Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX NOTE ON EVOLUTIONARY FREE PISTON PROBLEM FOR STOKES EQUATIONS WITH SLIP BOUNDARY CONDITIONS Boris Muha Department of Mathematics, University of Zagreb Bijeniˇ cka cesta 30, 10000 Zagreb, Croatia Zvonimir Tutek Department of Mathematics, University of Zagreb Bijeniˇ cka cesta 30, 10000 Zagreb, Croatia (Communicated by the associate editor name) 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. 1. Introduction. The main goal of this paper is to prove the existence of a solution of the free piston problem for viscous fluid flow. Unlike the classical piston problem where flow is generated by the known movement of the piston, in the free piston problem movement of the piston is unknown. Therefore, this is an example of a free boundary fluid-structure interaction problem. We consider fluid flow through the junction of two pipes in the gravity field. Inside the ”vertical” pipe there is a piston which can slide along that pipe. The piston is modeled as a rigid body and the fluid is modeled as an incompressible Newtonian fluid. Fluid-rigid body (and solids in general) interaction problems have been intensively studied from late 90s (see for example [3], [8], [6], [7], [4] and references within). However, in these papers the rigid body is fully immersed in the fluid, so the rigid body is not a part of the boundary of fluid-solid domain. Indeed, there are results that state that no contact will occur in finite time if the rigid body is smooth and no-slip boundary condition for the fluid are prescribed (see [9] and references within). However Neustupa and Penel ([16]) showed that if no-slip boundary conditions are replaced with Navier’s slip boundary conditions, collision may occur. In our case the rigid body (the piston) is part of the boundary, so there is permanent contact between the rigid body and the rigid boundary. In 2000 Mathematics Subject Classification. Primary: 74F10, 35Q30, 76D03; Secondary: 76D05. Key words and phrases. fluid-rigid body interaction, moving boundary, Navier-Stokes equa- tions, weak solution. The research of B. Muha has been supported by the Texas Higher Education Board, Advanced Research Project, post-doctoral support under grant number 003652-0023-2009 and MZOS grant number 0037-0693014-2765. The research of Z. Tutek has been supported by MZOS grant number 0037-0693014-2765. 1