 Solution

Free transport equation is the measure-preserving shear $(x,v)\mapsto (x-vs,v)$. Its composition operator preserves the $L_{x,v}^2$ norm. Since $a\ge 0$, the damping multiplier has modulus at most one:Consequently the damped free-transport evolution is contractive:This needs no upper bound on $a$. Nonnegative integrals may be interpreted in the extended sense with $e^{-\infty }=0$; local integrability along characteristics additionally gives the usual continuous initial trace.