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\(L^{p}\) Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

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Abstract

We consider a viscous incompressible fluid governed by the Navier–Stokes system written in a domain where a part of the boundary can deform. We assume that the corresponding displacement follows a damped beam equation. Our main results are the existence and uniqueness of strong solutions for the corresponding fluid-structure interaction system in an \(L^p\)-\(L^q\) setting for small times or for small data. An important ingredient of the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped plate equation. We show that this linear system possesses the maximal regularity property by proving the \({\mathcal {R}}\)-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.

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Authors and Affiliations

  1. Centre for Applicable Mathematics, TIFR, Post Bag No. 6503,, GKVK Post Office, Bangalore, 560065, India

    Debayan Maity

  2. Université de Lorraine, CNRS, Inria, IECL, 54000, Nancy, France

    Takéo Takahashi

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  1. Debayan Maity
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Communicated by C. Grandmont.

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Debayan Maity was partially supported by INSPIRE faculty fellowship (IFA18-MA128) and by Department of Atomic Energy, Government of India, under Project No. 12-R & D-TFR-5.01-0520.

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Maity, D., Takahashi, T. \(L^{p}\) Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate. J. Math. Fluid Mech. 23, 103 (2021). https://doi.org/10.1007/s00021-021-00628-5

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