For smooth periodic fields, the <Holder inequality> and $H^1\hookrightarrow L^6$ give
$$
\left|\int_\Omega
(u\mathbin\cdot\nabla v)\mathbin\cdot w\,dx\right|
\leq\|u\|_{L^3}\|\nabla v\|_{L^2}\|w\|_{L^6}
\leq c|u|^{1/2}\|u\|^{1/2}\|v\|\|w\|.
$$
Similarly,
$$
\left|\int_\Omega
(\nabla\mathbin\cdot u)(v\mathbin\cdot w)\,dx\right|
\leq\|\nabla u\|_{L^2}\|v\|_{L^3}\|w\|_{L^6}
\leq c\|u\||v|^{1/2}\|v\|^{1/2}\|w\|.
$$
Adding the bounds proves that the <skew-symmetrized transport form> extends continuously from smooth fields to $\widetilde V^3$ and
$$
\boxed{
|\langle\widetilde B(u,v),w\rangle|
\leq c\left(
|u|^{1/2}\|u\|^{1/2}\|v\|
+|v|^{1/2}\|v\|^{1/2}\|u\|
\right)\|w\|}.
$$
The periodic <Poincare inequality> gives
$$
|u|\leq\mu_1^{-1/2}\|u\|,
\qquad
|v|\leq\mu_1^{-1/2}\|v\|.
$$
Hence
$$
\boxed{
|\langle\widetilde B(u,v),w\rangle|
\leq c\mu_1^{-1/4}\|u\|\|v\|\|w\|}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / a / i / Solution

For a velocity field that need not be divergence free, the corrected transport
$$
\widetilde B(u,v)=(u\mathbin\cdot\nabla)v+\frac12(\nabla\mathbin\cdot u)v
$$
satisfies $\langle\widetilde B(u,v),w\rangle=-\langle\widetilde B(u,w),v\rangle$ and therefore preserves the quadratic energy cancellation.


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