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Lack of collision between solid bodies in a 2D incompressible viscous flow. (English) Zbl 1221.35279
The motion of a solid body \(B\) inside a cavity \(\Omega\subset \mathbb{R}^2\) filled with an incompressible constant-density viscous fluid is under consideration. In absence of external forces, the action of the fluid causes the body to move. Usually, the incompressible Navier-Stokes equations (resp. the solid mechanics equations) in order to describe the behaviour of the fluid (resp. the body) are used. Here this system named full Navier-Stokes for short interactions between the body and the fluid holds in two ways: the incompressible Navier-Stokes equations are supplemented with Dirichlet boundary conditions over the interface fluid/body and the force and torque applied by the fluid on the body are taken into account in solid mechanics relations.
The main result in the study is: Assume \(\Omega\) is a half-space in \(\mathbb{R}^2\) and \(B\) is a disk with radius 1. Furthermore, assume there is no external force. Then, any strong solution is global. In particular, for any strong solution, there is no collision between \(B\) and the boundary of \(\Omega\).

MSC:
35Q30 Navier-Stokes equations
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35R35 Free boundary problems for PDEs
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[1] DOI: 10.1080/03605300008821540 · Zbl 0954.35135 · doi:10.1080/03605300008821540
[2] DOI: 10.1007/s002050050136 · Zbl 0943.35063 · doi:10.1007/s002050050136
[3] DOI: 10.1023/A:1023245704966 · Zbl 1090.35137 · doi:10.1023/A:1023245704966
[4] DOI: 10.1007/s00205-002-0242-5 · Zbl 1090.76061 · doi:10.1007/s00205-002-0242-5
[5] DOI: 10.1007/s00028-003-0110-1 · Zbl 1039.76071 · doi:10.1007/s00028-003-0110-1
[6] Grandmont C., C. R. Acad. Sci. Paris Sér. I Math. 326 pp 525– (1998) · Zbl 0924.76022 · doi:10.1016/S0764-4442(97)89804-7
[7] DOI: 10.1007/PL00000954 · Zbl 0970.35096 · doi:10.1007/PL00000954
[8] Hesla , T. I. ( 2005 ). Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation . PhD thesis, University of Minnesota, revised version .
[9] Hillairet M., Asymptotic Analysis 43 (2005) · Zbl 1108.35013
[10] Hillairet , M. ( 2005b ). Interactive Features in Fluid Mechanics . PhD thesis, Ecole normale supérieure de Lyon .
[11] Hoffmann K.-H., Adv. Math. Sci. Appl. 9 pp 633– (1999)
[12] DOI: 10.1007/s002050100172 · Zbl 1018.76012 · doi:10.1007/s002050100172
[13] Starovoitov V. N., Internat. Ser. Numer. Math. 147 pp 313– (2002)
[14] DOI: 10.1007/s10958-005-0384-8 · Zbl 1148.35347 · doi:10.1007/s10958-005-0384-8
[15] Takahashi T., C. R. Math. Acad. Sci. Paris 336 pp 453– (2003) · Zbl 1044.35062 · doi:10.1016/S1631-073X(03)00081-5
[16] DOI: 10.1007/s00021-003-0083-4 · Zbl 1054.35061 · doi:10.1007/s00021-003-0083-4
[17] Temam R., Navier–Stokes Equations (1977)
[18] DOI: 10.1142/S0218202506001303 · Zbl 1387.35634 · doi:10.1142/S0218202506001303
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