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://archive.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://archive.numdam.org/articles/10.1016/j.anihpc.2016.11.003/

Bibliographie
Cité par

[1] Babenko, K.I. On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Math. USSR Sb., Volume 20 (1973), pp. 1–25 | DOI | Zbl

[2] Bogovskiĭ, M.E. Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Sov. Math. Dokl., Volume 20 (1979), pp. 1094–1098 | MR | Zbl

[3] Borchers, W.; Sohr, H. On the equations rot $v = g$ and div $u = f$ with zero boundary conditions, Hokkaido Math. J., Volume 19 (1990), pp. 67–87 | DOI | MR | Zbl

[4] Fabre, C.; Lebeau, G. Prolongement unique des solutions de l'équation de Stokes, Commun. Partial Differ. Equ., Volume 21 (1996) no. 3–4, pp. 573–596 | MR | Zbl

[5] Farwig, R. The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., Volume 211 (1992), pp. 409–447 | DOI | MR | Zbl

[6] Farwig, R.; Galdi, G.P.; Kyed, M. Asymptotic structure of a Leray solution to the Navier–Stokes flow around a rotating body, Pac. J. Math., Volume 253 (2011), pp. 367–382 | DOI | MR | Zbl

[7] Farwig, R.; Hishida, T. Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscr. Math., Volume 136 (2011), pp. 315–338 | DOI | MR | Zbl

[8] Farwig, R.; Hishida, T. Leading term at infinity of steady Navier–Stokes flow around a rotating obstacle, Math. Nachr., Volume 284 (2011), pp. 2065–2077 | DOI | MR | Zbl

[9] Farwig, R.; Hishida, T.; Müller, D. $L^{q}$ -theory of a singular “winding” integral operator arising from fluid dynamics, Pac. J. Math., Volume 215 (2004), pp. 297–312 | DOI | MR | Zbl

[10] Finn, R. Estimates at infinity for stationary solutions of the Navier–Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumanie, Volume 3 (1959), pp. 387–418 | MR | Zbl

[11] Finn, R. On the exterior stationary problem for the Navier–Stokes equations, and associated perturbation problems, Arch. Ration. Mech. Anal., Volume 19 (1965), pp. 363–406 | DOI | MR | Zbl

[12] Fursikov, A.V.; Gunzburger, M.D.; Hou, L.S. Optimal boundary control for the evolutionary Navier–Stokes system: the three-dimensional case, SIAM J. Control Optim., Volume 43 (2005) no. 6, pp. 2191–2232 | DOI | MR | Zbl

[13] Galdi, G.P. On the steady, translational self-propelled motion of a symmetric body in a Navier–Stokes fluid, Quaderni di Matematica della II Universita di Napoli, vol. 1, 1997, pp. 97–169 | MR | Zbl

[14] Galdi, G.P. On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., Volume 148 (1999), pp. 53–88 | DOI | MR | Zbl

[15] Galdi, G.P. On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, vol. 1, 2002, pp. 655–679 | DOI | MR | Zbl

[16] Galdi, G.P. An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problem, Springer, 2011 | MR | Zbl

[17] Galdi, G.P.; Kyed, M. Steady-state Navier–Stokes flows past a rotating obstacle: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal., Volume 200 (2011), pp. 21–58 | DOI | MR | Zbl

[18] Galdi, G.P.; Silvestre, A.L. The steady motion of a Navier–Stokes liquid around a rigid body, Arch. Ration. Mech. Anal., Volume 184 (2007), pp. 371–400 | DOI | MR | Zbl

[19] Galdi, G.P.; Silvestre, A.L. Further results on steady-state flow of a Navier–Stokes liquid around a rigid body. Existence of the wake, RIMS Kôkyûroku Bessatsu, Volume B1 (2007), pp. 127–143 | MR | Zbl

[20] Hishida, T. $L^{q}$ estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Jpn., Volume 58 (2006), pp. 743–767 | DOI | MR | Zbl

[21] Korolev, A.; Šverák, V. On the large-distance asymptotics of steady state solutions of the Navier–Stokes equations in 3D exterior domains, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011), pp. 303–313 | DOI | Numdam | MR | Zbl

[22] Kracmar, S.; Novotny, A.; Pokorny, M. Estimates of Oseen kernels in weighted $L^{p}$ spaces, J. Math. Soc. Jpn., Volume 53 (2001), pp. 59–111 | DOI | MR | Zbl

[23] Kyed, M. On the asymptotic structure of a Navier–Stokes flow past a rotating body, J. Math. Soc. Jpn., Volume 66 (2014), pp. 1–16 | DOI | MR | Zbl

[24] San Martín, J.; Takahashi, T.; Tucsnak, M. A control theoretic approach to the swimming of microscopic organisms, Q. Appl. Math., Volume 65 (2007), pp. 405–424 | MR | Zbl

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