Solution
= Solution
The <sobolev embedding theorem> in three dimensions and the <periodic elliptic estimate> for the <Stokes operator> give
$$
\|(v\mathbin\cdot\nabla)\eta\|_2
\leq \|v\|_\infty\|\nabla\eta\|_2
\leq C\|v\|_{H^2}\|\eta\|_{H^1}
\leq C|Av|\|\eta\|_{H^1}.
$$
The last step uses the absence of the zero Fourier mode: on mean-zero periodic fields, the <Poincaré inequality> makes the homogeneous $H^2$ norm controlled by $\|\Delta v\|_2=|Av|$.