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Nonlinear partial differential equations. An algebraic view of generalized solutions. (English) Zbl 0717.35001

North-Holland Mathematics Studies, 164. Amsterdam etc.: North-Holland. xxi, 380 p. Dfl. 210.00 (1990).
This is an important book. In addition to providing a lot of fascinating mathematical information, there is an underpinning of revealing philosophy to establish the mathematical construction in a meaningful context. The level of discourse is extremely high, dealing with questions about the very nature of nonlinear partial differential equations (NLPDE) and their solutions in the context of suitable chains of differential algebras containing the Schwartz distributions \({\mathcal D}'.\)
One begins with the fundamental conflict between discontinuity, multiplication, and differentiation. As an example take H to be the Heaviside function and naively write \(H^ m=H\) with \(mH^{m-1}(DH)=DH\). Then \(DH/p=DH/q\) for \(p\neq q\) implies \(DH=0\) but one wants \(DH=\delta\), so there is a problem. Another example, based on the famous Schwartz “impossibility” of multiplication result, shows that algebraic structure \(+\) customary differentiation is incompatible with \(x\delta =0\). In particular differential algebras \(A\supset {\mathcal D}'\) will be constructed where D on A is not the same as the normal D on \({\mathcal D}'\). Also, in the language of Colombeau’s algebra G, the objects representing H and \(H^ 2\) in \(G\subset {\mathcal D}'\) have the same shadow on \({\mathcal D}'\) but are different in G. In fact one needs chains of algebras with D: \(A\to \tilde A\) in order to deal effectively with NLPDE. The Colombeau calculus, using one algebra G and a coupled calculus on G, is also discussed at length in Chapter 8 and some recent important technique of Oberguggenberger is developed in Chapter 4. We omit details of the algebraic constructions which are carefully worked out in the book.
To give a flavor of the book we extract from the table of contents as follows. Chapter 1, Conflict between discontinuity, multiplication, and differentiation, discusses: Limits to compatibility; construction of algebras; the neutrix condition; stability, generality, and exactness; algebraic solution to nonlinear stability paradoxes (e.g. \(O^ 2=1)\); general NLPDE; notions of solution for NLPDE; etc. Chapter 2, Global version of the Cauchy Kovalevskaja theorem on analytic NLPDE, discusses: Nowhere dense ideals; nonlinear partial differential operators (NLPDO) on spaces of generalized functions; closed nowhere dense singularities with zero Lebesgue measure; strange phenomena; etc. Chapter 3, Algebraic characterization for the solvability of NLPDE, discusses: the notion of generalized solution; solvability problems for NLPDE; neutrix characterization for the solvability of NLPDE; the neutrix condition as a densely vanishing condition on ideals; etc. Chapter 4, Generalized solutions of semilinear wave equations with rough initial values, discusses: General existence and uniqueness result; coherence with locally \(L^ 1\) solutions; the delta wave space, etc. Chapter 5, Discontinuous, shock, weak and generalized solutions of basic NLPDE, discusses: The need for nonclassical solutions; concepts of generalized solution; the Lewy inexistence result; etc.
Chapter 6, Chains of algebras of generalized functions, discusses: Restrictions on embeddings of the distributions into quotient algebras; regularizations; neutrix characterization of regular ideals; NLPDO in chains of algebras; etc. Chapter 7, Resolution of singularities of weak solutions for polynomial NLPDE, discusses: Resolution of singularities; nonlinear shock waves; Klein-Gordon type nonlinear waves; junction conditions etc. for equations of magnetohydrodynamics and general relativity; resoluble systems of polynomial NLPDE; global version of the Cauchy-Kovalevskaja theorem in chains of algebras of generalized functions; etc. Chapter 8, The particular case of Colombeau’s algebras, discusses: Smooth approximations and representations; properties of the differential algebra G; Colombeau’s algebra G as a collapsed case of chains of algebras; integrals of generalized functions; coupled calculus in G; generalized solutions of NLPDE is quantum field interactions; etc.
The book is rich with results and ideas and has good referencing to other work and points of view. One point of emphasis here is the natural role of algebraic methods instead of functional analysis in many situations. There is indeed a strong case for this and it is easy to believe that this book, \(+\) other books of Rosinger on generalized functions and NLPDE, \(+\) Colombeau’s books and papers, \(+\) Biagoni’s book, \(+\) work of Oberguggenberger, \(+\)... (I apologize for omissions), signals another revolution in thinking about partial differential equations and nonlinear phenomena (to go along with all the other revolutions taking place in mathematics and physics).
Reviewer: R.Carroll

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
46F10 Operations with distributions and generalized functions
35G20 Nonlinear higher-order PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L67 Shocks and singularities for hyperbolic equations
35Q40 PDEs in connection with quantum mechanics
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