Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 76 / 1 / i / 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 givesand $⟨\Theta ({\Theta }_x^j{)}_x⟩=-⟨{\Theta }_x^{j+1}⟩$. Thus the energy method yieldsThe energy square completion for long-wave convection starts fromPut $v={\Theta }_x$. The energy identity becomesSince $sv-v^2=s^2/4-(v-s/2{)}^2\le s^2/4$ pointwise,Therefore $\mu <2-s^2/4$ 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.