Navier-Stokes system I have to study this system which name is Navier-Stokes. Can you explain please what means that $p$, $u$ and $(u \cdot \nabla)u$. What represents in reality ? Tell me please, how shoul I read the factor: $(u \cdot \nabla)u$? "$u$ mutiplied with gradient  applied to u " ? 
$ (N-S)\begin{cases} -\mu \Delta u +(u \cdot \nabla)u+\nabla{p}=f &\mbox{in  } \Omega, \\
\mbox{div }u=0 & \mbox{in } \Omega,\\
u_{\mid{\Gamma}}=0. 
\end{cases} $
thanks:)
 A: $p$ is pressure. $\mu$ is viscosity. $u$ is the velocity vector.  $\mu \Delta u$ is the resistance of the fluid to flow due to intermolecular forces.  $\nabla p$ is the force exerted by pressure differentials.  $f$ is force applied by the environment onto the fluid.  The remainder is described as the convective acceleration and can be interpreted as $u\cdot (\nabla u)$ if that is easier for you.  I usually understand this as the force applied to the fluid by the current motion of the fluid but I doubt that is a truly accurate explanation.
While I do not use the same notation (I am an engineer not a mathematician), the millennium puzzle explanation of the function is quite thorough at describing it in mathematical terms:
http://www.claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf
A: Navier-Stokes equations are main equations of fluid dynamics. Since major part of it comes from physics, it has some terms that are rarely used in mathematics. One of them is $\left( \mathbf u \cdot \nabla\right) \mathbf u$. It can be simplified. Let's say you're dealing with 3D case, so $\mathbf u = (u, v, w)$, then
$$
\mathbf u \cdot \nabla = u \partial x+v \partial y+w \partial z
$$
and 
$$
\left( \mathbf u \cdot \nabla \right) \mathbf u = \left( \begin{array}{c}
u u_x + v u_y + w u_z \\
u v_x + v v_y + w v_z \\
u w_x + v w_y + w w_z
\end{array}\right)
$$
where subscript means partial derivative $u_x = \frac {\partial u}{\partial x}$.
They are applied to so called conservation of momentum equations. You may find more information here Navier–Stokes equations: Cartesian coordinates
