Hyperbolic partial differential equations and geometric optics (Q2888015)

The book under consideration is an introduction to hyperbolic partial differential equations (PDEs) and systems of PDEs, and deals with the Cauchy problem (existence of solutions, well-posedness, finite speed of the waves), the propagation of singularities, the asymptotic analysis of short wave length solutions, the nonlinear interaction of such waves, etc. The author very carefully presents linear (elliptic and hyperbolic) and nonlinear geometric optics, the latter being synonymous with short wave length asymptotic analysis of solutions of nonlinear systems of PDEs. Damping of waves, resonances, dispersive decay and solutions to the compressible Euler equations with dense oscillations, created by resonant interactions, are investigated in detail. As far as the reviewer knows, for the first time in a textbook, such interesting problems as the existence and stability for the three wave interaction equations are included.NEWLINENEWLINE The machinery of geometric optics is a very powerful tool in the linear microlocal analysis. In fact, the two main results there -- the microlocal elliptic regularity theorem and the propagation of singularities for symmetric hyperbolic operators of constant multiplicity are proved via the formal asymptotic approach.NEWLINENEWLINE The book is divided into eleven chapters. Chapter 2 is devoted to the linear Cauchy problem for symmetric hyperbolic systems. Chapter 3 deals with the dispersive behaviour of the solutions to the these systems. Chapters 4 and 5 are devoted to linear elliptic geometric optics, respectively, linear hyperbolic geometric optics, including the Lax parametrix, Fourier integral operators and results on the small time (Lax, Hörmander) and global (Ludwig, Hörmander) propagation of singularities. The nonlinear Cauchy problem for quasilinear symmetric hyperbolic systems is treated in Chapter 6, while Chapter 7 deals with one phase nonlinear geometric optics and Chapter 8 is devoted to the stability problems for one phase geometric optics. Chapter 9 deals with resonance interaction and quasilinear systems; several examples (semilinear and quaslinear) of resonances in one-dimensional space are proposed in Chapter 10. As mentioned above, the book contains several results on dense oscillations for the compressible Euler equations, which can be found in Chapter 11.NEWLINENEWLINE The book is well written and contains an exuberance of interesting theorems equipped with the corresponding proofs. Moreover, the ideas are stressed and the proofs (some of them complicated) are clearly motivated. Many nice examples illustrate the nontrivial results and make the reader's acceptance easier.NEWLINENEWLINE In conclusion, the book will be useful to both researchers and graduate students interested or working in the domain of hyperbolic PDEs and the reviewer highly recommends it to them.