• Previous Article
    Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions
  • EECT Home
  • This Issue
  • Next Article
    Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller
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.

[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.

[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.

[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.

[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.

[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.

[7]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type,, Memoirs Amer. Math. Soc., (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. 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. 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. 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. doi: 10.3934/dcdss.2013.6.1193.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[31]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier Multiplier and Problems of Elliptic and Parabolic Type,, Memoirs Amer. Math. Soc., (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. 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. 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. 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. doi: 10.3934/dcdss.2013.6.1193.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[31]

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

[1]

Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539

[2]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133

[3]

George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557

[4]

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

[5]

Stuart S. Antman, David Bourne. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 123-142. doi: 10.3934/cpaa.2009.8.123

[6]

Hugo Beirão da Veiga. Turbulence models, $p-$fluid flows, and $W^{2, L}$ regularity of solutions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 769-783. doi: 10.3934/cpaa.2009.8.769

[7]

Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439

[8]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633

[9]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635

[10]

I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759

[11]

Johannes Elschner, George C. Hsiao, Andreas Rathsfeld. An inverse problem for fluid-solid interaction. Inverse Problems & Imaging, 2008, 2 (1) : 83-120. doi: 10.3934/ipi.2008.2.83

[12]

Peter Monk, Virginia Selgas. An inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2009, 3 (2) : 173-198. doi: 10.3934/ipi.2009.3.173

[13]

Fabien Caubet, Marc Dambrine, Djalil Kateb, Chahnaz Zakia Timimoun. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid. Inverse Problems & Imaging, 2013, 7 (1) : 123-157. doi: 10.3934/ipi.2013.7.123

[14]

Francesca Bucci, Irena Lasiecka. Regularity of boundary traces for a fluid-solid interaction model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 505-521. doi: 10.3934/dcdss.2011.4.505

[15]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[16]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199

[17]

Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787

[18]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

[19]

Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122

[20]

Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55

2016 Impact Factor: 0.826

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]