Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < C_k$. My question is: which estimates can be derived for the flow $\phi_t(x)$ as a function of $x$ and its derivatives wrt $x$?

I was thinking about this: suppose $t$ is small then we have $$ \phi_t(x)=x+tv(x)+\frac 1 2 t^2 Dv(x)v(x)+... $$ so apparently I should have nontrivial estimates in x which involve all the derivatives of $v$... is this reasonable?