The uniform order convergence structure on \(\mathcal {ML} (X)\). (English) Zbl 1145.54003

van der Walt, J. H.

Quaest. Math. 31, No. 1, 55-77 (2008).

If \(X\) is a topological space and \(u: X\to\overline{\mathbb{R}}\) is an extended real-valued function, one defines \(I(u), S(u): X\to\overline{\mathbb{R}}\) by \[ I(u)(x)= \sup\{\text{inf}\{u(y): y\in U\}: U\in{\mathcal U}(x)\} \] and \[ S(u)(x)= \text{inf}\{\sup\{u(y): y\in U\}: U\in{\mathcal U}(x)\}, \] where \({\mathcal U}(x)\) denotes the neighbourhood filter of \(x\). The function \(u\) is called normal lower-semicontinuous if \(I(S(u))= u\) and nearly finite if \(\{x\in X: u(x)\in\mathbb{R}\}\) is open and dense in \(X\). The set of all nearly finite, normal lower-semicontinuous functions on \(X\) is denoted by \({\mathcal N}{\mathcal L}(X)\) and \({\mathcal M}{\mathcal L}(X)\) denotes the set of all \(u\in{\mathcal N}{\mathcal L}(X)\) which are real-valued and continuous when restricted to the complement of a closed nowhere dense subset.
The author defines a uniform convergence structure on \({\mathcal N}{\mathcal L}(X)\). If \({\mathcal M}{\mathcal L}(X)\) carries the subspace uniform convergence structure, this structure induces the order convergence. He also proves that \({\mathcal N}{\mathcal L}(X)\) is the completion of \({\mathcal M}{\mathcal L}(X)\).
In the second part of the paper he uses the completion of the quotient of a suitable subspace to \({\mathcal M}{\mathcal L}(X)\) in order to construct generalized solutions of the Navier-Stokes euqations in three dimensions subject to an initial condition, which contain the classical ones.

Reviewer: Heinz-Peter Butzmann (Mannheim)

MSC:

54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
46E05	Lattices of continuous, differentiable or analytic functions
06F30	Ordered topological structures
35G25	Initial value problems for nonlinear higher-order PDEs

Keywords:

uniform convergence space; completion; nonlinear partial differential equations