The Liouville equation in L¹ spaces

We consider the first order equation ∂u/∂t = a · ∇u in the Banach lattice L¹ (R^N). By requiring a minimal amount of Sobolev regularity on the vector-field a, we show that a · ∇ generates a C_o-group, thereby generalizing a result of [1]. From there, we conclude the well-posedness of Liouville equation ∂u/∂t = ξ · ∇cursive Greek chiu + ∇cursive Greek chiV · ∇ξu, for a given potential V The comparison between the general and force-free Liouville evolution yields the existence of the wave and scattering operators, which in turn are used to prove that the spectrum of the Liouville operator is purely residual in L¹ (R⁶).