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Dot the momentum equation with $\mathbf{u}$, multiply the temperature equation by $\sigma \mathrm{Ra}\theta$, and integrate over $V$. The pressure term vanishes by incompressibility, while skew-symmetry of incompressible transport removes both nonlinear advection terms. Integration by parts and the boundary conditions givewhereFor $\mathrm{Ra}=0$, this reduces toso viscosity monotonically dissipates kinetic energy.

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-Lindelöf theorem gives a unique maximal local solution.

Taking the $H$ inner product with $u_m$ givesThe advecting field $P_N{u}_m$ is divergence free. Periodicity and the skew-symmetry of incompressible transport therefore make the nonlinear term zero. HenceA 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 $[0,T]$.

Because $v$ is a divergence-free vector field, the product rule givesThe integral of a divergence over the periodic domain vanishes. Hence integration by parts yieldsThis is the skew-symmetry of incompressible transport.

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 givesSubtracting the temperature equations yieldsPair this equation with $\Theta$. The term transported by $u_1$ vanishes by the skew-symmetry of incompressible transport, while the other nonlinear term satisfiesThe forcing difference is at most $C\Vert \Theta {\Vert }_2^2$. ConsequentlyThe 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.

Take the $L^2$ inner product of the first Galerkin equation with ${\omega }_m$. The orthogonal projection may be removed against ${\omega }_m\in {\tilde{H}}_m$, and skew-symmetry of incompressible transport cancels the nonlinear term. HenceThe Cauchy-Schwarz inequality and Young inequality giveThus both quantities requested in the first estimate are bounded by, for example,

Next take the inner product with $-\Delta {\omega }_m$. Periodicity and incompressibility giveThe given curl identity and the two-dimensional Gagliardo-Nirenberg inequality implyApplying Young's inequality to this term and to $(g,-\Delta {\omega }_m)$ yieldsThe first estimate bounds the right-hand side independently of $m$. Therefore one may choose a constant $R_2=R_2(\Vert g{\Vert }_2,\gamma ,\nu ,{\mu }_1)$ such that