= Solution
Let $(u_1,\theta_1)$ and $(u_2,\theta_2)$ be two weak solutions with the same initial data, and set $U=u_1-u_2$ and $\Theta=\theta_1-\theta_2$. The diagnostic Stokes equation gives
$$
|AU|\leq\frac{\alpha}{\nu}\|\Theta\|_2,
\qquad
\|U\|_\infty\leq C\|\Theta\|_2.
$$
Subtracting the temperature equations yields
$$
\partial_t\Theta-\kappa\Delta\Theta
+(u_1\mathbin\cdot\nabla)\Theta
+(U\mathbin\cdot\nabla)\theta_2
=\beta U\mathbin\cdot e_3.
$$
Pair this equation with $\Theta$. The term transported by $u_1$ vanishes by the <skew-symmetry of incompressible transport>, while the other nonlinear term satisfies
$$
\left|\int_\Omega(U\mathbin\cdot\nabla\theta_2)\Theta\,dx\right|
\leq C\|\nabla\theta_2\|_2\|\Theta\|_2^2.
$$
The forcing difference is at most $C\|\Theta\|_2^2$. Consequently
$$
\frac d{dt}\|\Theta\|_2^2
\leq C\bigl(1+\|\nabla\theta_2\|_2\bigr)\|\Theta\|_2^2.
$$
The coefficient is integrable on $[0,T]$ because $\theta_2\in L^2(0,T;H^1)$. Since $\Theta(0)=0$, the <Gronwall inequality> gives $\Theta=0$, and the Stokes equation then gives $U=0$. The weak solution is unique.
Solved by gpt-5.6-sol high.
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