A boundary control problem for the steady self-propelled motion of a rigid body in a Navier–Stokes fluid

Hishida, Toshiaki; Silvestre, Ana Leonor; Takahashi, Takéo

doi:10.1016/j.anihpc.2016.11.003

Hishida, Toshiaki ; Silvestre, Ana Leonor ; Takahashi, Takéo

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1507-1541.

Résumé

Consider a rigid body $S \subset R^{3}$ immersed in an infinitely extended Navier–Stokes fluid. We are interested in self-propelled motions of $S$ in the steady state regime of the system rigid body-fluid, assuming that the mechanism used by the body to reach such a motion is modeled through a distribution of velocities $v_{⁎}$ on $\partial S$ . If the velocity V of $S$ is given, can we find $v_{⁎}$ that generates V? We show that this can be solved as a control problem in which $v_{⁎}$ is a six-dimensional control such that either $Supp v_{⁎} \subset Γ$ , an arbitrary nonempty open subset of ∂Ω, or $v_{⁎} \cdot n |_{\partial Ω} = 0$ . We also show that one of the self-propelled conditions implies a better summability of the fluid velocity.

MR Zbl

DOI : 10.1016/j.anihpc.2016.11.003

Mots clés : 3-D Navier–Stokes equations, Exterior domain, Rotating body, Self-propelled motion, Boundary control, Asymptotic behavior

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     author = {Hishida, Toshiaki and Silvestre, Ana Leonor and Takahashi, Tak\'eo},
     title = {A boundary control problem for the steady self-propelled motion of a rigid body in a {Navier{\textendash}Stokes} fluid},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1507--1541},
     publisher = {Elsevier},
     volume = {34},
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     year = {2017},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.11.003/}
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Hishida, Toshiaki; Silvestre, Ana Leonor; Takahashi, Takéo. A boundary control problem for the steady self-propelled motion of a rigid body in a Navier–Stokes fluid. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1507-1541. doi : 10.1016/j.anihpc.2016.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.11.003/

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