Reconstruction of velocity and pressure field
 

Obviously, given the velocity vector u we can compute the rate of expansion(the divergence of u)and the vorticity(the cross product of del operator and u).

Suppose instead that we know the functions rate of expansion(the divergence of u)and the vorticity(the cross product of del operator and u) everywhere, can we reconstruct the velocity field and the pressure? If so, why not just focus on measuring these functions (fluid kinematics) and forget about fluid mechanics?

Thanks