Navier-Stokes equations in 2D: what do we know? (Millennium problems)

If i understood correctly,

one of the millennium problems state that for the time-dependent 3d NS equations you have to prove that

• either the solution is bounded in a certain space
• or find a counter example for which the solution is not bounded.

I was wondering what has been proven in 2d. Do you have any reference?

If someone is into the topic, why cannot the result be extended to 3d? Is it related to embedding theorems?