Let $v(t,x(t))$ be a velocity vector field and $[0,T]$ an interval. My question would be the following: If I want $$ L:=\sup_{t\in[0,T], x\in \mathbb{R}} \mid v'(t,x(t))\mid$$ to be finite, i.e. $L<\infty$, what conditions would I need to have for $v(t,x(t))$?
Would it be enough to assume that $v(t,x(t))$ is Lipschitz continuous? Because that would mean I have an $K\geq 0$ with $$\mid v(t,x)-v(t,x(t))\mid \leq K\mid x-x(t)\mid, $$ and therefore have an upper bound for the derivative $$v'(t,x(t))\leq K .$$ How would I show that $L$ is finite?