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A matrix approach to the energy-norm bisection in wave motion. (English) Zbl 0599.73028

In this work we explore a systematic procedure, leading to norm-bisection for the cases of acoustic, electromagnetic and elastic wave propagation. The characteristics of all three systems are that they are governed by hyperbolic systems of partial differential equations having smooth Cauchy data with compact support and they all preserve the energy imposed by the initial data. In all three cases a similar result holds: the appropriate energy-norms are bisected in finite time for Cauchy data with compact support and asymptotically in time for Cauchy data with finite energy- norm. After an extensive analysis of the algebraic structures of the systems of differential equations involved, we bring the relative problems in a form that is appropriate for the application of harmonic analysis and the results follow immediately. The parallel matrix analysis of the transformed problems points out the resemblance of the mathematical structures that govern as much diverse physical phenomena as the propagation of acoustical, electromagnetic and elastic waves.

MSC:

74J99 Waves in solid mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
35L05 Wave equation
78A40 Waves and radiation in optics and electromagnetic theory
76Q05 Hydro- and aero-acoustics
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