Lectures on Nonlinear Hyperbolic Differential Equations

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Springer Science & Business Media, Jul 17, 1997 - Mathematics - 290 pages
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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.
 

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Contents

Preface
1
Scalar first order equations with one space variable
13
Scalar first order equations with several variables
36
First order systems of conservation laws with
46
Compensated compactness
72
Nonlinear perturbations of the wave equation
89
Nonlinear perturbations of the KleinGordon
145
Microlocal analysis
186
Pseudodifferential operators of type 11
211
Paradifferential calculus
235
Propagation of singularities
246
Appendix on pseudoRiemannian geometry
270
References
283
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