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Systems with strong damping and their spectra
Author(s) -
Jacob Birgit,
Tretter Christiane,
Trunk Carsten,
Vogt Hendrik
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5166
Subject(s) - mathematics , hilbert space , compressibility , mathematical analysis , viscoelasticity , range (aeronautics) , space (punctuation) , flow (mathematics) , ideal (ethics) , oscillation (cell signaling) , state space , spectral analysis , steady state (chemistry) , physics , geometry , mechanics , quantum mechanics , linguistics , philosophy , materials science , genetics , epistemology , statistics , chemistry , biology , spectroscopy , composite material , thermodynamics
We establish a new method for obtaining nonconvex spectral enclosures for operators associated with second‐order differential equations z ¨ ( t ) + D z ˙ ( t ) + A 0 z ( t ) = 0 in a Hilbert space. In particular, we succeed in establishing the existence of a spectral gap, which is the first result of this kind since the seminal results of Krein and Langer for oscillations of damped systems. While the latter and other spectral bounds are confined to dampings D that are symmetric and dominated by A 0 , we allow for accretive D of equal strength as A 0 . To achieve these results, we prove new abstract spectral inclusion results that are much more powerful than classical numerical range bounds. Two different applications, small transverse oscillations of a horizontal pipe carrying a steady‐state flow of an ideal incompressible fluid and wave equations with strong (viscoelastic and frictional) damping, illustrate that our new bounds are explicit.