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Lectures on Nonlinear Hyperbolic Differential Equations

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Synopses & Reviews

Publisher Comments:

This introduction to the theory of nonlinear hyperbolic differential equations, a revised and extended version of widely circulated lecture notes from 1986, starts from a very elementary level with standard existence and uniqueness theorems for ordinary differential equations, but they are at once supplemented with less well-known material, required later on. A detailed and explicit study of discontinuous solutions of a model equation, Burgers' equation, is then followed by a general study of solutions of conservation laws, with one unknown or one space variable. Asymptotic properties of solutions of the linear wave equation and the Klein-Gordon equation are studied in detail as a preparation for the study of solutions of nonlinear perturbations with small and smooth initial data. Existence of solutions for all times is proved for large space dimensions and lower bounds for the "lifespan" of the solutions are given in low space dimensions. The last four chapters are devoted to microlocal analysis of singularities of solutions of nonlinear differential equations by means of the paradifferential techniques of J.-M. Bony.

Synopsis:

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.

Table of Contents

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.

Product Details

ISBN:
9783540629214
Author:
Hormander, Lars
Author:
Hormander, L.
Author:
Hormander
Author:
Hormander, L.
Publisher:
Springer
Location:
Berlin ;
Subject:
Mathematics
Subject:
Differential Equations
Subject:
Differential equations, Hyperbolic
Subject:
Mathematics-Differential Equations
Subject:
nicht lineare hyperbolische Differentialgleichungen
Subject:
nonlinear hyperbolic differential equations
Subject:
PARTIAL DIFFERENTIAL EQUATIONS
Subject:
Theoretical, Mathematical and Computational Physics
Copyright:
Edition Description:
Book
Series:
Mathématiques et Applications
Series Volume:
26
Publication Date:
20031217
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
298
Dimensions:
235 x 155 mm 960 gr

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Related Subjects

Reference » Science Reference » Technology
Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » Geometry » Algebraic Geometry

Lectures on Nonlinear Hyperbolic Differential Equations New Trade Paper
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Product details 298 pages Springer-Verlag - English 9783540629214 Reviews:
"Synopsis" by , 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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