In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
Preface.- Contents.- Chap. I: Ordinary differential equations.- Chap. II: Scalar first order equations with one space variable.- Chap. III: Scalar first order equations with several variables.- Chap. IV: First order systems of conservation laws with one space.- Chap. V: Compensated compactness.- Chap. VI: Nonlinear perturbations of the wave equation.- Chap. VII: Nonlinear perturbations of the Klein-Gordon equation.- Chap. VIII: Microlocal analysis.- Chap. IX: Pseudo-differential operators of type 1,1.- Chap. X: Paradifferential calculus.- Chap. XI: Propagation of singularities.- Appendix on pseudo-Riemannian geometry.- References.- Index of notations.- Index.