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In this paper, we consider the Cauchy problem for the incompressibleNavier-Stokes equations in $\mathbb{R}^n$ for $n\geq 3 $ with smooth periodicinitial data and derive a priori estimtes of the maximum norm of allderivatives 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 alsogeneralizes the main result of their paper to higher dimension.

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