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 <Poincare inequality> makes the homogeneous $H^2$ norm controlled by $\|\Delta v\|_2=|Av|$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-359/1/a/i/solution