This is not a transport equation. It is a conservation law. The difference between these class is that a TE is of the form $\partial_tu+a(t,x)\cdot\nabla_xu=0$, for which the essential supremum/infimum in the space variable remains constant as time varies. On the contrary, the space integrals of the positive/negative parts of the solution of a CL remain constant as time varies.

Remark that the classes are dual to each other: the adjoint of $\partial_t+a\cdot\nabla_x$ is $-\partial_t-{\rm div}_x(a\cdot)$.