Spaces of Solution of the N–S Equations

Abstract

In this chapter we prove that if each component of the vector u on the right-hand side of (1.11) is divergence-free and belongs to the space of functions A _α, d, T ³, then the same is true of the operation N u on the right-hand side of (1.11). We also derive some bilinear form expressions for the operation N u thus paving the way for our proof of existence of a solution to (1.11). We then give precise conditions for convergence of successive approximations to the solution of (1.11) based on the contraction mapping principle.