I have the following question:
let's suppose we have some continous vector field $X$ on a compact surface and let's suppose that it is locally Lipschitz. Moreover, let's suppose that its flow $\Phi^X_t$ is space-smooth at $t=1$.
Now let's modify $X$ through another locally Lipschitz vector field $Y$: we define a new field $T_t$ through the transport equation $$\frac{d}{dt} T_t = \mathcal{L}_Y T_t, \quad T_0 = X$$
Let's suppose that also $Y$ has the property that $\Phi_t^Y$ is space-smooth at $t = 1$. Is it true that the flow of $T_t$ is space-smooth for $t = 1$?
My knowledge is very basic, I'm a geometer. I need the previous fact in order to construct an isotopy on some manifold.
