Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach
A. C. L. Ashton · Studies in Applied Mathematics · 2010
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 C0‐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.