Is the vorticity equation + mass conservation equation equivalent to original Navier-Stokes equation for incompressible fluids For incompressible fluids, we can derive vorticity equation from moment conservation equation , but if they are(plus mass conservation equation) equivalent to original NS equation? (my understanding is the pressure is not there, so it may not equivalent)
 A: For sufficiently nice vector fields in all of $R^3$, 
the vorticity equation is equivalent to Navier-Stokes plus ${\rm div\,}u=0$.
The curl of $u_t+u\cdot\nabla u = -\nabla p+\Delta u$, 
with ${\rm div\,}u = 0$ gives
the vorticity equation 
 $\omega_t+u\cdot\nabla\omega-\omega\cdot\nabla u = \Delta\omega$, where
$u$ is to be replaced by the Biot-Savart integral of $\omega$. 
Working backwards from the vorticity equation to Navier-Stokes, define
$u$ to be the Biot-Savart integral of $\omega$. Then $u$ has divergence
zero, and the curl of $u_t+u\cdot\nabla u -\Delta u$ is zero, therefore
$u_t+u\cdot\nabla u -\Delta u$ 
is the gradient of some function which you define to be minus the pressure.
So the systems appear to be equivalent.
But, there is a problem. The  curl of Navier-Stokes
involves 3rd derivatives of $u$. We don't know under what
circumstances $u$ has 3rd derivatives.
A: There is another issue. What are the boundary conditions for the vorticity equation? Inability to provide these severely limits the practical application
