# American Institute of Mathematical Sciences

October  2013, 6(5): 1193-1213. doi: 10.3934/dcdss.2013.6.1193

## On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces

 1 CEA, DAM, DIF, F-91297 Arpajon, France 2 Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic

Received  December 2011 Revised  April 2012 Published  March 2013

The global existence of weak solutions is proved for the problem of the motion of several rigid bodies either in a non-Newtonian fluid of power law type or in a barotropic compressible fluid, under the influence of gravitational forces.
Citation: Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193
##### References:
 [1] M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. doi: 10.1016/j.anihpc.2008.02.004. [2] C. Bost, G.-H. Cottet and E. Maitre, Convergence analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid, SIAM J. Numer. Anal., 48 (2010), 1313-1337. doi: 10.1137/090767856. [3] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540. [4] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. [5] B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: Compressible and incompressible models, Comm. Partial Differential Equations, 25 (2000), 1399-1413. doi: 10.1080/03605300008821553. [6] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. [7] B. Ducomet, E. Feireisl, H. Petzeltov\'a and I. Stra\v skraba, Global in time weak solutions for compressible barotropic self-gravitating fluid, Discrete Contin. Dyn. Syst., 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113. [8] B. Ducomet and Š. Nečasová, On the motion of several rigid bodies in an incompressible viscous fluid under the influence of selfgravitating forces, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 167-192. [9] B. Ducomet, Š. Nečasová and A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities, Z. Angew. Math. Phys., 61 (2010), 479-491. doi: 10.1007/s00033-009-0035-x. [10] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [11] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [12] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal., 167 (2003), 281-308. doi: 10.1007/s00205-002-0242-5. [13] E. Feireisl, M. Hillairet and Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian fluid, Nonlinearity, 21 (2008), 1349-1366. doi: 10.1088/0951-7715/21/6/012. [14] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0. [15] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. [16] J. Frehse, J. Málek and M. Růžička, Large data existence result for unsteady flows of inhomogeneous heat-conducting incompressible fluids, Comm. Partial Differential Equations, 35 (2010), 1891-2191. doi: 10.1080/03605300903380746. [17] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1. [18] G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88. doi: 10.1007/s002050050156. [19] G. P. Galdi, On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications, in "Handbook of Mathematical Fluid Dynamics," Vol. I, Elsevier Sci., Amsterdam, 2002. [20] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407. doi: 10.1007/s00205-008-0202-9. [21] M. D. Gunzburger, H. C. Lee and A. Seregin, Global existence of weak solutions for viscous incompressible flow around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. [22] T. I. Hesla, "Collision of Smooth Bodies in a Viscous Fluid: A Mathematical Investigation," PhD Thesis, Minnesota, 2005. [23] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations, 32 (2007), 1345-1371. doi: 10.1080/03605300601088740. [24] M. Hillairet and T. Takahashi, Collisions in three dimensional fluid structure interaction problems, SIAM J. Math. Anal., 40 (2009), 2451-2477. doi: 10.1137/080716074. [25] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. [26] K.-H. Hoffmann and V. N. Starovoitov, Zur Bewegung einer Kugel in einer zähen Flüssigkeit, (German) [On the motion of a sphere in a viscous fluid], Doc. Math., 5 (2000), 15-21. [27] N. V. Judakov, The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid, (Russian) Dinamika Splošn. Sredy Vyp., 18 (1974), 249-253, 255. [28] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [29] E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, AMS, Providence, RI, 2001. [30] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968. [31] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [32] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press, Oxford, 2004. [33] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [34] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Jap. J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757. [35] V. N. Starovoĭtov, On the nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid, J. Math. Sci., 130 (2005), 4893-4898. doi: 10.1007/s10958-005-0384-8. [36] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary, in "Free Boundary Problems" (Trento, 2002), International Series of Numerical Mathematics, 147, Birkhäuser, Basel, (2004), 313-327. [37] H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid, in "Partial Differential Equations" (Proc. Sympos. Pure Math., Vol XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, RI, (1973), 421-439. [38] H. F. Weinberger, Variational properties of steady fall in Stokes flow, J. Fluid Mech., 52 (1972), 321-344. [39] J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.

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##### References:
 [1] M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. doi: 10.1016/j.anihpc.2008.02.004. [2] C. Bost, G.-H. Cottet and E. Maitre, Convergence analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid, SIAM J. Numer. Anal., 48 (2010), 1313-1337. doi: 10.1137/090767856. [3] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540. [4] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. [5] B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: Compressible and incompressible models, Comm. Partial Differential Equations, 25 (2000), 1399-1413. doi: 10.1080/03605300008821553. [6] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. [7] B. Ducomet, E. Feireisl, H. Petzeltov\'a and I. Stra\v skraba, Global in time weak solutions for compressible barotropic self-gravitating fluid, Discrete Contin. Dyn. Syst., 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113. [8] B. Ducomet and Š. Nečasová, On the motion of several rigid bodies in an incompressible viscous fluid under the influence of selfgravitating forces, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 167-192. [9] B. Ducomet, Š. Nečasová and A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities, Z. Angew. Math. Phys., 61 (2010), 479-491. doi: 10.1007/s00033-009-0035-x. [10] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [11] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3 (2003), 419-441. doi: 10.1007/s00028-003-0110-1. [12] E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid, Arch. Ration. Mech. Anal., 167 (2003), 281-308. doi: 10.1007/s00205-002-0242-5. [13] E. Feireisl, M. Hillairet and Š. Nečasová, On the motion of several rigid bodies in an incompressible non-Newtonian fluid, Nonlinearity, 21 (2008), 1349-1366. doi: 10.1088/0951-7715/21/6/012. [14] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids," Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8843-0. [15] E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976. [16] J. Frehse, J. Málek and M. Růžička, Large data existence result for unsteady flows of inhomogeneous heat-conducting incompressible fluids, Comm. Partial Differential Equations, 35 (2010), 1891-2191. doi: 10.1080/03605300903380746. [17] J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1. [18] G. P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Ration. Mech. Anal., 148 (1999), 53-88. doi: 10.1007/s002050050156. [19] G. P. Galdi, On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications, in "Handbook of Mathematical Fluid Dynamics," Vol. I, Elsevier Sci., Amsterdam, 2002. [20] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407. doi: 10.1007/s00205-008-0202-9. [21] M. D. Gunzburger, H. C. Lee and A. Seregin, Global existence of weak solutions for viscous incompressible flow around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. [22] T. I. Hesla, "Collision of Smooth Bodies in a Viscous Fluid: A Mathematical Investigation," PhD Thesis, Minnesota, 2005. [23] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations, 32 (2007), 1345-1371. doi: 10.1080/03605300601088740. [24] M. Hillairet and T. Takahashi, Collisions in three dimensional fluid structure interaction problems, SIAM J. Math. Anal., 40 (2009), 2451-2477. doi: 10.1137/080716074. [25] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. [26] K.-H. Hoffmann and V. N. Starovoitov, Zur Bewegung einer Kugel in einer zähen Flüssigkeit, (German) [On the motion of a sphere in a viscous fluid], Doc. Math., 5 (2000), 15-21. [27] N. V. Judakov, The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid, (Russian) Dinamika Splošn. Sredy Vyp., 18 (1974), 249-253, 255. [28] P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998. [29] E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, AMS, Providence, RI, 2001. [30] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968. [31] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996. [32] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press, Oxford, 2004. [33] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. [34] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Jap. J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757. [35] V. N. Starovoĭtov, On the nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid, J. Math. Sci., 130 (2005), 4893-4898. doi: 10.1007/s10958-005-0384-8. [36] V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary, in "Free Boundary Problems" (Trento, 2002), International Series of Numerical Mathematics, 147, Birkhäuser, Basel, (2004), 313-327. [37] H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid, in "Partial Differential Equations" (Proc. Sympos. Pure Math., Vol XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, RI, (1973), 421-439. [38] H. F. Weinberger, Variational properties of steady fall in Stokes flow, J. Fluid Mech., 52 (1972), 321-344. [39] J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.
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