  The aim of this paper is to set up appropriate uniform convergence spaces in
which to reformulate and enrich the Order Completion Method for nonlinear 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, as well as in the theory of nonlinear PDEs. We
define a uniform convergence structure on ML(X) such that the induced
convergence structure is the order convergence structure. The uniform
convergence space completion of ML(X) is constructed as the set of normal lower
semi-continuous functions. It is then shown how these ideas 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.
