  This article develops mathematical formalisms and provides numerical methods
for studying the evolution of measures in nonsmooth dynamical systems using the
continuity equation. The nonsmooth dynamical system is described by an
evolution variational inequality and we derive the continuity equation
associated with this system class using three different formalisms. The first
formalism consists of using the {superposition principle} to describe the
continuity equation for a measure that disintegrates into a probability measure
supported on the set of vector fields and another measure representing the
distribution of system trajectories at each time instant. The second formalism
is based on the regularization of the nonsmooth vector field and describing the
measure as the limit of a sequence of measures associated with the
regularization parameter. In doing so, we obtain quantitative bounds on the
Wasserstein metric between measure solutions of the regularized vector field
and the limiting measure associated with the nonsmooth vector field. The third
formalism uses a time-stepping algorithm to model a time-discretized evolution
of the measures and show that the absolutely continuous trajectories associated
with the continuity equation are recovered in the limit as the sampling time
goes to zero. We also validate each formalism with numerical examples. For the
first formalism, we use polynomial optimization techniques and the moment-SOS
hierarchy to obtain approximate moments of the measures. For the second
formalism, we illustrate the bounds on the Wasserstein metric for an academic
example for which the closed-form expression of the Wasserstein metric can be
calculated. For the third formalism, we illustrate the time-stepping based
algorithm for measure evolution on an example that shows the effect of the
concentration of measures.
