Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

Abstract

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ${ℝ}_{+}^n$ for $n\ge 3$ with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in ${\mathbb{T}}^n$ and ${ℝ}^n$ respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.