ON THE EXISTENCE AND UNIQUENESS OF THE ‘WEAK’ SOLUTION TO THE SIXTH PROBLEM OF THE MILLENNIUM

Joe Hlomuka

Received January 6, 2010

Abstract

In the follow-up to our analysis of the non-homogeneous case of the problem in [3], in this paper, we present the analysis of the homogeneous case, leading to the existence and uniqueness of the solution to Option A in the statement of the problem as it appears in [2]. As in [3], our analysis occurs in the space,  In this space, we select appropriate test functions with a compact support in the open bounded domain W. Our analysis differs to the one proposed by Ladyzhenskaya in [7], in the selection of some function spaces. Obviously, our test functions are ‘candidates’ for a ‘weak’ solution (in the sense of distributions), to the problem. Using ‘energy methods’, and some results from our previous research papers, we proceed to confirm the existence and uniqueness of the ‘weak’ solution to Option A of the problem as communicated in [2].

Keywords and phrases: sixth problem, millennium, existence and uniqueness, solution.