The energy equality shows that
$$
t\longmapsto |u(t)|^2
=|u_0|^2-2\nu\int_0^t\|u(\tau)\|^2\,d\tau
$$
is continuous. Part i gives weak continuity in $H$. In a Hilbert space, weak convergence together with convergence of norms implies strong convergence. Applying this whenever $t_n\to t$ proves the :
$$
\boxed{u\in C([0,T];H)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / c / iii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-359/1/c/iii/solution