Initial-boundary value problems for second order systems of partial differential equations∗

Heinz-Otto Kreiss; Omar E. Ortiz; N. Anders Petersson

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 3, page 559-593
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kreiss, Heinz-Otto, Ortiz, Omar E., and Anders Petersson, N.. "Initial-boundary value problems for second order systems of partial differential equations∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 559-593. <http://eudml.org/doc/222162>.

@article{Kreiss2012,
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.},
author = {Kreiss, Heinz-Otto, Ortiz, Omar E., Anders Petersson, N.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Well-posed 2nd-order hyperbolic equations; surface waves; glancing waves; elastic wave equation; Maxwell equations; Laplace and Fourier transforms},
language = {eng},
month = {1},
number = {3},
pages = {559-593},
publisher = {EDP Sciences},
title = {Initial-boundary value problems for second order systems of partial differential equations∗},
url = {http://eudml.org/doc/222162},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Kreiss, Heinz-Otto
AU - Ortiz, Omar E.
AU - Anders Petersson, N.
TI - Initial-boundary value problems for second order systems of partial differential equations∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/1//
PB - EDP Sciences
VL - 46
IS - 3
SP - 559
EP - 593
AB - 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.
LA - eng
KW - Well-posed 2nd-order hyperbolic equations; surface waves; glancing waves; elastic wave equation; Maxwell equations; Laplace and Fourier transforms
UR - http://eudml.org/doc/222162
ER -

References

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  1. M.S. Agranovich, Theorem on matrices depending on parameters and its applications to hyperbolic systems. Funct. Anal. Appl.6 (1972) 85–93.  Zbl0253.35060
  2. B. Gustafsson, H.-O. Kreiss and J. Oliger, Time dependent problems and difference methods. Wiley-Interscience (1995).  Zbl0843.65061
  3. H.-O. Kreiss, Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math.23 (1970) 277–298.  Zbl0188.41102
  4. H.-O. Kreiss and J. Lorenz, Initial-boundary value problems and the Navier-Stokes equations. Academic Press, San Diego (1989).  Zbl0689.35001
  5. H.-O. Kreiss and J. Winicour, Problems which are well posed in a generalized sense with applications to the Einstein equations. Class. Quantum Grav.23 (2006) 405–420.  Zbl1191.83010

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