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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 & 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, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar [2] J. Bemelmanns, G. P. Galdi and M. Kyed, On steady motion of a coupled system solid-liquid,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00678-8.  Google Scholar [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.  doi: 10.1016/j.anihpc.2008.02.004.  Google Scholar [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.  doi: 10.1080/03605300008821540.  Google Scholar [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.  doi: 10.1007/s10587-008-0063-2.  Google Scholar [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.  doi: 10.1002/mma.702.  Google Scholar [7] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type,, Memoirs Amer. Math. Soc., (2003).   Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [9] B. Desjardins and M. Esteban, Existence of weak solutions for rigid bodies in a viscous fluid,, Arch. Ration. Mech. Anal., 146 (1999), 59.  doi: 10.1007/s002050050136.  Google Scholar [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.  doi: 10.1080/03605300008821553.  Google Scholar [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.  doi: 10.3934/dcdss.2013.6.1193.  Google Scholar [12] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [13] 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 [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.  doi: 10.1088/0951-7715/21/6/012.  Google Scholar [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), (2002), 653.   Google Scholar [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, (2002), 121.  doi: 10.1007/978-1-4615-0777-2_8.  Google Scholar [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.  doi: 10.1090/S0002-9947-2012-05652-2.  Google Scholar [18] K. Götze, Maximal $L^p$-regularity for 2D fluid-solid interaction problem,, Operator Theory: Advances and Applications, 221 (2012), 373.  doi: 10.1007/978-3-0348-0297-0_19.  Google Scholar [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.  doi: 10.1007/PL00000954.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.na.2014.04.012.  Google Scholar [23] A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flows,, Oxford University Press, (2004).   Google Scholar [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.  doi: 10.1007/s00205-007-0092-2.  Google Scholar [25] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  Google Scholar [26] Y. Shibata, On the global well-posedness of some free boundary problem for compressible barotoropic viscous fluid flow,, Preprint., ().   Google Scholar [27] P. E. Sobolevskii, Fractional powers of coercively positive sums of operators,, Dokl. Akad. Nauk SSSR., 225 (1975), 1271.   Google Scholar [28] G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equation,, Analysis, 9 (1989), 1.  doi: 10.1524/anly.1989.9.12.1.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar [31] H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid,, Proc. Symp. Pure Math., 23 (1973), 421.   Google Scholar

show all references

##### References:
 [1] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar [2] J. Bemelmanns, G. P. Galdi and M. Kyed, On steady motion of a coupled system solid-liquid,, Mem. Amer. Math. Soc., 226 (2013).  doi: 10.1090/S0065-9266-2013-00678-8.  Google Scholar [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.  doi: 10.1016/j.anihpc.2008.02.004.  Google Scholar [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.  doi: 10.1080/03605300008821540.  Google Scholar [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.  doi: 10.1007/s10587-008-0063-2.  Google Scholar [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.  doi: 10.1002/mma.702.  Google Scholar [7] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type,, Memoirs Amer. Math. Soc., (2003).   Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [9] B. Desjardins and M. Esteban, Existence of weak solutions for rigid bodies in a viscous fluid,, Arch. Ration. Mech. Anal., 146 (1999), 59.  doi: 10.1007/s002050050136.  Google Scholar [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.  doi: 10.1080/03605300008821553.  Google Scholar [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.  doi: 10.3934/dcdss.2013.6.1193.  Google Scholar [12] E. Feireisl, Dynamics of Viscous Compressible Fluids,, Oxford University Press, (2004).   Google Scholar [13] 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 [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.  doi: 10.1088/0951-7715/21/6/012.  Google Scholar [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), (2002), 653.   Google Scholar [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, (2002), 121.  doi: 10.1007/978-1-4615-0777-2_8.  Google Scholar [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.  doi: 10.1090/S0002-9947-2012-05652-2.  Google Scholar [18] K. Götze, Maximal $L^p$-regularity for 2D fluid-solid interaction problem,, Operator Theory: Advances and Applications, 221 (2012), 373.  doi: 10.1007/978-3-0348-0297-0_19.  Google Scholar [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.  doi: 10.1007/PL00000954.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.na.2014.04.012.  Google Scholar [23] A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flows,, Oxford University Press, (2004).   Google Scholar [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.  doi: 10.1007/s00205-007-0092-2.  Google Scholar [25] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99.  doi: 10.1007/BF03167757.  Google Scholar [26] Y. Shibata, On the global well-posedness of some free boundary problem for compressible barotoropic viscous fluid flow,, Preprint., ().   Google Scholar [27] P. E. Sobolevskii, Fractional powers of coercively positive sums of operators,, Dokl. Akad. Nauk SSSR., 225 (1975), 1271.   Google Scholar [28] G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equation,, Analysis, 9 (1989), 1.  doi: 10.1524/anly.1989.9.12.1.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00021-003-0083-4.  Google Scholar [31] H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid,, Proc. Symp. Pure Math., 23 (1973), 421.   Google Scholar
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