Let $v(t,x): \mathbb{R}^n \mapsto \mathbb{R}^n$ be Lipschitz with bounded divergence $\nabla \cdot v$. How do I prove that the the solution $X_t$ of the initial-value problem
$$\dot{X_t} = v(t, X_t) , \quad X_0 = id ,$$
exists and constitutes a one-parameter family of diffeomorphisms?