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Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach
Author(s) -
Ashton A. C. L.
Publication year - 2010
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00490.x
Subject(s) - mathematics , sobolev space , mathematical analysis , matrix (chemical analysis) , cauchy problem , initial value problem , infinitesimal , semigroup , boundary value problem , cauchy distribution , stability (learning theory) , boundary (topology) , materials science , machine learning , computer science , composite material
We consider the motion of a collection of fluid loaded elastic plates, situated horizontally in an infinitely long channel. We use a new, unified approach to boundary value problems, introduced by A.S. Fokas in the late 1990s, and show the problem is equivalent to a system of one‐parameter integral equations. We give a detailed study of the linear problem, providing explicit solutions and well‐posedness results in terms of standard Sobolev spaces. We show that the associated Cauchy problem is completely determined by a matrix, which depends solely on the mean separation of the plates and the horizontal velocity of each of the driving fluids. This matrix corresponds to the infinitesimal generator of the C 0 ‐semigroup for the evolution equations in Fourier space. By analyzing the properties of this matrix, we classify necessary and sufficient conditions for which the problem is asymptotically stable.