Global existence for the inflow-outflow problem for the Navier–Stokes equations in a cylinder

Piotr Kacprzyk¹

¹ Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 36 (2009) no. 2, pp. 195-212

Résumé

Global existence of regular solutions to the Navier–Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. To prove the long time existence we need smallness of derivatives, with respect to the variable along the axis of the cylinder, of the external force and of the initial velocity in $L_2$-norms. Moreover, we need smallness of derivatives of inflow and outflow with respect to tangent directions to the boundary and with respect to time in some norms. The global existence is proved step by step using the existence on the time interval $[0,T]$, with $T$ sufficiently large.

Détail

DOI : 10.4064/am36-2-7

Keywords: global existence regular solutions navier stokes equations describing motion incompressible viscous fluid cylindrical pipe large inflow outflow shown prove long time existence smallness derivatives respect variable along axis cylinder external force initial velocity norms moreover smallness derivatives inflow outflow respect tangent directions boundary respect time norms global existence proved step step using existence time interval sufficiently large