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Über dieses Buch

This book deals with evolutionary systems whose equation of state can be formulated as a linear Volterra equation in a Banach space. The main feature of the kernels involved is that they consist of unbounded linear operators. The aim is a coherent presentation of the state of art of the theory including detailed proofs and its applications to problems from mathematical physics, such as viscoelasticity, heat conduction, and electrodynamics with memory. The importance of evolutionary integral equations ‒ which form a larger class than do evolution equations​ ‒ stems from such applications and therefore special emphasis is placed on these. A number of models are derived and, by means of the developed theory, discussed thoroughly. An annotated bibliography containing 450 entries increases the book’s value as an incisive reference text. --- This excellent book presents a general approach to linear evolutionary systems, with an emphasis on infinite-dimensional systems with time delays, such as those occurring in linear viscoelasticity with or without thermal effects. It gives a very natural and mature extension of the usual semigroup approach to a more general class of infinite-dimensional evolutionary systems. This is the first appearance in the form of a monograph of this recently developed theory. A substantial part of the results are due to the author, or are even new. (…) It is not a book that one reads in a few days. Rather, it should be considered as an investment with lasting value. (Zentralblatt MATH) In this book, the author, who has been at the forefront of research on these problems for the last decade, has collected, and in many places extended, the known theory for these equations. In addition, he has provided a framework that allows one to relate and evaluate diverse results in the literature. (Mathematical Reviews) This book constitutes a highly valuable addition to the existing literature on the theory of Volterra (evolutionary) integral equations and their applications in physics and engineering. (…) and for the first time the stress is on the infinite-dimensional case. (SIAM Reviews)



Equations of Scalar Type


1. Resolvents

The concept of the resolvent which is central for the theory of linear Volterra equations is introduced and discussed. It is applied to the inhomogeneous equation to derive various variation of parameters formulas. The main tools for proving existence theorems for the resolvent are described in detail; these methods are the operational calculus in Hilbert spaces, perturbation arguments, and the Laplace-transform method. The generation theorem, the analog of the Hille-Yosida theorem of semigroup theory for Volterra equations, proved in Section 1.5, is of fundamental importance in later chapters. The theory is completed with several counterexamples, and with a discussion of the integral resolvent.
Jan Prüss

2. Analytic Resolvents

This section is devoted to the theory of analytic resolvents, the analog of analytic semigroups for Volterra equations of scalar type. A complete characterization of such resolvents in terms of Laplace transforms is given. In contrast to the general generation theorem of Section 1, the main result of this section, Theorem 2.1, requires conditions which are much simpler to check; this is done in several illustrating examples. The spatial regularity of analytic resolvents is studied and a characterization of analytic semigroups in these terms is derived. It is shown that analytic resolvents lead to improved perturbation results and stronger properties of the variation of parameter formulas.
Jan Prüss

3. Parabolic Equations

As a continuation of Section 2, the concept of parabolicity for Volterra equations of scalar type is introduced and resolvents for such equations are discussed in detail. If the kernel a(t ) has some extra regularity property, like convexity, then the resolvent exists, and exhibits the same stability under perturbations as analytic resolvents. Again the maximal regularity property of type is valid, even for the perturbed equation. In Section 3.6 we derive a representation formula for the resolvent in case A is the generator of a C0-semigroup.
Jan Prüss

4. Subordination

The class of completely positive kernels plays a prominent role in the theory of vector-valued Volterra equations, and appears in applications quite naturally. This class of kernels, its properties and associated creep functions are discussed thoroughly in this section. By means of the principle of subordination it is possible to construct new resolvents from a given one, e.g. from a C0-semigroup or from a cosine family. The new resolvent can be explicitly represented in terms of the given one, and of the propagation function associated with a completely positive kernel. This representation is particularly useful for the understanding of the regularity and the asymptotic behaviour of the resolvent.
Jan Prüss

5. Linear Viscoelasticity

A rich source for vector-valued Volterra equations is the continuum mechanics for materials with memory, i.e. the theory of viscoelastic materials. In this section the basic concepts of this theory are introduced and the resulting boundary value problems are formulated. The discussion of the involved material functions shows that the notion of creep functions introduced in Section 4 appears here naturally. The well-posedness of some special problems which lead to Volterra equations of scalar type will be discussed, but also the limits of this class of equations become apparent.
Jan Prüss

Nonscalar Equations


6. Hyperbolic Equations of Nonscalar Type

The basic properties of Volterra equations of nonscalar type are discussed in this section. Resolvents for such problems are introduced and their relations to wellposedness and variation of parameters formulae are studied. The latter are then used for perturbation results which yield several well-known existence theorems. The generation theorem for the nonscalar case is proved and then applied to the convergence of resolvents and to existence theorems for equations in Hilbert spaces involving operator-valued kernels of positive type.
Jan Prüss

7. Nonscalar Parabolic Equations

The results on existence of analytic resolvents in Section 2 and on resolvents for parabolic problems in Section 3 as well as the corresponding results on maximal regularity of type are here extended to nonscalar equations. Particularly easy to verify are the conditions in the variational approach in Section 7.3 which continues the discussion begun in Section 6.7. The remaining subsections are devoted to a far reaching improvement of the perturbation theorem from Section 6.3 in the parabolic case.
Jan Prüss

8. Parabolic Problems in L p-Spaces

The subject of this section is the Lp-theory for parabolic equations with main part. The first three subsections prepare the approach via sums of commuting linear operators; the two basic results, i.e. a vector-valued Fourier-multiplier theorem and the Dore-Venni theorem, are stated without proof.
Jan Prüss

9. Viscoelasticity and Electrodynamics with Memory

The discussion of problems in linear viscoelasticity leading to linear abstract Volterra equations which was begun in Section 5 is continued here. Models for viscoelastic beams and plates are introduced and their well-posedness is studied by means of the results on Volterra equations of scalar type from Chapter I but also by those on equations of nonscalar type from this chapter. In Sections 9.3 and 9.4 two approaches to general linear thermoviscoelasticity based on the results of Sections 6, 7, and 8 are carried through. Under physically reasonable assumptions these yield well-posedness in the variational setting, but also in the strong if the material in question is almost separable. In Sections 9.5 and 9.6 memory effects in isotropic linear electrodynamics are discussed and via the perturbation method well-posedness of the whole space problem as well as of a transmission problem are proved.
Jan Prüss

Equations on the Line


10. Integrability of Resolvents

The connections between stability properties of Volterra equations and integrability of the corresponding resolvent are discussed in Section 10.1; this discussion motivates the study of integrability of resolvents. For the classes of equations of scalar type introduced in Sections 2, 3, and 4 a complete characterization of integrability of S(t) in terms of spectral conditions is derived. For nonscalar parabolic problems sufficient conditions are presented in a fairly general setting, while for nonscalar hyperbolic problems the analysis is valid in Hilbert spaces only, as counterexamples show.
Jan Prüss

11. Limiting Equations

The solvability behaviour of Volterra equations on the line will be studied in this section. The central concept which corresponds to well-posedness is that of admissibility of homogeneous spaces of functions on the line. Necessary conditions for admissibility are derived and some consequences of this property are studied. Thereafter the connections between the equations on the line and on the halfline are studied and the notion of limiting equation is justified.
Jan Prüss

12. Admissibility of Function Spaces

This section is concerned with sufficiency of the frequency domain conditions derived in Proposition 11.5 for admissibility of evolutionary integral equations on the line as well as on the existence of Λ-kernels and their properties. The main results cover subordinated equations, hyperbolic problems in Hilbert spaces, and parabolic problems in arbitrary spaces. The discussion includes also perturbation problems and maximal regularity on the line for parabolic problems.
Jan Prüss

13. Further Applications and Complements

The first three subsections are devoted to applications of the results of Sections 10, 11, and 12 to some of the problems introduced in Sections 5 and 9. These include the hyperbolic viscoelastic Timoshenko beam, heat conduction in isotropic materials with memory, and boundary value problems for electrodynamics with memory.
Jan Prüss


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