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Full Description
This book presents a solution of the harder part of the problem of defining globally arbitrary Lie group actions on such nonsmooth entities as generalised functions. Earlier, in part 3 of Oberguggenberger & Rosinger, Lie group actions were defined globally - in the projectable case - on the nowhere dense differential algebras of generalised functions An, as well as on the Colombeau algebras of generalised functions, and also on the spaces obtained through the order completion of smooth functions, spaces which contain the solutions of arbitrary continuous nonlinear PDEs. Further details can be found in Rosinger & Rudolph, and Rosinger & Walus [1,2]. To the extent that arbitrary Lie group actions are now defined on such nonsmooth entities as generalised functions, this result can be seen as giving an ans wer to Hilbert's fifth problem, when this problem is interpreted in its original full gener- ality, see for details chapter 11.
Contents
1. Introduction.- 2. Actions on functions, difficulties.- 3. Parametric representation of functions.- 4. Action on parametric representations.- 5. Parametric functions as solutions.- 6. Rarefaction waves and Riemann solvers of the nonlinear shock wave equation.- 7. Arbitrary nonlinear Lie group actions on generalised functions.- 8. Nonprojectable Lie group symmetries of rarefaction waves and Riemann solvers.- 9. General parametric approach to Lie symmetries.- 10. Projectable Lie group actions and Hilbert's fifth problem.- 11. Nonprojectable Lie group actions and an answer to Hilbert's fifth problem.- 12. Singularities and the nowhere dense algebras of generalised functions.- 13. Lie semigroup actions and semisymmetries.
