Coercive energy estimate for kinetic transport ($\Vert f(t_2){\Vert }_2\le e^{-\delta (t_2-t_1)}\Vert f(t_1){\Vert }_2$)

If a velocity operator satisfies $\mathrm{Re}⟨Lh,h⟩\ge \delta \Vert h{\Vert }_2^2$, then sufficiently regular solutions of ${\partial }_{t}f+v\cdot {\nabla }_{x}f+Lf=0$ obey the displayed estimate when spatial boundary fluxes vanish. The free-transport part contributes zero to the energy derivative. Integrating the resulting differential inequality avoids dividing by a possibly zero norm.