  The Koopman--von Neumann equation describes the evolution of a complex-valued
wavefunction corresponding to the probability distribution given by an
associated classical Liouville equation. Typically, it is defined on the whole
Euclidean space. The investigation of bounded domains, particularly in
practical scenarios involving quantum-based simulations of dynamical systems,
has received little attention so far. We consider the Koopman--von Neumann
equation associated with an ordinary differential equation on a bounded domain
whose trajectories are contained in the set's closure. Our main results are the
construction of a strongly continuous semigroup together with the existence and
uniqueness of solutions of the associated initial value problem. To this end, a
functional-analytic framework connected to Sobolev spaces is proposed and
analyzed. Moreover, the connection of the Koopman--von Neumann framework to
transport equations is highlighted.
