Maximum principle and linear transport Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
What happens to the maximum of $u$ along the evolution? Is it true that $\|u(t,\cdot)\|_{L^\infty} = \|u_0\|_{L^\infty}$ for $t \ge 0$ or does the inequality hold?
 A: 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 $-\partial_t-{\rm div}_x(a\cdot)$.
Edit. The confusion comes from the fact that one often considers divergence-free vector fields (${\rm div}_xa\equiv0$), in which case both equations are identical. But in general, TE are related to ODEs, because the solution of the Cauchy problem is given by
$$u(t,\cdot)=u_0\circ\phi^0_t$$
where $\phi$ stands for the flow of the ODE
$$\frac{dx}{dt}=a(t,x).$$
On the contrary, CL govern the evolution of densities (= volume forms), and the solution of Cauchy problem is given by a pullback operation
$$u(t,\cdot)dx=\left(\phi^0_t\right)_\sharp(u_0dx).$$
