= Solution
On the finite-dimensional space $H_m$, the Galerkin equation is an autonomous system of ordinary differential equations whose right-hand side is a quadratic polynomial in the coefficients of $u_m$. It is locally Lipschitz, so the <Picard-Lindelof theorem> gives a unique maximal local solution.
Taking the $H$ inner product with $u_m$ gives
$$
\frac12\frac d{dt}|u_m|^2+\nu\|u_m\|^2
+\langle B(P_Nu_m,u_m),u_m\rangle=0.
$$
The advecting field $P_Nu_m$ is divergence free. Periodicity and the <skew-symmetry of incompressible transport> therefore make the nonlinear term zero. Hence
$$
|u_m(t)|\leq|P_mu_0|\leq|u_0|.
$$
A finite-dimensional solution can cease to exist only if its norm diverges. This uniform bound prevents such blow-up, so the solution extends uniquely through every interval $[0,T]$.
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