Solvability of a fluid-structure interaction problem with semigroup theory

Maxime Krier; Julia Orlik; Maxime Krier; Julia Orlik

doi:10.3934/math.20231510

AIMS Mathematics

2023, Volume 8, Issue 12: 29490-29516. doi: 10.3934/math.20231510

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Solvability of a fluid-structure interaction problem with semigroup theory

Maxime Krier ^,,
Julia Orlik

Department of Flow and Material Simulation, Fraunhofer ITWM, 67663 Kaiserslautern, Germany

Received: 26 September 2023 Revised: 25 October 2023 Accepted: 25 October 2023 Published: 31 October 2023
MSC : 74F10, 76S05, 35B27

Continuous semigroup theory is applied to proof the existence and uniqueness of a solution to a fluid-structure interaction (FSI) problem of non-stationary Stokes flow in two bulk domains, separated by a 2D elastic, permeable plate. The plate's curvature is proportional to the jump of fluid stresses across the plate and the flow resistance is modeled by Darcy's law. In the weak formulation of the considered physical problem, a linear operator in space is associated with a sum of two bilinear forms on the fluid and the interface domains, respectively. One attains a system of equations in operator form, corresponding to the weak problem formulation. Utilizing the sufficient conditions in the Lumer-Phillips theorem, we show that the linear operator is a generator of a contraction semigroup, and give the existence proof to the FSI problem.
- fluid-structure interaction,
- asymptotic analysis,
- homogenization,
- dimension reduction,
- semigroup theory
Citation: Maxime Krier, Julia Orlik. Solvability of a fluid-structure interaction problem with semigroup theory[J]. AIMS Mathematics, 2023, 8(12): 29490-29516. doi: 10.3934/math.20231510

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Abstract

Continuous semigroup theory is applied to proof the existence and uniqueness of a solution to a fluid-structure interaction (FSI) problem of non-stationary Stokes flow in two bulk domains, separated by a 2D elastic, permeable plate. The plate's curvature is proportional to the jump of fluid stresses across the plate and the flow resistance is modeled by Darcy's law. In the weak formulation of the considered physical problem, a linear operator in space is associated with a sum of two bilinear forms on the fluid and the interface domains, respectively. One attains a system of equations in operator form, corresponding to the weak problem formulation. Utilizing the sufficient conditions in the Lumer-Phillips theorem, we show that the linear operator is a generator of a contraction semigroup, and give the existence proof to the FSI problem.

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