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July  2014, 13(4): 1629-1639. doi: 10.3934/cpaa.2014.13.1629

Note on evolutionary free piston problem for Stokes equations with slip boundary conditions

Boris Muha 1, and  Zvonimir Tutek 2,

1. 

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
2. 

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Received  April 2012 Revised  October 2012 Published  February 2014

In this paper we study a free boundary fluid-rigid body interaction problem, the free piston problem. We consider an evolutionary incompressible viscous fluid flow through a junction of two pipes. Inside the "vertical" pipe there is a heavy piston which can freely slide along the pipe. On the lateral boundary of the "vertical" pipe we prescribe Navier's slip boundary conditions. We prove the existence of a local in time weak solution. Furthermore, we show that the obtained solution is a strong solution.
Keywords: Navier-Stokes equations, moving boundary, weak solution., Fluid-rigid body interaction.
Mathematics Subject Classification: Primary: 74F10, 35Q30, 76D03; Secondary: 76D0.
Citation: Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629
References:
[1]

A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[2]

C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.), 20 (1994), 279-318.

[3]

C. Conca, J. San Martín H. and M. Tucsnak, Motion of a rigid body in a viscous fluid, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 473-478. doi: 10.1016/S0764-4442(99)80193-1.

[4]

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.

[5]

B. D'Acunto and S. Rionero, A note on the existence and uniqueness of solutions to a free piston problem, Rend. Accad. Sci. Fis. Mat. Napoli, 66 (1999), 75-84.

[6]

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.

[7]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, in Hemodynamical Flows, volume 37 of Oberwolfach Semin., pages 121-273. Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-7806-6_3.

[8]

M. Hillairet and D. Serre, Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 779-803. doi: 10.1016/S0294-1449(02)00028-8.

[9]

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.

[10]

E. Marušić-Paloka., Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid, Asymptot. Anal., 33 (2003), 51-66.

[11]

V. Maz'ya and J. Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr., 280 (2007), 751-793. doi: 10.1002/mana.200610513.

[12]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528.

[13]

B. Muha and Z. Tutek, On a free piston problem for Stokes and Navier-Stokes equations,, To appear in \emph{Glasnik Matemati\v cki}., (). 

[14]

B. Muha and Z. Tutek, Numerical analysis of a free piston problem, Math. Commun., 15 (2010), 573-585.

[15]

B. Muha and Z. Tutek, On a stationary and evolutionary free piston problem for potential ideal fluid flow, Math. Meth. Appl. Sci., 35 (2012), 1721-1736, DOI:10.1002/mma.2555. doi: 10.1002/mma.2555.

[16]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall, in Advances in Mathematical Fluid Mechanics, pages 385-407. Springer, Berlin, 2010. doi: 10.1007/978-3-642-04068-9_24.

[17]

V. G. Osmolovskiĭ, Linear and Nonlinear Perturbations of the Operator div, volume 160 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1995 Russian original by Tamara Rozhkovskaya.

[18]

S. Takeno, Free piston problem for isentropic gas dynamics, Japan J. Indust. Appl. Math., 12 (1995), 163-194. doi: 10.1007/BF03167287.

[19]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, 1977. Studies in Mathematics and its Applications, Vol. 2.

show all references

References:
[1]

A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[2]

C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.), 20 (1994), 279-318.

[3]

C. Conca, J. San Martín H. and M. Tucsnak, Motion of a rigid body in a viscous fluid, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 473-478. doi: 10.1016/S0764-4442(99)80193-1.

[4]

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.

[5]

B. D'Acunto and S. Rionero, A note on the existence and uniqueness of solutions to a free piston problem, Rend. Accad. Sci. Fis. Mat. Napoli, 66 (1999), 75-84.

[6]

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.

[7]

G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, in Hemodynamical Flows, volume 37 of Oberwolfach Semin., pages 121-273. Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-7806-6_3.

[8]

M. Hillairet and D. Serre, Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 779-803. doi: 10.1016/S0294-1449(02)00028-8.

[9]

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.

[10]

E. Marušić-Paloka., Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid, Asymptot. Anal., 33 (2003), 51-66.

[11]

V. Maz'ya and J. Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr., 280 (2007), 751-793. doi: 10.1002/mana.200610513.

[12]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528.

[13]

B. Muha and Z. Tutek, On a free piston problem for Stokes and Navier-Stokes equations,, To appear in \emph{Glasnik Matemati\v cki}., (). 

[14]

B. Muha and Z. Tutek, Numerical analysis of a free piston problem, Math. Commun., 15 (2010), 573-585.

[15]

B. Muha and Z. Tutek, On a stationary and evolutionary free piston problem for potential ideal fluid flow, Math. Meth. Appl. Sci., 35 (2012), 1721-1736, DOI:10.1002/mma.2555. doi: 10.1002/mma.2555.

[16]

J. Neustupa and P. Penel, A weak solvability of the Navier-Stokes equation with Navier's boundary condition around a ball striking the wall, in Advances in Mathematical Fluid Mechanics, pages 385-407. Springer, Berlin, 2010. doi: 10.1007/978-3-642-04068-9_24.

[17]

V. G. Osmolovskiĭ, Linear and Nonlinear Perturbations of the Operator div, volume 160 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1995 Russian original by Tamara Rozhkovskaya.

[18]

S. Takeno, Free piston problem for isentropic gas dynamics, Japan J. Indust. Appl. Math., 12 (1995), 163-194. doi: 10.1007/BF03167287.

[19]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam, 1977. Studies in Mathematics and its Applications, Vol. 2.

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