= Solution
For the <long-wave convection equation with broken Boussinesq symmetry>, use a sufficiently smooth real temperature field. To make the spatial average and integrations meaningful, take a periodic pattern, or an existing long-interval average with bounded derivatives and vanishing averaged endpoint fluxes. Boundedness of the temperature alone does not guarantee all those averaging properties. Multiply the evolution equation by $\Theta$ and average. <Integration by parts> gives
$$
\langle\Theta\Theta_{xx}\rangle=-\langle\Theta_x^2\rangle,\qquad
\langle\Theta\Theta_{xxxx}\rangle=\langle\Theta_{xx}^2\rangle,
$$
and $\langle\Theta(\Theta_x^j)_x\rangle=-\langle\Theta_x^{j+1}\rangle$. Thus the <energy method> yields
$$
\boxed{\frac12\frac d{dt}\langle\Theta^2\rangle
=-\langle\Theta^2\rangle+\mu\langle\Theta_x^2\rangle
-\langle\Theta_{xx}^2\rangle+s\langle\Theta_x^3\rangle
-\langle\Theta_x^4\rangle.}
$$
The <energy square completion for long-wave convection> starts from
$$
\langle(\Theta+\Theta_{xx})^2\rangle
=\langle\Theta^2\rangle-2\langle\Theta_x^2\rangle
+\langle\Theta_{xx}^2\rangle\geq0.
$$
Put $v=\Theta_x$. The energy identity becomes
$$
\frac12\frac d{dt}\langle\Theta^2\rangle
=-\langle(\Theta+\Theta_{xx})^2\rangle
+\langle(\mu-2)v^2+sv^3-v^4\rangle.
$$
Since $sv-v^2=s^2/4-(v-s/2)^2\leq s^2/4$ pointwise,
$$
\boxed{\frac12\frac d{dt}\langle\Theta^2\rangle
\leq\langle(\mu-2)\Theta_x^2+s\Theta_x^3-\Theta_x^4\rangle
\leq(\mu-2+s^2/4)\langle\Theta_x^2\rangle.}
$$
Therefore \b[$\mu<2-s^2/4$ excludes growth of the mean-square temperature], for arbitrary amplitude within this smooth averaging class. This is a nonlinear energy-stability criterion, not a proof of pointwise monotonicity at each position. A spatially constant component instead decays through the $-\Theta$ term. The criterion is sufficient; it need not coincide with the linear instability threshold.
Back to article page
