The uniform order convergence structure on ML (X)

Abstract

The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the order completion method [18] for nonlinear Partial Differential Equations (PDEs). In this regard, we consider an appropriate space ML(X) of normal lower semi-continuous functions. The space ML(X) appears in the ring theory of C (X) and its various extensions [13], as well as in the theory of nonlinear PDEs [18] and [20]. We define a uniform convergence structure on ML(X) such that the induced convergence structure is the order convergence structure introduced in [6] and [23]. The uniform convergence space completion of ML(X) is constructed as the space all normal lower semi-continuous functions on X. It is then shown how these results may be applied to solve nonlinear PDEs. In particular, we construct generalized solutions to the Navier-Stokes equations in three spatial dimensions, subject to an initial condition.

Keywords: General topology, uniform convergence structures, nonlinear PDEs, poset

Quaestiones Mathematicae 31(2008), 55–77