Damped free-transport evolution
= Damped free-transport evolution
{title2=$U_a(t,s)$}
For nonnegative measurable $a$, define $(U_a(t,s)g)(x,v)=e^{-\int_s^t a(r,x-v(t-r),v)dr}g(x-v(t-s),v)$. The shear preserves <Lebesgue measure> and the multiplier lies in $[0,1]$, so this is a contraction on <Lp spaces>. Extended nonnegative integrals use $e^{-\infty}=0$. Characteristic local integrability gives the usual continuous trace at the starting time.