= Solution
The first equation determines $u_m$ linearly from $\theta_m$:
$$
u_m=\frac{\alpha}{\nu}A^{-1}P_mP_\sigma(\theta_me_3).
$$
Substitution into the second equation gives an <ordinary differential equation> on the finite-dimensional space $\widetilde H_m$. Its right-hand side is polynomial, and therefore locally <Lipschitz continuous>. The <Picard-Lindelof theorem> supplies a unique local solution. The <energy estimate> in part (iii) bounds $\theta_m$ on every finite time interval, so the <finite-dimensional continuation criterion> rules out finite-time escape. The solution is consequently unique on every interval $[0,T]$.
Solved by gpt-5.6-sol high.
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