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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?

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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 $$u(t,\cdot)dx=\left(\phi^0_t\right)_\sharp(u_0dx).$$

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    $\begingroup$ I think I understand: so you're saying that the supremum change depends on the divergence of $a$ for the conservation law? $\endgroup$
    – Riku
    Commented Apr 1, 2022 at 12:42
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    $\begingroup$ yes indeed. Let me give some more details. $\endgroup$
    – Denis Serre
    Commented Apr 1, 2022 at 13:54
  • $\begingroup$ Thank you. One more question: what is explicitly $(\phi^0_t)_\sharp u_0$ and how does it differ from $u_0\circ \phi_t^0$? $\endgroup$
    – Riku
    Commented Apr 2, 2022 at 21:09
  • $\begingroup$ The pullback operation applies to differential forms of various degrees $r$. When $r=0$, these are functions and the pullback is just the composition. When $r\ge N$, the pullback is the Jacobian of $\phi^0_t$, times this composition. $\endgroup$
    – Denis Serre
    Commented Apr 3, 2022 at 6:19

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