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On the problem of small motions and normal oscillations of a viscous fluid in a partially filled container
Author(s) -
Azizov Tomas Ya.,
Hardt Volker,
Kopachevsky Nikolay D.,
Mennicken Reinhard
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310001
Subject(s) - mathematics , eigenfunction , hilbert space , operator (biology) , mathematical analysis , container (type theory) , space (punctuation) , spectrum (functional analysis) , boundary value problem , metric (unit) , eigenvalues and eigenvectors , pure mathematics , physics , biochemistry , chemistry , linguistics , philosophy , repressor , quantum mechanics , transcription factor , gene , mechanical engineering , operations management , engineering , economics
The famous classical S. Krein problem of small motions and normal oscillations of a viscous fluid in a partially filled container is investigated by a new approach based on a recently developed theory of operator matrices with unbounded entries. The initial boundary value problem is reduced to a Cauchy problem$$ {dy \over dt} + {\cal A}y = f(t), \qquad y(0) = y^{0}, $$in some Hilbert space. The operator matrix is a maximal uniformly accretive operator which is selfadjoint in this space with respect to some indefinite metric. The theorem on strong solvability of the hydrodynamic problem is proved. Further, the spectrum of normal oscillations, basis properties of eigenfunctions and other questions are studied.