Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs: Including a Solution to Hilbert’s Fifth Problem

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This text presents global actions of arbitrary Lie groups on large classes of generalized functions by using a parametric approach. This method extends and completes earlier results of the author and collaborators, in which global Lie group actions on generalised functions were only defined in the case of projectable or fibre-preserving Lie group actions. The parametric method opens the possibility of dealing with vastly larger classes of Lie semigroup actions which still transform solutions into solutions. These Lie semigroups can contain arbitrary noninvertible smooth mappings. Thus, they cannot be sub-semigroups of Lie groups.