Why maximum principle holds for scalar conservation law? I wonder why the maximum principle holds for scalar conservation law.
Consider the PDE below.
$u_t+f(u)_x=u_{xx}, \quad u_0(x)=u(0,x)$
(where $f$ is a smooth real function)
The paper which I am reading says $||u||_\infty \le ||u_0||_\infty$.
(We can assume $u$ is bounded weak solution.)
I try to prove this.
Please tell me why the maximum principle holds.
 A: What you are looking for follows from the maximum principle for parabolic equations.
I guess with $\Vert u \Vert_\infty$ you denote the infinity norm both over space $x$ and time $t$.
So let's assume that $u(x, t)$ attains a strict maximum at $x^\star, t^\star > t_0 = 0$. Then (necessary conditions for optimality in multiple dimensions)
$$u_t(x^\star, t^\star) = 0 = u_x(x^\star, t^\star)$$
and since we assume that we have a strict maximum
$$u_{xx}(x^\star, t^\star) < 0$$
Rewriting the PDE $\big($ which holds of course also at the point $(x^\star, t^\star) \big) $as
$$ u_t + f'(u) u_x - u_{xx} = 0$$ we see that our assumption lead to a contradiction (by the conditions of optimality):
$$ \underbrace{u_t\Big \vert_{(x^\star, t^\star)}}_{=0} + f'(u)\Big \vert_{(x^\star, t^\star)} \underbrace{u_x\Big \vert_{(x^\star, t^\star)}}_{= 0} - \underbrace{u_{xx}\Big \vert_{(x^\star, t^\star)}}_{< 0 } \neq 0$$
The arguments hold also for strict minima.
Thus, no strict maximum/minimum can be attained for $t^\star > t_0$ and we have
$$ \Vert u(x, t) \Vert_{L^\infty(\mathbb{R}, \mathbb{R}_+)} \leq \Vert u_0(x) \Vert_{L^\infty(\mathbb{R})}$$
