Consider the scalar conservation law
$$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a convex function ($f''>0$).
Usually, this kind of equation admits several weak solutions and additional conditions have been imposed to select the "physically" relevant solution among the others.
One of those conditions is the so called $\textbf{"Oleinik's entropy condition"}$ which states that if $x=x(t)$ is a curve in which a solution $u$ is discontinuous, then there is a unique solution that satisfy:
$$ \frac{u(x+a,t)-u(x,t)}{a} \leq \frac{E}{t} \hspace{0.7 cm} a>0,t>0,$$
where $E$ is independent of $x,t$ and $a$.
It is stated that this condition express the growth of entropy along the curve $x$. I'm not really familiarized with these topics and I would like to have a sort of "physical" explanation of this assertion.
$\textbf{Remark}$
When $f$ is convex and $u_l$, $u_r$ denote the values of the function $u$ at the "left" and "right" of the discontinuity, then the Oleinik's entropy condition can also be stated as:
$$f'(u_r) < s < f'(u_l)$$
where $s=x'(t)$.
Best Answer
Note that the mathematical notion of entropy in the theory of conservation laws may not be related to the physical concept of entropy. Conceptually, both notions are disconnected in general. The condition introduced in OP leads to the vanishing viscosity weak solution, which is the physically relevant one (don't know any intuitive formulation of this property). Nevertheless, for some hyperbolic systems, the physical negentropy defines a mathematical entropy... See also these related posts here and there.