Considering the following equation,
$$
u\_t + \operatorname{div} \, (u \, \mathbf{b}(\mathbf{x},t)) = 0
$$
in a cylinder $K = \{(\mathbf{x},t) \in \Omega \times (0,T) \}$ where $\Omega \subset \mathbb{R}^d$ is regular (whatever we need), with initial condition 
$$u |\_{t=0} = u\_0$$
and 
$$\mathbf{b} \cdot \mathbf{n} = 0$$ at the boundary of $\Omega$, 
what are the conditions (necessary or sufficient) on $\mathbf{b}$ for this homogeneous equation to have unique solution in $L^2 (K)$?


I have seen something like $\operatorname{div} \, ( \mathbf{b}(\mathbf{x},t) ) \geq 0$ but I cannot find a reference to be sure if it is right. As far as I remember, we need the characteristics not to intersect each other but how to ensure that?


Any precise statements or references to them are welcome.


Thanks!



Uniqueness conditions for linear transport equation with nonconstant velocity
Well, the answer comes from a quite simple energy argument.
After multiplying the equation by $u$, integrating by parts and using
$$
\pmb{b} \cdot \nabla u = \operatorname{div} ( \pmb{b} u ) - \left( \operatorname{div} \pmb{b} \right) u, 
$$
one can get
$$
\frac{1}{2} \frac{d}{dt} \int\_{\Omega} u^2 d\Omega = - \int\_{\Omega} \left(\operatorname{div} \pmb{b} \right) u^2 d\Omega \leq 0,
$$
if $\operatorname{div} \pmb{b} \geq 0$. So this condition implies uniqueness which I actually sought for.


