Long-wave convection equation with broken Boussinesq symmetry (${\Theta }_t=-\Theta -\mu {\Theta }_{xx}-{\Theta }_{xxxx}-s({\Theta }_x^2{)}_x+({\Theta }_x^3{)}_x$)

A reduced long-wave convection model can combine linear damping, destabilizing second derivatives, stabilizing fourth derivatives and gradient nonlinearities. In the displayed normalization the quadratic coefficient $s$ breaks Boussinesq symmetry, while the cubic gradient term dissipates the mean-square temperature. The linear dispersion relation is $\lambda (k)=-1+\mu k^2-k^4$, with critical wavenumber $k=1$ at $\mu =2$. Smoothness and spatial averaging hypotheses must be specified before applying the energy method on an infinite interval.

In a weakly nonlinear expansion of the long-wave convection equation with broken Boussinesq symmetry at $\mu =2$, a critical Fourier mode ${\Theta }_1=A{e}^{ix}+c.c.$ generates ${\Theta }_2=(2is/9)A^2{e}^{2ix}+c.c.$. The quadratic interaction of these first and second harmonics feeds back $8s^2|A{|}^{2}A/9$ into the critical Fourier mode, while the cubic gradient nonlinearity contributes $-3|A{|}^{2}A$. The Fredholm solvability condition is consequently ${\mu }_{2}A-g|A{|}^{2}A=0$. The sign of $g$ distinguishes supercritical and subcritical branches; $g=0$ requires a higher-order amplitude equation.

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 identityThus $\mu <{\mu }_E$ 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.