D. Natroshvili, G. Sadunishvili, I. Sigua, Z. Tediashvili

Fluid-Solid Interaction: Acoustic Scattering by an Elastic Obstacle with Lipschitz Boundary

The potential method is developed for the three-dimensional interface problems of the theory of acoustic scattering by an elastic obstacle which are also known as fluid-solid (fluid-structure) interaction problems. It is assumed that the obstacle has a Lipschitz boundary. The sought for field functions belong to spaces having $L_2$ integrable nontangential maximal functions on the interface and the transmission conditions are understood in the sense of nontangential convergence almost everywhere. The uniqueness and existence questions are investigated. The solutions are represented by potential type integrals. The solvability of the direct problem is shown for arbitrary wave numbers and for arbitrary  incident wave functions. It is established that the scalar acoustic (pressure) field in the exterior domain is defined uniquely, while the elastic (displacement) vector field in the interior domain is defined modulo Jones modes, in general. On the basis of the results obtained it is proved that the inverse fluid-structure interaction problem admits at most one solution.

Mathematics Subject Classification: 35J05, 35J25, 35J55, 35P25, 47A40, 74F10, 74J20

Key words and phrases: Fluid-solid interaction, elasticity theory, Helmholtz equation, potential theory, interface problems, steady state oscillations