I am now considering the vorticity formulation of the 2D incompressible Navier-Stokes equation in some regular, bounded domain $D$: \begin{align*} D_t\omega &= \nu\Delta \omega,\\ \omega(x,0) &=\omega_0(x) \in C^\infty(D), \end{align*} where we recall that the velocity $u$ is given by the Biot-Savart law: $u(x,t) = \nabla^\perp (-\Delta_D)^{-1}\omega(x,t)$.
It follows from straightforward computation that given $\omega(x,t)$ a smooth solution with enough decay to the 2D problem on the whole $\mathbb{R}^2$, we have $$ \|\omega(t,\cdot)\|_p \le \|\omega_0\|_p,\; p \ge 1. $$ I am now wondering if this still holds on a bounded domain (assuming enough boundary regularity). Indeed, if we integrate by parts, there will be boundary terms involving $\nabla \omega \cdot n,$ where $n$ is the outward normal on $\partial D$. Is there away to get around this term? Or is it not even true that such decay in $L^p$ norm of vorticity occurs in the case of bounded domains? Thanks in advance!