For a hyperbolic PDE of the form
$$u_t + f(u)_x = 0$$
it turns out that the right notion of solution is entropy solution. Now, the notion of classical solutions are obviously very natural, and also the notion of weak solutions (or solutions as distributions) also seems natural to me. On the other hand, the notion of entropy solutions is completely unnatural to me. I am seeking some intuition about why entropy solutions are the right "physical" solutions, and how to interpret the definition. (For instance, one very naive comment is that it feels strange to me that the solutions are defined by an inequality.)