tag removed - Estimating the flow when we know the vector field

Let $X=sum_{1le jle n}a_j(x)partial_{x_j}$ be a Lipschitz-continuous vector field on some open subset of $mathbb R^n$. The flow is then Lipschitz-continuous: it is a consequence of Gronwall's inequality. In fact, with
$$
dot Phi(t,y)=X(Phi(t,y)),quad Phi(0,y)=y,
$$
we have
$
Phi(t,y_1)-Phi(t,y_2)=y_1-y_2+int_0^tBigl(X(Phi(s,y_1))-X(Phi(s,y_2))Bigr) ds
$
and consequently for $tge 0$
$$
vert Phi(t,y_1)-Phi(t,y_2)vertle vert y_1-y_2 vert+
int_0^t Lvert Phi(s,y_1))-Phi(s,y_2)vert ds=R(t).
$$
As a result, we get
$
dot R(t)le L R(t),quad R(0)=vert y_1-y_2 vert
$
so that
$$vert Phi(t,y_1)-Phi(t,y_2)vertle
R(t)le vert y_1-y_2 vert e^{tL}.
$$
When the vector field is $C^1$, the flow is also $C^1$ with respect to $x$, but the proof is not so simple (the previous argument is somehow a first step). The Birkhoff-Rota book on ODE provides a nice proof.

Bazin.