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

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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

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

Abstract
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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 & 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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar
[2]	C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure,, \emph{Japan. J. Math. (N.S.)}, 20 (1994), 279. Google Scholar
[3]	C. Conca, J. San Martín H. and M. Tucsnak, Motion of a rigid body in a viscous fluid,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 328 (1999), 473. doi: 10.1016/S0764-4442(99)80193-1. Google Scholar
[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,, \emph{Czechoslovak Math. J.}, 58 (2008), 961. doi: 10.1007/s10587-008-0063-2. Google Scholar
[5]	B. D'Acunto and S. Rionero, A note on the existence and uniqueness of solutions to a free piston problem,, \emph{Rend. Accad. Sci. Fis. Mat. Napoli}, 66 (1999), 75. Google Scholar
[6]	B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1399. doi: 10.1080/03605300008821553. Google Scholar
[7]	G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics,, in \emph{Hemodynamical Flows}, (2008), 121. doi: 10.1007/978-3-7643-7806-6_3. Google Scholar
[8]	M. Hillairet and D. Serre, Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 20 (2003), 779. doi: 10.1016/S0294-1449(02)00028-8. Google Scholar
[9]	M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2451. doi: 10.1137/080716074. Google Scholar
[10]	E. Marušić-Paloka., Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid,, \emph{Asymptot. Anal.}, 33 (2003), 51. Google Scholar
[11]	V. Maz'ya and J. Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains,, \emph{Math. Nachr.}, 280 (2007), 751. doi: 10.1002/mana.200610513. Google Scholar
[12]	T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain,, \emph{Hiroshima Math. J.}, 12 (1982), 513. Google Scholar
[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}., (). Google Scholar
[14]	B. Muha and Z. Tutek, Numerical analysis of a free piston problem,, \emph{Math. Commun.}, 15 (2010), 573. Google Scholar
[15]	B. Muha and Z. Tutek, On a stationary and evolutionary free piston problem for potential ideal fluid flow,, \emph{Math. Meth. Appl. Sci.}, 35 (2012), 1721. doi: 10.1002/mma.2555. Google Scholar
[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 \emph{Advances in Mathematical Fluid Mechanics}, (2010), 385. doi: 10.1007/978-3-642-04068-9_24. Google Scholar
[17]	V. G. Osmolovskiĭ, Linear and Nonlinear Perturbations of the Operator div, volume 160 of "Translations of Mathematical Monographs,", American Mathematical Society, (1997). Google Scholar
[18]	S. Takeno, Free piston problem for isentropic gas dynamics,, \emph{Japan J. Indust. Appl. Math.}, 12 (1995), 163. doi: 10.1007/BF03167287. Google Scholar
[19]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland Publishing Co., (1977). Google Scholar

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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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar
[2]	C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure,, \emph{Japan. J. Math. (N.S.)}, 20 (1994), 279. Google Scholar
[3]	C. Conca, J. San Martín H. and M. Tucsnak, Motion of a rigid body in a viscous fluid,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 328 (1999), 473. doi: 10.1016/S0764-4442(99)80193-1. Google Scholar
[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,, \emph{Czechoslovak Math. J.}, 58 (2008), 961. doi: 10.1007/s10587-008-0063-2. Google Scholar
[5]	B. D'Acunto and S. Rionero, A note on the existence and uniqueness of solutions to a free piston problem,, \emph{Rend. Accad. Sci. Fis. Mat. Napoli}, 66 (1999), 75. Google Scholar
[6]	B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1399. doi: 10.1080/03605300008821553. Google Scholar
[7]	G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics,, in \emph{Hemodynamical Flows}, (2008), 121. doi: 10.1007/978-3-7643-7806-6_3. Google Scholar
[8]	M. Hillairet and D. Serre, Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 20 (2003), 779. doi: 10.1016/S0294-1449(02)00028-8. Google Scholar
[9]	M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems,, \emph{SIAM J. Math. Anal.}, 40 (2009), 2451. doi: 10.1137/080716074. Google Scholar
[10]	E. Marušić-Paloka., Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid,, \emph{Asymptot. Anal.}, 33 (2003), 51. Google Scholar
[11]	V. Maz'ya and J. Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains,, \emph{Math. Nachr.}, 280 (2007), 751. doi: 10.1002/mana.200610513. Google Scholar
[12]	T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain,, \emph{Hiroshima Math. J.}, 12 (1982), 513. Google Scholar
[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}., (). Google Scholar
[14]	B. Muha and Z. Tutek, Numerical analysis of a free piston problem,, \emph{Math. Commun.}, 15 (2010), 573. Google Scholar
[15]	B. Muha and Z. Tutek, On a stationary and evolutionary free piston problem for potential ideal fluid flow,, \emph{Math. Meth. Appl. Sci.}, 35 (2012), 1721. doi: 10.1002/mma.2555. Google Scholar
[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 \emph{Advances in Mathematical Fluid Mechanics}, (2010), 385. doi: 10.1007/978-3-642-04068-9_24. Google Scholar
[17]	V. G. Osmolovskiĭ, Linear and Nonlinear Perturbations of the Operator div, volume 160 of "Translations of Mathematical Monographs,", American Mathematical Society, (1997). Google Scholar
[18]	S. Takeno, Free piston problem for isentropic gas dynamics,, \emph{Japan J. Indust. Appl. Math.}, 12 (1995), 163. doi: 10.1007/BF03167287. Google Scholar
[19]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, North-Holland Publishing Co., (1977). Google Scholar

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