2007/08/21 by Jan Harm van der Walt, van der Walt, Jan Harm
Mathematics · #06F30 #46E05 #54A20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #General Mathematics (math.GM) #General Topology (math.GN) #Iterative Methods for Nonlinear Equations #Mathematical and Theoretical Analysis #math.AP #math.GM #math.GN #msc:06F30 #msc:46E05 #msc:54A20
paper · pdf · doi:10.48550/arxiv.0708.2785
32 pages
openalex publication_date 2007/08/21 · arxiv created 2007/11/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.