George Chkadua
abstract:
In the paper, we consider a three-dimensional model of fluid-solid acoustic
interaction when an electro-magneto-elastic body occupying a bounded region
$\Omega^{+}$ is embedded in an unbounded fluid domain $\Omega^{-}=\mathbb{R}^3
\setminus \overline{\Omega^+}$. In this case, we have a five-dimensional
electro-magneto-elastic field (the displacement vector with three components,
electric potential and magnetic potential) in the domain $\Omega^{+}$, while we
have a scalar acoustic pressure field in the unbounded domain $\Omega^{-}$. The
physical kinematic and dynamic relations are mathematically described by the
appropriate boundary and transmission conditions. We consider less restrictions
on a matrix differential operator of electro-magneto-elasticity by introducing
asymptotic classes, in particular, we allow the corresponding characteristic
polynomial of the matrix operator to have multiple real zeros.
In the paper, we consider mixed type interaction problem. In particular, except
transmission conditions, electric and magnetic potentials are given on one part
of the boundary of $\Omega^+$ (the Dirichlet type condition), while on the other
part, normal components of electric displacement and magnetic induction are
given (the Neumann type condition).
We derive asymptotic expansion of solutions near the line where different
boundary conditions change, and on the basis of asymptotic analysis, we
establish optimal H\"{o}lder's smoothness results for solutions of the problem.
Mathematics Subject Classification: 35J47, 74F15, 31B10, 34L25, 35B40, 35C20, 35S15
Key words and phrases: Boundary-transmission problems, fluid-solid interaction, potential method, pseudodifferential equations, Helmholtz equation, steady state oscillations, asymptotic analysis