Jump to ContentJump to Main Navigation
Multi-dimensional hyperbolic partial differential equationsFirst-order systems and applications$
Users without a subscription are not able to see the full content.

Sylvie Benzoni-Gavage and Denis Serre

Print publication date: 2006

Print ISBN-13: 9780199211234

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199211234.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 07 December 2019

LINEAR CAUCHY PROBLEM WITH VARIABLE COEFFICIENTS

LINEAR CAUCHY PROBLEM WITH VARIABLE COEFFICIENTS

Chapter:
(p.50) 2 LINEAR CAUCHY PROBLEM WITH VARIABLE COEFFICIENTS
Source:
Multi-dimensional hyperbolic partial differential equations
Author(s):

Sylvie Benzoni-Gavage

Denis Serre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199211234.003.0002

This chapter emphasizes energy estimates, that is, a priori estimates without loss of derivatives in Sobolev spaces. The appropriate tools are functional symmetrizers, obtained merely as Fourier multipliers for Friedrichs symmetrizable operators, and constructed by means of symbolic symmetrizers for more general operators. In particular, symbolic symmetrizers are shown to exist for constantly hyperbolic operators. Subsequently, well-posedness is proved in Sobolev spaces Hs , for any s in the case of infinitely smooth coefficients, and also for s large enough in the case of Hs coefficients. The existence of solutions relies on the Hahn-Banach and Riesz theorems, using energy estimates for the adjoint backward Cauchy problem. Uniqueness follows from energy estimates for the direct Cauchy problem. Regularity is shown by means of the weak equals strong argument, which is shown as the weak solutions obtained by the duality method are necessarily strong solutions, that is, suitable limits of infinitely smooth solutions of regularized problems.

Keywords:   energy estimates, functional symmetrizers, symbolic symmetrizers, symbolic calculus, Holmgren's principle

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .