Dynamics of viscous compressible fluids.

*(English)*Zbl 1080.76001
Oxford Lecture Series in Mathematics and its Applications 26. Oxford: Oxford University Press (ISBN 0-19-852838-8/hbk). xi, 212 p. (2004).

The book is devoted to the mathematical theory of viscous compressible fluids. In Chapter 1 the author presents a review of the underlying physical theory. Chapters 2 and 3 contain the basic mathematical concepts used in the book. The author presents some function spaces that are related to the problem, and then establishes fundamental a priori estimates that form the mathematical background of the theory of variational solutions developed in Chapter 4. The discussion of variational solutions is further given in Chapter 5, where more delicate estimates are obtained for pressure and temperature

The most advanced Chapter 6 introduces the concept of oscillation defect measure and establishes its relation to the propagation of oscillations and the renormalized continuity equation. With these technical tools, the author focuses on the central issue of this chapter, namely, on the propagation of density oscillations in a sequence of solutions. Chapter 7 contains a complete proof of the existence of global-in-time variational solutions for Navier-Stokes system describing the motion of a viscous compressible heat conducting fluid. Each chapter is completed by bibliographical remarks. The book will be of interest to students and researchers in the dynamics of fluids.

The most advanced Chapter 6 introduces the concept of oscillation defect measure and establishes its relation to the propagation of oscillations and the renormalized continuity equation. With these technical tools, the author focuses on the central issue of this chapter, namely, on the propagation of density oscillations in a sequence of solutions. Chapter 7 contains a complete proof of the existence of global-in-time variational solutions for Navier-Stokes system describing the motion of a viscous compressible heat conducting fluid. Each chapter is completed by bibliographical remarks. The book will be of interest to students and researchers in the dynamics of fluids.

Reviewer: Valeriu Al. Sava (Iaşi)