Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t - \Delta \omega + u\cdot\nabla \omega = f$$ My question is: what is the parabolic regularity result here? Is it similar to that of heat equation $u_t - \Delta u=f$? $$\Vert u_t\Vert_{L^{q}_{t}L^{q}_{t}} + \Vert \Delta u\Vert_{L^{q}_{t}L^{q}_{t}} \leq C(\Vert f\Vert_{L^{q}_{t}L^{q}_{t}} + \Vert u_0\Vert_{L^{q}_{t}L^{q}_{t}} ) $$

1