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

TL;DR

This paper derives maximum norm estimates for solutions of the Navier-Stokes equations with periodic initial data, extending previous results to higher dimensions and providing bounds on derivatives based on initial data norms.

Contribution

It provides a priori maximum norm estimates for all derivatives of Navier-Stokes solutions in higher dimensions with periodic initial data, generalizing prior work.

Findings

Maximum norm estimates for derivatives of solutions

Extension of results to higher dimensions

Generalization of previous estimates

Abstract

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^n$ for $n\geq 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.