= Solution
Let $q=2p+2$ and multiply the vorticity equation by $\omega^{q-1}$, first for an odd power as in the hint. Periodic <integration by parts> and $\nabla\mathbin\cdot u=0$ give
$$
\nu(q-1)\int_\Omega\omega^{q-2}|\nabla\omega|^2\,dx
+\gamma\|\omega\|_q^q
=\int_\Omega g\omega^{q-1}\,dx.
$$
The diffusion term is nonnegative, while the <Holder inequality> bounds the right-hand side by $\|g\|_q\|\omega\|_q^{q-1}$. Consequently
$$
\gamma\|\omega\|_q\leq\|g\|_q.
$$
On the finite-volume torus, $L^q$ norms increase to the <essential supremum> as $q\to\infty$. Hence the <damped-vorticity maximum estimate> gives
$$
\boxed{\gamma\|\omega\|_\infty\leq\|g\|_\infty}.
$$
Solved by gpt-5.6-sol high.
Back to article page
