# American Institute of Mathematical Sciences

December  2014, 3(4): 557-578. doi: 10.3934/eect.2014.3.557

## A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, United States 2 Department of Mathematics, Statistics, and Computer Science, Dordt College, Sioux Center, IA 51250, United States

Received  February 2014 Revised  May 2014 Published  October 2014

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ coupled to a fourth order plate equation, possibly with rotational inertia parameter $\rho >0$. This plate PDE evolves on a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on $\Omega$ is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on $\Omega$. We note that as the Stokes fluid velocity does not vanish on $\Omega$, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational (inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.
Citation: George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
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