Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D

On the interaction problem between a compressible viscous fluid and a nonlinear thermoelastic plate

“…Then in [41], we studied the interaction between in which the nonlinear plate with the nonlinear elastic force F(w) satisfying the assumptions (A1) and (A2) given in section 2.2 interacts with a viscous incompressible fluid and constructed a hybrid approximation scheme where the fluid sub-problem is stationary and the structure sub-problem is continuous in time and in a finite basis. We later extended this result in [42] by studying the interaction between an incompressible viscous fluid and a nonlinear thermoelastic plate, where we also included an additional quasilinear plate model with cubic nonlinear elastic force. Now, in this paper, we choose the fluid sub-problem to be continuous in time as well.…”

“…Later, the authors considered in [41] a general semilinear plate model that generalizes the Kirchhoff, von Karman and Berger plates 1 and constructed a hybrid splitting scheme (stationary for the fluid and time-continuous in a finite base for the plate) in order to deal with the general form of the nonlinearity in the plate equation. Recently, we extended this result in [42] to the problem where a thermoelastic semilinear/quasilinear 2 plate interacted with an incompressible viscous fluid. Muha and Schwarzacher [34] proved the existence of a weak solution for the interaction problem of a nonlinear (quasilinear) Koiter shell and incompressible viscous fluid.…”

“…To formulate the problem on the fixed reference domain Ω (as it was done in the context of incompressible fluids in [36,37,35,41,42]), we introduce a family of the following arbitrary Lagrangian-Eulerian (ALE) transformations:…”

“…In this paper, we aim to study the existence of a weak solution for the interaction problem between a compressible viscous flow and a thermoelastic plate, by constructing a novel decoupling scheme (first such in the compressible case) that splits the fluid and the structure. This scheme was inspired by schemes in [36,37,35,41,42], which are used to construct weak solutions for the incompressible case. However, unlike in the incompressible case, here the both approximate sub-problems, corresponding to the fluid and the structure, are constructed to be continuous in time.…”

Controllability of a simplified fluid-structure interaction system

“…Using similar techniques they also analyze the case of the wave equation (α 1 = δ = 0 and α 2 > 0) in [1] showing in particular that the semigroup of the linearized system is analytic. Let us mention also some results for more complex models: [29,28] (linear elastic Koiter shell), [38] (dynamic pressure boundary conditions), [39,40] (3D cylindrical domain with nonlinear elastic cylindrical Koiter shell), [49] and [50] (nonlinear elastic and thermoelastic plate equations), [32], [34] (compressible fluids), etc.…”