Is the vorticity equation + mass conservation equation equivalent to original Navier-Stokes equation for incompressible fluids

Question

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)

Answer 1

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.