On the stationary Navier-Stokes system with nonhomogeneous boundary data

On the stationary Navier-Stokes system with nonhomogeneous boundary data

Mon, 08/11/2010
17:00 		Konstantin Pileckas (Vilnius University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
On the stationary Navier-Stokes system with nonhomogeneous boundary data
We study the nonhomogeneous boundary value problem for the Navier–Stokes equations
\[ \left\{ \begin{array}{rcl} -\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=\qquad \hbox{\rm in }\;\;\Omega,\\[4pt] {\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt] {\bf u}&=\qquad \hbox{\rm on }\;\;\partial\Omega \end{array}\right \eqno(1) \]
in a bounded multiply connected domain $ \Omega\subset\mathbb{R}^n $ with the boundary $ \partial\Omega $, consisting of $ N $ disjoint components $ \Gamma_j $. Starting from the famous J. Leray's paper published in 1933, problem (1) was a subject of investigation in many papers. The continuity equation in (1) implies the necessary solvability condition
$$ \int\limits_{\partial\Omega}{\bf a}\cdot{\bf n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\eqno(2) $$
where $ {\bf n} $ is a unit vector of the outward (with respect to $ \Omega $) normal to $ \partial\Omega $. However, for a long time the existence of a weak solution $ {\bf u}\in W^{1,2}(\Omega) $ to problem (1) was proved only under the stronger condition
$$ {\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad j=1,2,\ldots,N. \eqno(3) $$
During the last 30 years many partial results concerning the solvability of problem (1) under condition (2) were obtained. A short overview of these results and the detailed study of problem (1) in a two–dimensional bounded multiply connected domain $ \Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset \Omega_1 $ will be presented in the talk. It will be proved that this problem has a solution, if the flux $ {\cal F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS $ of the boundary datum through $ \partial\Omega_2 $ is nonnegative (outflow condition).