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Parametric Lie group actions on global generalized solutions of nonlinear PDEs. Including a solution to Hilbert’s fifth problem. (English) Zbl 0934.35003
Mathematics and its Applications (Dordrecht). 452. Dordrecht: Kluwer Academic Publishers. xvii, 234 p. Dfl. 195.00; \$ 106.00; £67.00 (1998).

From the author’s presentation: “A novel approach to Lie group actions on usual and generalized functions, based on a parametric representation, is introduced. This allows, for the first time, an easy global definition of arbitrary nonlinear, including nonprojectable, Lie group actions on functions, as well as on generalized functions, and in particular, on the traditional Schwartz distributions. The present book is addressed to those who are interested in solving nonlinear PDEs, and in particular, in studying the Lie group symmetries of their classical or generalized solutions. Indeed, as it happens, quite basic nonlinear PDEs, such as for instance, the shock wave equation, have nonprojectable Lie group symmetries, as well as generalized solutions of physical interest, like the rarefaction waves and Riemann solvers.”

The first five chapters motivate, introduce and present basic applications of the parametric approach to global Lie group actions on smooth functions.

The extension of the parametric approach to arbitrary and global Lie group actions on generalized functions is studied in the 7th chapter with the aim to extend the arbitrary nonlinear Lie group actions to global generalized solutions of nonlinear PDEs.

One considers so called nowhere dense differential algebras of generalized functions, ${A}_{{\Omega }}$, ${\Omega }$ a nonvoid open set from ${ℝ}^{n}$.

The reasons to consider generalized functions from such algebras ${A}_{{\Omega }}$ as for example the algebra of Colombeau generalized functions, are mainly related to the fact that they enjoy a rather simple interplay between ring theoretic and topological type properties without however requiring any kind of growth type conditions, which in applications bring with them significant difficulties, and, also, to the fact that these algebras are the only ones known so far which have a flabby sheaf structure. This last property is fundamental in order to be able to accomodate the largest class of singularities which the generalized functions can have. Namely these singularities can be on arbitrary closed and nowhere dense subsets, subsets which can have arbitrary large positive Lebesgue measure and, in whose neighbourhood, the generalized functions are restricted in any way. The unique position of the algebras ${A}_{{\Omega }}$ among all other known differential algebras of generalized functions is described in some details in chapter 12.

The free use of the parametric approach is presented in the 9th chapter where, essentially, one shows that, as well as in the case of classical solutions, the action of the usual Lie group symmetries of nonlinear PDEs upon their generalized solutions will again give generalized solutions.

Chapters 6 and 8 apply the parametric approach to nonprojectable Lie group symmetries of the refaction waves and Riemann solvers of the nonlinear shock wave equation. The simplicity and the power of the application of the parametric approach is presented in section 3 of 9-th chapter with regard to some of complexities of the approach if trying not to depart to far from the usual nonparametric approach illustrated in chapter 8.

The extension of arbitrary Lie group actions to large classes of generalized functions allows to obtain an answer to the Hilbert’s fifth problem in its full generality which is presented in chapters 11 and 12.

The last chapter, 13, tries to motivate the researchers in Lie groups to pursue what the author appears to be the theory, or rather the future theory, of genuine Lie semigroups of actions as a more globally minded approach to large classes of singularities of classical smooth or nonsmooth generalized functions.

Finally, let underline the fact that most of the results from the book are personal author’s results and have been published starting from 1966 in 16 papers, much of them representing fundamental contributions as well as for the study of the algebras ${A}_{{\Omega }}$ and the Lie group actions on global generalized solutions of nonlinear PDEs.

##### MSC:
 35-02 Research monographs (partial differential equations) 46F30 Generalized functions for nonlinear analysis 58-02 Research monographs (global analysis) 58J70 Invariance and symmetry properties