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March  2015, 4(1): 69-87. doi: 10.3934/eect.2015.4.69

The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids

1. 

Department of Mathematics, TU Darmstadt, Schlossgartenstr, 7, D-64289 Darmstadt, Germany

2. 

Department of Pure and Applied Mathematics, Graduate School of Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

Received  December 2014 Revised  January 2015 Published  February 2015

Consider the system of equations describing the motion of a rigid body immersed in a viscous, compressible fluid within the barotropic regime. It is shown that this system admits a unique, local strong solution within the $L^p$-setting.
Citation: Matthias Hieber, Miho Murata. The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids. Evolution Equations and Control Theory, 2015, 4 (1) : 69-87. doi: 10.3934/eect.2015.4.69
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Grundlehren, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. Bemelmanns, G. P. Galdi and M. Kyed, On steady motion of a coupled system solid-liquid, Mem. Amer. Math. Soc., 226 (2013), vi+89 pp. doi: 10.1090/S0065-9266-2013-00678-8.

[3]

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.

[4]

C. Conca, J. San Martín and M. Tucsnak, Existence of solutions for equations modeling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540.

[5]

P. Cumsille and T. Takahashi, Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid, Czechoslovak Math. J., 58 (2008), 961-992. doi: 10.1007/s10587-008-0063-2.

[6]

P. Cumsille and M. Tucsnak, 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.

[7]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type, Memoirs Amer. Math. Soc., No. 788, 2003.

[8]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$ estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[9]

B. Desjardins and M. Esteban, Existence of weak solutions for rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136.

[10]

B. Desjardins and M. 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.

[11]

B. Ducomet and S. Nečsová, On the motion of rigid bodies in a compressible viscous fluid under the action of gravitation forces, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1193-1213. doi: 10.3934/dcdss.2013.6.1193.

[12]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[13]

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.

[14]

E. Feireisl, M. Hillairet and S. Necasova, 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.

[15]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in Handbook of Mathematical Fluid Dynamics. Vol. I (in S. J. Friedlander and D. Serre), North-Holland, Amsterdam, 2002, 653-791.

[16]

G. P. Galdi and A. Silvestre, Strong solutions to the problem of motion of a rigid body in a Navier-stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics, I, Int. Math. Ser. (N. Y.), 1, Kluwer, New York, 2002, 121-144. doi: 10.1007/978-1-4615-0777-2_8.

[17]

M. Geissert, K. Götze and M. Hieber, $L^p$-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Trans. Amer. Math. Soc., 365 (2013), 1393-1439. doi: 10.1090/S0002-9947-2012-05652-2.

[18]

K. Götze, Maximal $L^p$-regularity for 2D fluid-solid interaction problem, Operator Theory: Advances and Applications, 221 (2012), 373-384. doi: 10.1007/978-3-0348-0297-0_19.

[19]

M. D. Gunzburger, H. -C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954.

[20]

K.-H. Hoffmann and V. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648.

[21]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 24 (1977), 303-319.

[22]

M. Murata, On a maximal $L_p$-$L_q$ approach to the compressible viscous fluid flow with slip boundary condition, Nonlinear Anal., 106 (2014), 86-109. doi: 10.1016/j.na.2014.04.012.

[23]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flows, Oxford University Press, Oxford, 2004.

[24]

J. San Martín, J. Scheid, T. Takahashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming, Arch. Ration. Mech. Anal., 188 (2008), 429-455. doi: 10.1007/s00205-007-0092-2.

[25]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.

[26]

Y. Shibata, On the global well-posedness of some free boundary problem for compressible barotoropic viscous fluid flow,, Preprint., (). 

[27]

P. E. Sobolevskii, Fractional powers of coercively positive sums of operators, Dokl. Akad. Nauk SSSR., 225 (1975), 1271-1274.

[28]

G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equation, Analysis, 9 (1989), 1-39. doi: 10.1524/anly.1989.9.12.1.

[29]

T. Takahashi, Analysis of strong solutions for equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.

[30]

T. Takahashi and M. Tucsnak, 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.

[31]

H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid, Proc. Symp. Pure Math., 23 (1973), 421-439.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Grundlehren, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. Bemelmanns, G. P. Galdi and M. Kyed, On steady motion of a coupled system solid-liquid, Mem. Amer. Math. Soc., 226 (2013), vi+89 pp. doi: 10.1090/S0065-9266-2013-00678-8.

[3]

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.

[4]

C. Conca, J. San Martín and M. Tucsnak, Existence of solutions for equations modeling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. doi: 10.1080/03605300008821540.

[5]

P. Cumsille and T. Takahashi, Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid, Czechoslovak Math. J., 58 (2008), 961-992. doi: 10.1007/s10587-008-0063-2.

[6]

P. Cumsille and M. Tucsnak, 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.

[7]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type, Memoirs Amer. Math. Soc., No. 788, 2003.

[8]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$ estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[9]

B. Desjardins and M. Esteban, Existence of weak solutions for rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136.

[10]

B. Desjardins and M. 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.

[11]

B. Ducomet and S. Nečsová, On the motion of rigid bodies in a compressible viscous fluid under the action of gravitation forces, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1193-1213. doi: 10.3934/dcdss.2013.6.1193.

[12]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[13]

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.

[14]

E. Feireisl, M. Hillairet and S. Necasova, 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.

[15]

G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, in Handbook of Mathematical Fluid Dynamics. Vol. I (in S. J. Friedlander and D. Serre), North-Holland, Amsterdam, 2002, 653-791.

[16]

G. P. Galdi and A. Silvestre, Strong solutions to the problem of motion of a rigid body in a Navier-stokes liquid under the action of prescribed forces and torques, in Nonlinear Problems in Mathematical Physics and Related Topics, I, Int. Math. Ser. (N. Y.), 1, Kluwer, New York, 2002, 121-144. doi: 10.1007/978-1-4615-0777-2_8.

[17]

M. Geissert, K. Götze and M. Hieber, $L^p$-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Trans. Amer. Math. Soc., 365 (2013), 1393-1439. doi: 10.1090/S0002-9947-2012-05652-2.

[18]

K. Götze, Maximal $L^p$-regularity for 2D fluid-solid interaction problem, Operator Theory: Advances and Applications, 221 (2012), 373-384. doi: 10.1007/978-3-0348-0297-0_19.

[19]

M. D. Gunzburger, H. -C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954.

[20]

K.-H. Hoffmann and V. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648.

[21]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 24 (1977), 303-319.

[22]

M. Murata, On a maximal $L_p$-$L_q$ approach to the compressible viscous fluid flow with slip boundary condition, Nonlinear Anal., 106 (2014), 86-109. doi: 10.1016/j.na.2014.04.012.

[23]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flows, Oxford University Press, Oxford, 2004.

[24]

J. San Martín, J. Scheid, T. Takahashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming, Arch. Ration. Mech. Anal., 188 (2008), 429-455. doi: 10.1007/s00205-007-0092-2.

[25]

D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110. doi: 10.1007/BF03167757.

[26]

Y. Shibata, On the global well-posedness of some free boundary problem for compressible barotoropic viscous fluid flow,, Preprint., (). 

[27]

P. E. Sobolevskii, Fractional powers of coercively positive sums of operators, Dokl. Akad. Nauk SSSR., 225 (1975), 1271-1274.

[28]

G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equation, Analysis, 9 (1989), 1-39. doi: 10.1524/anly.1989.9.12.1.

[29]

T. Takahashi, Analysis of strong solutions for equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.

[30]

T. Takahashi and M. Tucsnak, 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.

[31]

H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid, Proc. Symp. Pure Math., 23 (1973), 421-439.

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