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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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INTIAL BOUNDARY VALUE PROBLEM IN A HALF-SPACE WITH CONSTANT COEEFICIENTS

INTIAL BOUNDARY VALUE PROBLEM IN A HALF-SPACE WITH CONSTANT COEEFICIENTS

Chapter:
(p.99) 4 INTIAL BOUNDARY VALUE PROBLEM IN A HALF-SPACE WITH CONSTANT COEEFICIENTS
Source:
Multi-dimensional hyperbolic partial differential equations
Author(s):

Sylvie Benzoni-Gavage

Denis Serre

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199211234.003.0004

This chapter drops the assumption of symmetry, or at least it does not assume the dissipativity in a classical sense. The search for a necessary condition for maximal estimates (strong well-posedness in Kreiss' sense) yields the so-called uniform Kreiss-Lopatinskii condition. The chapter investigates the case of a characteristic boundary. It gives practical devices to check the K.-L. condition, including the construction of a Lopatinskii determinant. It shows that the adjoint BVP shares with the direct one the K.-L. condition, a fact exploited in the duality method employed in the existence theory. The latter is carried out for the BVP in weighted (in time) spaces under the assumption that a Kreiss' dissipative boundary symmetrizer exists. Its existence is stated for a constantly hyperbolic operator, but the proof will be seen in the next chapter. The evolutionary property is shown with the use of the Paley-Wiener Theorem. Rauch's Theorem tells what the solution at a given time T can be estimated to; this extends the well-posedness to the full IBVP.

Keywords:   strong well-posedness, normal mode analysis, Kreiss-Lopatinskii condition, Lopatinskii determinant, adjoint IBVP, dissipative Kreiss symmetrizer

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