Global regular solutions to the Navier-Stokes equations in a cylinder with slip boundary conditions

Global regular solutions to the Navier-Stokes equations in a cylinder with slip boundary conditions

Mon, 08/03/2010
17:00 		Wojciech ZAJACZKOWSKI (Polish Academy of Sciences) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Global regular solutions to the Navier-Stokes equations in a cylinder with slip boundary conditions
We consider the motion of a viscous incompressible fluid described by the Navier-Stokes equations in a bounded cylinder with slip boundary conditions. Assuming that $ L_2 $ norms of the derivative of the initial velocity and the external force with respect to the variable along the axis of the cylinder are sufficiently small we are able to prove long time existence of regular solutions. By the regular solutions we mean that velocity belongs to $ W^{2,1}_2 (Dx(0,T)) $ and gradient of pressure to $ L_2(Dx(0,T)) $. To show global existence we prolong the local solution with sufficiently large T step by step in time up to infinity. For this purpose we need that $ L_2(D) $ norms of the external force and derivative of the external force in the direction along the axis of the cylinder vanish with time exponentially. Next we consider the inflow-outflow problem. We assume that the normal component of velocity is nonvanishing on the parts of the boundary which are perpendicular to the axis of the cylinder. We obtain the energy type estimate by using the Hopf function. Next the existence of weak solutions is proved.