Multidimensional hyperbolic partial differential equations : first-order systems and applications / Sylvie Benzoni-Gavage, Denis Serre

The book consists of five main parts, each of which consists of several chapters. The linear Cauchy problem is the topic of Part 1, and well-posedness (in suitable Sobolev spaces) is established for Friedrichs symmetrizable systems and/or constantly hyperbolic systems (the eigenvalues are semi-simple and of constant multiplicity). Symmetrizers constitute the main tool for deriving energy estimates and thus the well-posedness. The case of variable coefficients is treated by introducing symbolic symmetrizers and using pseudo- or para-differential calculus. Initial-boundary value problems for linear hyperbolic systems are the subject matter of Part 2 of this book. The discussion starts with the easiest case, namely that of Friedrichs symmetric systems with dissipative boundary conditions. Next, the more difficult problem of constantly hyperbolic operators with constant coefficients in a half-space is proved to be well-posed; along the way important concepts like the (uniform) Kreiss-Lopatinskiĭ condition and Kreiss symmetrizers are introduced. Extensions to initial-boundary value problems with variable coefficients are worked out through the use of pseudo- or para-differential calculus. In general, constructing Kreiss symmetrizers requires involved arguments, and therefore the authors devote a whole chapter to this subject. Part 2 also covers topics like domains with characteristic boundaries and homogeneous initial-boundary value problems. The third part of this book is devoted to quasilinear systems (in several space dimensions). First, local well-posedness of the Cauchy problem for Friedrichs symmetrizable systems (e.g., those admitting a strictly convex entropy) or, more generally, those admitting a symbolic symmetrizer, is proved; as usual, local well-posedness refers to a Sobolev space of high enough index. Weak and entropy solutions are defined, and a weak-strong uniqueness result is proved, showing that a single strictly convex entropy is sufficient to select the unique smooth solution whenever it exists. Then the local existence of a unique solution to the initial-boundary value problem for quasilinear systems is established through an iterative scheme and by showing the well-posedness of solutions to linearized equations. Finally, the existence and stability of multidimensional shocks are proved, in which the main step is to analyze an initial-boundary value problem for suitably linearized equations with Rankine-Hugoniot type boundary condition. The theory developed in the preceding parts of the book is applied to and exemplified on the Euler equations of gas dynamic in Part 4. Finally, Part 5 is an appendix collecting various tools (Laplace and Fourier transforms, Paley-Wiener theorems, pseudo- and para-differential calculus) used throughout the book. (MathScinet)

Bibliogr. p. [492]-503 . Index