Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ an observable, and $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ two vector fields such that for any $(x^1_t)_{t\geq 0}$ and $(x^2_t)_{t\geq 0}$ solutions of

\begin{align} & \dot{x}^1_t=v^1(x^1_t) \\ & \dot{x}^2_t=v^2(x^2_t), \end{align}

$\big(f(x^1_t)\big)_{t\geq 0}$ is increasing, and $\big(f(x^2_t)\big)_{t\geq 0}$ is non-decreasing.

Question: for any $(x_t)_{t\geq 0}$ solution of $\dot{x}_t=(v^1+v^2)(x_t)$, is $\big(f(x_t)\big)_{t\geq 0}$ increasing?

Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ and $(x^2_t)_{t\geq 0}$ solutions of

\begin{align} & \dot{x}^1_t=v^1(x^1_t) \\ & \dot{x}^2_t=v^2(x^2_t), \end{align}

$\big(f(x^1_t)\big)_{t\geq 0}$ is increasing, and $\big(f(x^2_t)\big)_{t\geq 0}$ is non-decreasing.

Question: for any $(x_t)_{t\geq 0}$ solution of $\dot{x}_t=(v^1+v^2)(x_t)$, is $\big(f(x_t)\big)_{t\geq 0}$ increasing?