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Elements of continuum mechanics and conservation laws. Translation from the 1998 Russian original. (English) Zbl 1031.74004
New York, NY: Kluwer Academic/Plenum Publishers. viii, 258 p. (2003).
From a modern point of view, a unified presentation of ideas and general principles common to all branches of solid and fluid mechanics is made under the general heading of continuum mechanics. This important subject plays a central role in modern applied mathematics and is at the forefront of current research in modern applied mathematics and engineering science.
This short research monograph has five chapters, one appendix, literature, and subject index. After a brief discussion of the basic ideas and elementary properties of deformations and stresses, the authors present a theory of effective elastic deformation. Included are many topics such as relaxation law, equations for metric effective elastic deformation tensor, compatibility conditions, and relaxation shear stresses by equations for effective distortion. Differential equations of dynamical processes are the main topics of chapter III. This chapter explains how conservation laws can be used to derive differential equations which describe the motion of continuous metric under deformations. Some mathematical models and equations for one-dimensional non-stationary processes and stationary waves in Maxwell medium are also discussed. Chapter IV is devoted to well-posedness conditions for equations of elasticity, for a system of two symmetric hyperbolic equations, and for equations of gas dynamics written in the form of a symmetric hyperbolic system. Several model problems are discussed in detail, including the influence of small viscosity on the behavior of solutions, weak solutions of general quasilinear equations, one-dimensional hydrodynamic equations with heat conductivity, the propagation of small perturbations in a medium at rest, and oscillations of a piston in a cylinder containing gas. These examples illustrate well-posedness, stability and laws of thermodynamics. The final chapter deals with multidimensional thermodynamically compatible conservation laws. Included are hydromagnetic and gas dynamics equations, equations of elasticity, compatible conservation laws, and structure of multidimensional equations.
Appendix is concerned with the latest research on structure of thermodynamically compatible systems of equations of mathematical physics performed by the first author. This is an interesting text describing the simplest Galilei-invariant thermodynamically compatible systems and the use of the theory of orthogonal representations of rotation group \(\text{SO}(3)\) and the group \(\text{O}(3)\) in the construction of model systems of thermodynamically compatible conservation laws.
Some specific comments are in order. First, the authors have made some important contributions to the subject matter of the book, and the authors’ strong research interest in continuum mechanics is reflected by the material selected in the chapters. Second, this is not a textbook on continuum mechanics as contents of the book do not meet the basic requirements of standard syllabus of a graduate level course designed for applied mathematics and engineering science. Third, this book was written to stimulate further interest in the advanced study and research in continuum mechanics. This book would be very useful as a reference on the subject.

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35L65 Hyperbolic conservation laws