| In this paper we consider the Cauchy problem for the 3D NS equations for incompressible flows, and their solutions. We will discuss the results of a paper by Otto Kreiss and Jens Lorenz on the role of the pressure term in the NS equations, and its relationship to the fluid field u( x,t). The focus here is to concentrate on solutions to the equation where the fluid field u lies in the space C infinity( R3 ) ∩ Linfinity( R3 ), and not necessarily in L2( R3 ). If u(x,0) = f( x), where f ∈ Cinfinity ( R3 ) ∩ Linfinity( R3 ) we will consider the solutions for all t in time interval 0 ≤ t < T(f). In the original paper, estimates for the derivatives of the pressure were proved, but the definition of the pressure proved unsatisfactory due to the possibility of the divergence of the pressure term. The main object of this paper is to use the theory of singular integrals and the space of functions of BMO to properly address the pressure. In doing so, we will provide estimates on pressure term itself. This will allow us to strengthen the results of the original paper, and rigorously extend all results from the original paper to any function u ∈ Cinfinity( R3 ) ∩ Linfinity( R3 ). |
