Asymptotics of the spectrum and eigenvectors of a problem of fluid mechanics

Abstract

We consider small motions of an inhomogeneous incompressible ideal fluid filling a motionless vessel with elastic walls and subject to the action of a stationary potential force field. We obtain an operational formulation of the problem. We find new, very precise estimates of the eigenvalues and eigenvectors of this problem.In studying the small motions of a fluid filling a vessel with elastic walls, an operator (generalized) statement of the problem has been obtained. In the process, instead of the "method of auxiliary problems" that is customarily used in problems' of this type [2,3], we have used the method of "embedding" the boundary-value problem in an integral identity [4]. In our view, the applicability of this last method in the problem considered below is very problematical.An inhomogeneous incompressible ideal fluid with density p(x), x = (xl, x2, x3) in a state of rest fills a (not necessarily connected) region f~ in R 3 with a piecewise-smooth boundary. In f~ a stationary force field F = -V~o (~o being the potential) acts on the fluid, while 'the boundary 0f~ is subject to the reactive force of the wails of the motionless vessel. The functions p(x) and ~o(x) axe smooth up to the boundary, and p(x) is bounded away from zero. It is assumed that the system is in equilibrium, so that the relationholds in fl, where Po is the equilibrium pressure. Part of the boundary of fl is made up of absolutely rigid walls of the vessel, and its complement F consists of elastic elements of various kinds. There may be elastic connections between the different parts of F. In P we distinguish the part Fo "wetted" by the fluid from both sides (i.e., Fo is made up of the elastic partitions within the fluid) from the part F1 that separates the fluid from the "void." It is assumed that a stationary external pressure acts on Px. We note, finally, that a mass of density rn(x) is distributed over P. The elastic properties of P can be expressed by the operator D that connects the displacement of the surface P and the load that causes this displacement. Here we shall consider the displacement ( to be small in what foUows (the linear theory) and orthogonal to the equihbrium surface F; by "load" we mean a pressure with density q. We can now state our assumptions on the linear operator D. We consider the space L2(F) with inner product J (2) (r P where ~ is Lebesgue measure on F. We shall assume that there is a rigging H+ C La(F) C H_ of the space L2(F) by Hilbert spaces H+ and H_, where the imbeddings are dense and compact. As usual, the definition of the inner product (2) is extended so that (q, () makes sense for q e H_ and ( E H+, and H+ and H_ are in a dual relation with respect to this bilinear form that is consistent with the topology of the two spaces. The space H+ is the energy space [5] of the elastic system F. It is made up of all possible displacements ( of the surface F having finite potential energy, and the square of the norm [[(1t+ equals the potential energy of the system under displacement (. The space H_ is made up o...