On the “viscous incompressible fluid+rigid body” system with Navier conditions
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 55-80.

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole system “viscous incompressible fluid+rigid body” is assumed to occupy the full space ${ℝ}^{3}$. We start by proving the existence of global weak solutions to the Cauchy problem. Then, we exhibit several properties of these solutions. First, we show that the added-mass effect can be computed which yields better-than-expected regularity (in time) of the solid velocity-field. More precisely we prove that the solid translation and rotation velocities are in the Sobolev space ${H}^{1}$. Second, we show that the case with the body fixed can be thought as the limit of infinite inertia of this system, that is when the solid density is multiplied by a factor converging to +∞. Finally we prove the convergence in the energy space of weak solutions “à la Leray” to smooth solutions of the system “inviscid incompressible fluid+rigid body” as the viscosity goes to zero, till the lifetime T of the smooth solution of the inviscid system. Moreover we show that the rate of convergence is optimal with respect to the viscosity and that the solid translation and rotation velocities converge in ${H}^{1}\left(0,T\right)$.

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title = {On the {\textquotedblleft}viscous incompressible fluid+rigid body{\textquotedblright} system with {Navier} conditions},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {55--80},
publisher = {Elsevier},
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Planas, Gabriela; Sueur, Franck. On the “viscous incompressible fluid+rigid body” system with Navier conditions. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 55-80. doi : 10.1016/j.anihpc.2013.01.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.004/

[1] C. Bardos, F. Golse, L. Paillard, The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition, Commun. Math. Sci. 10 no. 1 (2012), 159-190 | MR | Zbl

[2] C. Bardos, E.S. Titi, Euler equations for an ideal incompressible fluid, Uspekhi Mat. Nauk 62 no. 3(375) (2007), 5-46, Russian Math. Surveys 62 no. 3 (2007), 409-451 | MR | Zbl

[3] H. Beirao Da Veiga, F. Crispo, A missed persistence property for the Euler equations, and its effect on inviscid limits, Nonlinearity 25 (2012), 1661-1669 | MR | Zbl

[4] T. Chambrion, A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlinear Sci. 21 no. 3 (2011), 325-385 | MR | Zbl

[5] M. Chapouly, On the global null controllability of a Navier–Stokes system with Navier slip boundary conditions, J. Differential Equations 247 no. 7 (2009), 2094-2123 | MR | Zbl

[6] S. Childress, An Introduction to Theoretical Fluid Mechanics, Courant Lect. Notes Math. vol. 19, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI (2009) | MR | Zbl

[7] T. Clopeau, A. Mikelic, R. Robert, On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions, Nonlinearity 11 no. 6 (1998), 1625-1636 | MR | Zbl

[8] C. Conca, J.A. San Martin, M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 no. 5–6 (2000), 1019-1042 | Zbl

[9] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.) 44 no. 4 (2007), 603-621 | MR | Zbl

[10] F. Coron, Derivation of slip boundary conditions for the Navier–Stokes system from the Boltzmann equation, J. Stat. Phys. 54 no. 3–4 (1989), 829-857 | MR | Zbl

[11] J.-M. Coron, On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var. 1 (1995/1996), 35-75 | EuDML | Numdam | MR | Zbl

[12] J.-M. Coron, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim. 37 no. 6 (1999), 1874-1896 | MR | Zbl

[13] A.-L. Dalibard, D. Gérard-Varet, Effective boundary condition at a rough surface starting from a slip condition, J. Differential Equations 251 no. 12 (2011), 3450-3487 | MR | Zbl

[14] B. Desjardins, M.J. Esteban, On weak solutions for fluid–rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25 no. 7–8 (2000), 1399-1413 | MR | Zbl

[15] B. Desjardins, M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal. 146 no. 1 (1999), 59-71 | MR | Zbl

[16] W. E, Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 no. 2 (2000), 207-218 | MR | Zbl

[17] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3 no. 3 (2003), 419-441 | MR | Zbl

[18] E. Feireisl, On the motion of rigid bodies in a viscous fluid, Mathematical Theory in Fluid Mechanics Paseky, 2001 Appl. Math. 47 no. 6 (2002), 463-484 | EuDML | MR | Zbl

[19] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal. 167 no. 4 (2003), 281-308 | MR | Zbl

[20] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I, Springer, New York (1994) | MR | Zbl

[21] D. Gérard-Varet, E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 no. 2 (2010), 591-609 | MR | Zbl

[22] D. Gérard-Varet, M. Hillairet, Existence of weak solutions up to collision for viscous fluid–solid systems with slip, preprint, 2012, arXiv:1207.0469. | MR

[23] D. Gérard-Varet, N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys. 295 no. 1 (2010), 99-137 | MR | Zbl

[24] P. Gamblin, X. Saint Raymond, On three-dimensional vortex patches, Bull. Soc. Math. France 123 no. 3 (1995), 375-424 | EuDML | Numdam | MR | Zbl

[25] O. Glass, C. Lacave, F. Sueur, On the motion of a small body immersed in a two dimensional incompressible perfect fluid, preprint, 2011, arXiv:1104.5404, Bull. Soc. Math. France, in press. | MR

[26] O. Glass, F. Sueur, On the motion of a rigid body in a two-dimensional irregular ideal flow, SIAM J. Math. Anal. 44 no. 5 (2012), 3101-3126 | MR | Zbl

[27] O. Glass, F. Sueur, Low regularity solutions for the two-dimensional “rigid body+incompressible Euler” system, preprint, 2012, hal-00682976. | MR

[28] O. Glass, F. Sueur, The movement of a solid in an incompressible perfect fluid as a geodesic flow, Proc. Amer. Math. Soc. 140 no. 6 (2012), 2155-2168 | MR | Zbl

[29] O. Glass, F. Sueur, Uniqueness results for weak solutions of two-dimensional fluid–solid systems, preprint, 2012, arXiv:1203.2894. | MR

[30] O. Glass, F. Sueur, T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, Ann. Sci. École Norm. Sup. 45 no. 1 (2012), 1-51 | EuDML | Numdam | MR | Zbl

[31] C. Grandmont, Y. Maday, Existence for an unsteady fluid–structure interaction problem, M2AN Math. Model. Numer. Anal. 34 no. 3 (2000), 609-636 | EuDML | Numdam | MR | Zbl

[32] E. Grenier, Boundary layers, Handbook of Mathematical Fluid Dynamics, vol. III, Elsevier (2004), 245-309 | MR | Zbl

[33] Y. Guo, T.T. Nguyen, A note on the Prandtl boundary layers, Comm. Pure Appl. Math. 64 no. 10 (2011), 1416-1438 | MR | Zbl

[34] K.-H. Hoffmann, V.N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl. 9 no. 2 (1999), 633-648 | MR | Zbl

[35] J.-G. Houot, J. San Martin, M. Tucsnak, Existence and uniqueness of solutions for the equations modelling the motion of rigid bodies in a perfect fluid, J. Funct. Anal. 259 no. 11 (2010), 2856-2885 | MR | Zbl

[36] D. Iftimie, G. Planas, Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006), 899-918 | MR | Zbl

[37] D. Iftimie, F. Sueur, Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199 no. 1 (2011), 145-175 | MR | Zbl

[38] T. Kato, Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary, Seminar on Nonlinear Partial Differential Equations Math. Sci. Res. Inst. Publ. 2 (1984), 85-98 | MR | Zbl

[39] N. Masmoudi, F. Rousset, Uniform regularity for the Navier–Stokes equations with Navier boundary condition, Arch. Ration. Mech. Anal. 203 no. 2 (2012), 529-575 | MR | Zbl

[40] N. Masmoudi, L. Saint-Raymond, From the Boltzmann equation to the Stokes–Fourier system in a bounded domain, Comm. Pure Appl. Math. 56 no. 9 (2003), 1263-1293 | MR | Zbl

[41] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge (2000) | MR | Zbl

[42] C.-L. Navier, Mémoire sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Paris 6 (1823), 389-416

[43] J.H. Ortega, L. Rosier, T. Takahashi, Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid, M2AN Math. Model. Numer. Anal. 39 no. 1 (2005), 79-108 | EuDML | Numdam | MR | Zbl

[44] J.H. Ortega, L. Rosier, T. Takahashi, On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 no. 1 (2007), 139-165 | EuDML | MR | Zbl

[45] M. Paddick, Stability and instability of Navier boundary layers, preprint, 2011, arXiv:1103.5009. | MR

[46] C. Rosier, L. Rosier, Smooth solutions for the motion of a ball in an incompressible perfect fluid, J. Funct. Anal. 256 no. 5 (2009), 1618-1641 | MR | Zbl

[47] J.A. San Martin, V. Starovoitov, M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 no. 2 (2002), 113-147 | MR | Zbl

[48] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math. 4 no. 1 (1987), 99-110 | MR | Zbl

[49] J. Simon, Compact sets in ${L}^{p}\left(0,T;B\right)$, Ann. Mat. Pura Appl. (4) CXLVI (1987), 65-96 | MR | Zbl

[50] F. Sueur, A Kato type theorem for the inviscid limit of the Navier–Stokes equations with a moving rigid body, Comm. Math. Phys. 316 no. 3 (2012), 783-808 | MR | Zbl

[51] R. Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l'Informatique vol. 12, Gauthier–Villars (1983) | MR | Zbl

[52] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, Stud. Math. Appl. vol. 2, North-Holland, The Netherlands (1984) | MR | Zbl

[53] L. Wang, Z. Xin, A. Zang, Vanishing viscous limits for 3D Navier–Stokes equations with Navier-slip boundary conditions, J. Math. Fluid Mech. 14 no. 4 (2012), 791-825 | MR | Zbl

[54] Y. Wang, A. Zang, Smooth solutions for motion of a rigid body of general form in an incompressible perfect fluid, J. Differential Equations 252 (2012), 4259-4288 | MR | Zbl

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