Lectures on Nonlinear Evolution Equations

Initial Value Problem
Erscheinungsjahr 2014
ISBN: 978-3-663-10631-9
Verlag: Vieweg+Teubner Verlag

Buch, Englisch, Band 19, 260 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 421 g

Reihe: Aspects of Mathematics

The book in hand is based on lectures which were given at the University of Bonn in the winter semesters of 1989/90 and 1990/91. The aim of the lectures was to present an elementary, self-contained introduction into some important aspects of the theory of global, small, smooth solutions to initial value problems for non linear evolution equa­ tions. The addressed audience included graduate students of both mathematics and physics who were only assumed to have abasie knowledge of linear partial differential equations. Thus, in the spirit of the underlying series, this book is intended to serve as a detailed basis for lectures on the subject as weIl as for self-studies for students or for other newcomers to this field. The presentation of the theory is made using the classical method of continuation of local solutions with the help of apriori estimates obtained for small data. The corre­ sponding global existence theorems have been proved mainly in the last decade, focussing on fully nonlinear systems; Related questions concerning large data problems, the ex­ istence of weak solutions or the analysis of &.hock waves are not discussed. Also the question of optimal regularity assumptions on the coefficients is beyond the scope of the book and is touched only in part and exemplarily.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Racke, Reinhard

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1 Global solutions to wave equations — existence theorems.- 2 Lp-Lq-decay estimates for the linear wave equation.- 3 Linear symmetric hyperbolic systems.- 4 Some inequalities.- 5 Local existence for quasilinear symmetric hyperbolic systems.- 6 High energy estimates.- 7 Weighted a priori estimates for small data.- 8 Global solutions to wave equations — proofs.- 9 Other methods.- 10 Development of singularities.- 11 More evolution equations.- 12 Further aspects and questions.- A Interpolation.- B The Theorem of Cauchy-Kowalevsky.- C A local existence theorem for hyperbolic-parabolic systems.- References.- Notation.

Über Autor(innen)

Dr. Reinhard Racke, Institut für Angewandte Mathematik,Universität Bonn/ Prof. Dr. Klas Diederich, Mathematisches Institut der Universität Wuppertal.