Non-local approach in mathematical problems of fluid-structure interaction

Abstract

Three-dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω 1 ⊆ R 3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω 2 = R 3 \Ω 1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady-state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω 1 . The problems are studied by the so-called non-local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional-variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov-Poincare type operators and on the basis of the Garding inequality and the Lax-Milgram theorem. In particular, it is shown that the physical fluid-solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter.