= Energy square completion for long-wave convection
{title2=$\mu_E=2-s^2/4$}
For a real smooth solution of the <long-wave convection equation with broken Boussinesq symmetry>, assume a periodic spatial average or an existing long-interval average with vanishing endpoint fluxes. <Integration by parts> gives the exact <energy method> identity
$$
\frac12\partial_t\langle\Theta^2\rangle
=-\langle(\Theta+\Theta_{xx})^2\rangle
-\langle\Theta_x^2(\Theta_x-s/2)^2\rangle
+(\mu-2+s^2/4)\langle\Theta_x^2\rangle.
$$
Thus $\mu<\mu_E$ prevents growth of the mean-square temperature at arbitrary amplitude in this averaging class. This sufficient nonlinear bound need not equal the linear instability threshold, and it does not imply pointwise monotonicity of the temperature.
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