Notation

Through all this documnet we will use the following notation:

$\Omega \subset{\bf R}^d$  ...  bounded domain occupied by the fluid

d=2,3,  ...  space dimension,

$\left ( 0,T\right)$   ...  time interval, T>0

${\bf x}=(x_1,\dots,x_d)\in\Omega$  ...  any point from $\Omega$

$t\in(0,T)$  ...  time instant from time interval

$Q_T = \Omega \times\left( 0,T\right)$  ...  space-time cylinder

${\bf u=u(x},t)=(u_1,\dots,u_d)$  ...  velocity vector

$\rho=\rho ({\bf x},t)$  ...  density

$p=p({\bf x},t)$  ...  pressure

$\theta = \theta ({\bf x},t)$  ...  absolute temperature

$e = e({\bf x},t)$  ...  total energy

${\bf f=f(x},t)=(f_1,\dots,f_d)$  ...  external forces per unit mass

$q=q({\bf x},t)$  ...  rate of external heat sources

${\bf q=q(x},t) =(q_1,\dots,q_d)$  ...  heat flux