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Multi-dimensional hyperbolic partial differential equationsFirst-order systems and applications$
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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

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THE CAUCHY PROBLEM FOR QUASILINEAR SYSTEMS

THE CAUCHY PROBLEM FOR QUASILINEAR SYSTEMS

Chapter:
(p.291) 10 THE CAUCHY PROBLEM FOR QUASILINEAR SYSTEMS
Source:
Multi-dimensional hyperbolic partial differential equations
Author(s):

Sylvie Benzoni-Gavage

Denis Serre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199211234.003.0010

This chapter obtains local solutions for Hs -data when this space is contained in the differentiable functions. The proof proceeds from a contraction mapping principle. The stability is obtained in Hs , thanks to the linear estimates of Chapter 2 and to Moser estimates, while the contraction is established in the L2 -norm. The Hs regularity is a little bit elaborate. Smooth solutions usually blow-up in finite time. The chapter recalls blow-up or continuation criteria. The analysis beyond blow-up requires the system to be given in the form of conservation laws. The chapter then recalls the notions of weak entropy solution, shock waves, and the Rankine-Hugoniot relations.

Keywords:   local solutions, finite time blow-up, conservation laws, entropy solutions

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