Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases} $$ This is the prototype of non-uniqueness for ODEs in the classical sense. I'm tempted to say, however, that the Lagrangian flow in the sense of Ambrosio exists and should be made of trajectories that don't "stay" at zero. Is this intuition correct? How can the regular Lagrangian flow be computed explicitly in this example?

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases} $$ This is the prototype of non-uniqueness for ODEs in the classical sense. I'm tempted to say, however, that the Lagrangian flow in the sense of Ambrosio exists and should be made of trajectories that don't "stay" at zero. Is this intuition correct? How can the regular Lagrangian flow be computed explicitly in this example?

For the existence and uniqueness of RLF for this example (which is not covered by the classical DiPerna-Lions theory), see this paper.