Let $F:\mathbb{R}^2 \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = (0,\chi_{\{x \le F(y)\}})$.

Consider the ODE $$ \begin{cases} \partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\ \Phi(0,x) = x & x \in \mathbb R^2 \end{cases} $$

• How can we write the solution $\Phi$ explicitly?

Let $F:\mathbb{R} \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = (0,\chi_{\{x \le F(y)\}})$.

Consider the ODE $$ \begin{cases} \partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\ \Phi(0,x) = x & x \in \mathbb R^2 \end{cases} $$

• How can we write the solution $\Phi$ explicitly?

Let $F:\mathbb{R}^2 \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = \chi_{\{x \le F(y)\}}$.

Consider the ODE $$ \begin{cases} \partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\ \Phi(0,x) = x & x \in \mathbb R^2 \end{cases} $$

• How can we write the solution $\Phi$ explicitly?

Let $F:\mathbb{R}^2 \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = (0,\chi_{\{x \le F(y)\}})$.

Consider the ODE $$ \begin{cases} \partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\ \Phi(0,x) = x & x \in \mathbb R^2 \end{cases} $$

• How can we write the solution $\Phi$ explicitly?