Differential algebras of generalised functions with dense singularities

Abstract

It is shown how chains of algebras of generalised functions may be used to construct algebras of generalised functions that are able to deal with larger classes of singularities than each of the constituent algebras in the chain. The general method is applied to a chain of almost everywhere algebras, yielding an algebra that can handle certain densely singular functions. The embedding of the distributions into the mentioned algebra, as well as the existence of solutions of nonlinear PDEs, is considered.

Keywords: Generalised functions, differential algebras, nonlinear PDEs