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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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PERSISTENCE OF MULTIDIMENSIONAL SHOCKS

PERSISTENCE OF MULTIDIMENSIONAL SHOCKS

Chapter:
(p.329) 12 PERSISTENCE OF MULTIDIMENSIONAL SHOCKS
Source:
Multi-dimensional hyperbolic partial differential equations
Author(s):

Sylvie Benzoni-Gavage

Denis Serre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199211234.003.0012

This chapter is partly influenced by Majda's work. Viewing shock fronts as free boundaries, it addresses their stability and existence by fixing front locations through a solution-dependent change of variable. This leads to non-standard hyperbolic IBVPs, in that boundary conditions are also PDEs. The stability of a shock front and the well-posedness of the corresponding non-standard hyperbolic IBVP are strongly related, and depend on a generalized uniform K.-L. condition. When this condition holds true for a constantly hyperbolic system and a given shock front, the chapter adapts the method of Chapter 11 to prove the existence of nearby shock-front solutions. Similar to standard mixed problems, compatibility conditions are needed between the initial data on either side of the initial front.

Keywords:   free boundary value problems, generalized K.-L. condition, stability, local existence, well-posedness

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