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In this paper, we reprove the principal result of a paper by H-O Kreiss andJens Lorenz from a different approach than the method proposed in their paper.More precisely, we consider the Cauchy problem for the incompressibleNavier-Stokes equations in $\mathbb{R}^n$ for $n \ge 3$ with non-decayinginitial data and derive a priori estimates of the maximum norm of allderivatives of the solution in terms of the maximum norm of the initial data.This paper is also an extension of their paper to higher dimension.

$L^\infty$-Estimates of the Solution of the Navier-Stokes Equations for a Nondecaying Initial Data