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Initial-boundary value problems for second order systems ofpartial differential equations

Published online by Cambridge University Press:  11 January 2012

Heinz-Otto Kreiss
Affiliation:
Träsko-StoröInstitute of Mathematics, NADA, KTH, 100 44 Stockholm, Sweden. [email protected]
Omar E. Ortiz
Affiliation:
Facultad de Matemáticas, Astronomía y Física and IFEG, Universidad Nacional de Córdoba, Ciudad Universitaria, CP :X5000HUA, Córdoba, Argentina
N. Anders Petersson
Affiliation:
Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, California, USA
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Abstract

We develop a well-posedness theory for second order systems in bounded domains whereboundary phenomena like glancing and surface waves play an important role. Attempts havepreviously been made to write a second order system consisting of nequations as a larger first order system. Unfortunately, the resulting first order systemconsists, in general, of more than 2n equations which leads to manycomplications, such as side conditions which must be satisfied by the solution of thelarger first order system. Here we will use the theory of pseudo-differential operatorscombined with mode analysis. There are many desirable properties of this approach: (1) thereduction to first order systems of pseudo-differential equations poses no difficulty andalways gives a system of 2n equations. (2) We can localize the problem,i.e., it is only necessary to study the Cauchy problem and halfplaneproblems with constant coefficients. (3) The class of problems we can treat is much largerthan previous approaches based on “integration by parts”. (4) The relation betweenboundary conditions and boundary phenomena becomes transparent.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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References

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