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

Boris Muha; Zvonimir Tutek

doi:10.3934/cpaa.2014.13.1629

Article Contents

2014, Volume 13, Issue 4: 1629-1639. Doi: 10.3934/cpaa.2014.13.1629

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

Received: March 31, 2012

Revised: September 30, 2012

Published: June 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 74F10, 35Q30, 76D03; Secondary: 76D05.

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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, 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 Glasnik Matematički.
[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.