What is the difference between the notions of
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, where $b$ is a non-smooth (that is, non-Lipschitz) vector field?
In general, are they related in any way? Under which (non-trivial) assumptions are they equivalent? Which is the strongest notion of solutions between these?