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# Article

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Keywords:
incompressible fluid; rotating rigid body; strong solution
Summary:
We study a linear system of equations arising from fluid motion around a moving rigid body, where rotation is included. Originally, the coordinate system is attached to the fluid, which means that the domain is changing with respect to time. To get a problem in the fixed domain, the problem is rewritten in the coordinate system attached to the body. The aim of the present paper is the proof of the existence of a strong solution in a weighted Lebesgue space. In particular, we prove the existence of a global pressure gradient in \$L^2\$.
References:
[1] Borchers, W.: Zur Stabilität und Faktorisienrungsmethode für die Navier-Stokes Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift University of Paderborn (1992), German.
[2] Chen, Z.-M., Miyakawa, T.: Decay properties of weak solutions to a perturbed Navier-Stokes system in {\$\mathbb R^n\$}. Adv. Math. Sci. Appl. 7 (1997), 741-770. MR 1476275
[3] Cumsille, P., Takahashi, T.: Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid. Czech. Math. J. 58 (2008), 961-992. DOI 10.1007/s10587-008-0063-2 | MR 2471160 | Zbl 1174.35092
[4] Cumsille, P., Tucsnak, M.: Wellposedness for the Navier-Stokes flow in the exterior of a rotating obstacle. Math. Methods Appl. Sci. 29 (2006), 595-623. DOI 10.1002/mma.702 | MR 2205973
[5] Dintelmann, E., Geissert, M., Hieber, M.: Strong \$L^p\$-solutions to the Navier-Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity. Trans. Am. Math. Soc. 361 (2009), 653-669. DOI 10.1090/S0002-9947-08-04684-9 | MR 2452819 | Zbl 1156.76016
[6] Galdi, G. P.: On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications. Handbook of Mathematical Fluid Dynamics 1 Elsevier Amsterdam (2002), 653-791 S. Friedlander et al. MR 1942470 | Zbl 1230.76016
[7] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations I. Linearized Steady Problems. Springer Tracts in Natural Philosophy 38 Springer, New York (1994). MR 1284205
[8] Galdi, G. P., Silvestre, A. L.: Strong solutions to the Navier-Stokes equations around a rotating obstacle. Arch. Ration. Mech. Anal. 176 (2005), 331-350. DOI 10.1007/s00205-004-0348-z | MR 2185661 | Zbl 1081.35076
[9] Galdi, G. P., Silvestre, A. L.: Strong solutions to the problem of motion of a rigid body in a Navier-Stokes liquid under the action of prescribed forces and torques. Nonlinear Problems in Mathematical Physics and Related Topics I. Int. Math. Ser. (N. Y.) 1 Kluwer Academic/Plenum Publishers, New York (2002), 121-144 M. S. Birman et al. DOI 10.1007/978-1-4615-0777-2_8 | MR 1970608 | Zbl 1046.35084
[10] Geissert, M., Heck, H., Hieber, M.: \$L^p\$-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596 (2006), 45-62. MR 2254804 | Zbl 1102.76015
[11] Hishida, T.: An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 150 (1999), 307-348. DOI 10.1007/s002050050190 | MR 1741259 | Zbl 0949.35106
[12] Hishida, T., Shibata, Y.: \$L_p\$-\$L_q\$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193 (2009), 339-421. DOI 10.1007/s00205-008-0130-8 | MR 2525121 | Zbl 1169.76015
[13] Inoue, A., Wakimoto, M.: On existence of solutions of the Navier-Stokes equation in a time dependent domain. J. Fac. Sci., Univ. Tokyo, Sect. I A 24 (1977), 303-319. MR 0481649 | Zbl 0381.35066
[14] Ladyzhenskaya, O. A.: An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time. Semin. Math., V. A. Steklov Math. Inst., Leningrad 11 (1968), 35-46 translation from Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov. 11 (1968), 97-128 Russian. MR 0416222
[15] Neustupa, J.: Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method. Math. Methods Appl. Sci. 32 (2009), 653-683. DOI 10.1002/mma.1059 | MR 2504002 | Zbl 1160.35494
[16] Neustupa, J., Penel, P.: A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall. Advances in Mathematical Fluid Mechanics Springer, Berlin (2010), 385-407 R. Rannacher et al. MR 2665044
[17] Neustupa, J., Penel, P.: A weak solution to the Navier-Stokes system with Navier's boundary condition in a time varying domain. Accepted to ``Recent Developments of Mathematical Fluid Mechanics'', Series: Advances in Math. Fluid Mech. Birkhäuser G. P. Galdi, J. G. Heywood, R. Rannacher.
[18] Serre, D.: Free fall of a rigid body in an incompressible viscous fluid. Existence. Japan J. Appl. Math. 4 French (1987), 99-110. MR 0899206
[19] Takahashi, T.: Existence of strong solutions for the problem of a rigid-fluid system. C. R., Math., Acad. Sci. Paris 336 (2003), 453-458. DOI 10.1016/S1631-073X(03)00081-5 | MR 1979363 | Zbl 1044.35062
[20] Takahashi, T., Tucsnak, M.: Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6 (2004), 53-77. DOI 10.1007/s00021-003-0083-4 | MR 2027754 | Zbl 1054.35061

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