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 & 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. doi: 10.1016/j.anihpc.2008.02.004. Google Scholar

[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. doi: 10.1137/090767856. Google Scholar

[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. doi: 10.1080/03605300008821540. Google Scholar

[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. doi: 10.1007/s002050050136. Google Scholar

[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. doi: 10.1080/03605300008821553. Google Scholar

[6]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[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. doi: 10.3934/dcds.2004.11.113. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00033-009-0035-x. Google Scholar

[10]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 26 (2004). Google Scholar

[11]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. doi: 10.1007/s00028-003-0110-1. Google Scholar

[12]

E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid,, Arch. Ration. Mech. Anal., 167 (2003), 281. doi: 10.1007/s00205-002-0242-5. Google Scholar

[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. doi: 10.1088/0951-7715/21/6/012. Google Scholar

[14]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (2009). doi: 10.1007/978-3-7643-8843-0. Google Scholar

[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. doi: 10.1007/PL00000976. Google Scholar

[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. doi: 10.1080/03605300903380746. Google Scholar

[17]

J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids,, Math. Z., 260 (2008), 355. doi: 10.1007/s00209-007-0278-1. Google Scholar

[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. doi: 10.1007/s002050050156. Google Scholar

[19]

G. P. Galdi, On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications,, in, (2002). Google Scholar

[20]

D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction,, Arch. Ration. Mech. Anal., 195 (2010), 375. doi: 10.1007/s00205-008-0202-9. Google Scholar

[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. doi: 10.1007/PL00000954. Google Scholar

[22]

T. I. Hesla, "Collision of Smooth Bodies in a Viscous Fluid: A Mathematical Investigation,", PhD Thesis, (2005). Google Scholar

[23]

M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow,, Comm. Partial Differential Equations, 32 (2007), 1345. doi: 10.1080/03605300601088740. Google Scholar

[24]

M. Hillairet and T. Takahashi, Collisions in three dimensional fluid structure interaction problems,, SIAM J. Math. Anal., 40 (2009), 2451. doi: 10.1137/080716074. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998). Google Scholar

[29]

E. H. Lieb and M. Loss, "Analysis,", Second edition, 14 (2001). Google Scholar

[30]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 2, (1968). Google Scholar

[31]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996). Google Scholar

[32]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Series in Mathematics and its Applications, 27 (2004). Google Scholar

[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. doi: 10.1007/s002050100172. Google Scholar

[34]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Jap. J. Appl. Math., 4 (1987), 99. doi: 10.1007/BF03167757. Google Scholar

[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. doi: 10.1007/s10958-005-0384-8. Google Scholar

[36]

V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary,, in, 147 (2004), 313. Google Scholar

[37]

H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid,, in, (1973), 421. Google Scholar

[38]

H. F. Weinberger, Variational properties of steady fall in Stokes flow,, J. Fluid Mech., 52 (1972), 321. Google Scholar

[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. doi: 10.1007/s00021-006-0219-5. Google Scholar

show all references

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. doi: 10.1016/j.anihpc.2008.02.004. Google Scholar

[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. doi: 10.1137/090767856. Google Scholar

[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. doi: 10.1080/03605300008821540. Google Scholar

[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. doi: 10.1007/s002050050136. Google Scholar

[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. doi: 10.1080/03605300008821553. Google Scholar

[6]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511. doi: 10.1007/BF01393835. Google Scholar

[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. doi: 10.3934/dcds.2004.11.113. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00033-009-0035-x. Google Scholar

[10]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 26 (2004). Google Scholar

[11]

E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid,, J. Evol. Equ., 3 (2003), 419. doi: 10.1007/s00028-003-0110-1. Google Scholar

[12]

E. Feireisl, On the motion of rigid bodies in a viscous compressible fluid,, Arch. Ration. Mech. Anal., 167 (2003), 281. doi: 10.1007/s00205-002-0242-5. Google Scholar

[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. doi: 10.1088/0951-7715/21/6/012. Google Scholar

[14]

E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (2009). doi: 10.1007/978-3-7643-8843-0. Google Scholar

[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. doi: 10.1007/PL00000976. Google Scholar

[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. doi: 10.1080/03605300903380746. Google Scholar

[17]

J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids,, Math. Z., 260 (2008), 355. doi: 10.1007/s00209-007-0278-1. Google Scholar

[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. doi: 10.1007/s002050050156. Google Scholar

[19]

G. P. Galdi, On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications,, in, (2002). Google Scholar

[20]

D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction,, Arch. Ration. Mech. Anal., 195 (2010), 375. doi: 10.1007/s00205-008-0202-9. Google Scholar

[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. doi: 10.1007/PL00000954. Google Scholar

[22]

T. I. Hesla, "Collision of Smooth Bodies in a Viscous Fluid: A Mathematical Investigation,", PhD Thesis, (2005). Google Scholar

[23]

M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow,, Comm. Partial Differential Equations, 32 (2007), 1345. doi: 10.1080/03605300601088740. Google Scholar

[24]

M. Hillairet and T. Takahashi, Collisions in three dimensional fluid structure interaction problems,, SIAM J. Math. Anal., 40 (2009), 2451. doi: 10.1137/080716074. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,", Oxford Lecture Series in Mathematics and its Applications, 10 (1998). Google Scholar

[29]

E. H. Lieb and M. Loss, "Analysis,", Second edition, 14 (2001). Google Scholar

[30]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 2, (1968). Google Scholar

[31]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996). Google Scholar

[32]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,", Oxford Lecture Series in Mathematics and its Applications, 27 (2004). Google Scholar

[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. doi: 10.1007/s002050100172. Google Scholar

[34]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Jap. J. Appl. Math., 4 (1987), 99. doi: 10.1007/BF03167757. Google Scholar

[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. doi: 10.1007/s10958-005-0384-8. Google Scholar

[36]

V. N. Starovoitov, Behavior of a rigid body in an incompressible viscous fluid near boundary,, in, 147 (2004), 313. Google Scholar

[37]

H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid,, in, (1973), 421. Google Scholar

[38]

H. F. Weinberger, Variational properties of steady fall in Stokes flow,, J. Fluid Mech., 52 (1972), 321. Google Scholar

[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. doi: 10.1007/s00021-006-0219-5. Google Scholar

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