Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R}^n)$ be a vector field on $\mathbb{R}^n$ and consider the following PDE:

$$u_t = div[V(u)]$$

For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with initial conditions specified on the $n$-dimensional faces of the $n+1$-th cube minus the top face.

Are there nessasary and sufficient conditions one can put on $V$ for which the above problem will always be well-posed for some choice of $T$? (for initial conditions in reasonable function spaces).