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?



Maximum principle and linear transport
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
$$u(t,\cdot)dx=\left(\phi^0\_t\right)\_\sharp(u\_0dx).$$


