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Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity
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

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.
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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