The uniform order convergence structure on <i>ML</i> (<i>X</i>)

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