Finite-time Lp bound for Hamiltonian transport
= Finite-time Lp bound for Hamiltonian transport
{title2=$\|f(t)\|_p\leq\|f_0\|_p+\int_0^t\|h(s)\|_p\,ds$}
The <Duhamel formula for Hamiltonian transport>, <phase space> preservation and <Minkowski integral inequality> give the displayed bound. A time-independent source in an <Lp space> yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.