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Initial-boundary value problems for second order systems of partial 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. hokreiss@nada.kth.se
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 where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order system. Here we will use the theory of pseudo-differential operators combined with mode analysis. There are many desirable properties of this approach: (1) the reduction to first order systems of pseudo-differential equations poses no difficulty and always gives a system of 2n equations. (2) We can localize the problem, i.e., it is only necessary to study the Cauchy problem and halfplane problems with constant coefficients. (3) The class of problems we can treat is much larger than previous approaches based on “integration by parts”. (4) The relation between boundary conditions and boundary phenomena becomes transparent.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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References

Agranovich, M.S., Theorem on matrices depending on parameters and its applications to hyperbolic systems. Funct. Anal. Appl. 6 (1972) 8593. Google Scholar
B. Gustafsson, H.-O. Kreiss and J. Oliger, Time dependent problems and difference methods. Wiley-Interscience (1995).
Kreiss, H.-O., Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math. 23 (1970) 277298. Google Scholar
H.-O. Kreiss and J. Lorenz, Initial-boundary value problems and the Navier-Stokes equations. Academic Press, San Diego (1989).
Kreiss, H.-O. and Winicour, J., Problems which are well posed in a generalized sense with applications to the Einstein equations. Class. Quantum Grav. 23 (2006) 405420. Google Scholar

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