Affiliation:
1. Department of Mathematics, University of Utah Asia Campus, 119-3 Songdo Moonhwa-Ro Yeonsu-Gu, Incheon, 21985, Republic of Korea
Abstract
In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in
for
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
and
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.
Subject
Applied Mathematics,Analysis