Can we calculate a vector field from the pressure field with Navier Stokes?

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I've been looking for weeks and can't seem to wrap my head around CFD or the computation of fluid and its movement in 2D with Navier Stokes.

Without an initial velocity field, and given ONLY a pressure field, can you reverse engineer and find the velocity field from this, or is this impossible, or the literal point of Navier Stokes? I might be in the complete wrong for what I should be trying to find out because I simply don't know what I don't know. I just want to further my understanding and be able to make a small simulator for fluids with added pressure and see it's movement with time.

Navier-Stokes uses pressure and velocity fields to calculate the change in velocity over time. This is just something that I don't have in this case. Is it possible?

It's hard to get an answer on this because not many people from what I've seen have no initial velocity, and I'm not sure what I should be learning...

Any help would be extremely appreciated, thank you!