A priori estimates in terms of the maximum norm for the solution of the Navier-Stokes equations for the periodic initial data

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations in ${\mathbb{R}}^n$ for $n\ge 3$ with smooth periodic initial data and derive a priori estimtes of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a special case of a paper by H-O Kreiss and J. Lorenz which also generalizes the main result of their paper to higher dimension.