Consider the problem 
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi\_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the [indicator function of a set](https://en.wikipedia.org/wiki/Indicator_function).


* What is the unique [regular Lagrangian flow](http://cvgmt.sns.it/media/doc/paper/1612/crippa_bari.pdf) of $(\star)$?
* How can it be approximated numerically?




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Questions related to the second point has been asked on [Mathematica Stackexchange.](https://mathematica.stackexchange.com/questions/210496/use-mathematica-to-plot-the-flow-of-an-ode-with-discontinuity)



Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
The flow $X:[0,T]\times \mathbb{R}\to\mathbb{R}$ defined by $X(t,x)= x+t\chi\_{\mathbb{R}\_+}(x)$, for all $(t,x)\in [0,T]\times \mathbb{R}$, is a regular Lagrangian flow solution to $(\star)$ in the sense of ***Definition (4)*** of the linked paper (for a.e. initial datum $x\in \mathbb{R}$, in fact for all, one has $(x+t\chi\_{\mathbb{R}\_+}(x))'=\chi\_{\mathbb{R}\_+}(x)=\chi\_{\mathbb{R}\_+}(x+t\chi\_{\mathbb{R}\_+}(x))$; moreover $X(t,\cdot)\_\#\mathcal{L}^1\le\mathcal{L}^1$ for all $t\ge0$). 


Note that according to the above definition, one can always modify a regular Lagrangian flow for a vector field $b$ at least on a countable number of flow lines (even to non-solution curves), always yielding to a regular Lagrangian solution; indeed the uniqueness has to be intended for the flow as an element of $L^\infty([0,T]\times \mathbb{R}^d,\mathbb{R}^d)$. The vector field $b$ itself is given as an element of $L^\infty([0,T]\times \mathbb{R}^d,\mathbb{R}^d)$.


