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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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LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS

LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS

Chapter:
(p.3) 1 LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS
Source:
Multi-dimensional hyperbolic partial differential equations
Author(s):

Sylvie Benzoni-Gavage

Denis Serre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199211234.003.0001

For first-order systems, the well-posedness of the Cauchy problem is equivalent to an algebraic condition on the symbol of the operator: the hyperbolicity. The hyperbolic Cauchy problem has the property of finite velocity of propagation (support, singularities). Important classes are those of symmetric (or symmetrizable) and constantly hyperbolic operators (i.e., with constant multiplicity eigenvalues). Hyperbolic polynomials are interesting on their own. Dispersion can be studied through Strichartz estimates. The adjoint of an hyperbolic operator is hyperbolic.

Keywords:   Fourier analysis, well-posedness, hyperbolic, symmetric hyperbolic, strictly hyperbolic, constantly hyperbolic, dependence and influence cones, Strichartz estimates, hyperbolic polynomials

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