficient conditions that guarantee regularity of solutions to the generalized surface growth equation are also presented. As a byproduct, the global well-posedness of the generalized surface growth equation in the sense of Lions's classical work in J.L. Lions (1969) is obtained.(c) 2022 Published by Elsevier Ltd.