intuition for entropy solutions 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.)
 A: A weak solution is based on the fact that a very smooth solutions for this equation will satisfy an integral equation when multiplied and integrated with a test function.
The idea behind entropy solutions is of the same order: if we had a very smooth solution, it would have to satisfy some (integral) inequalities. Thus it is reasonable to ask for our notion of solutions to satisfy this inequality (since we want smooth entropy solutions to be classical solutions, and classical solutions to be entropy solutions). Also, it features more insight about the PDE studied, since it rules out some weak solutions one can not be happy with (see the example of the Burgers equation, for which one can construct an infinity of weak solutions). 
In terms of physical intuition I will quote the content of a lecture from Mouhot which can be find here (p.27, especially exercise 48, and example of pathological weak solutions is on the same page)

The inequalities with the entropic-flux pairs in the definition of
  entropic solution can be understood as the time-arrow information that
  should be retained from the microscopic dissipative mechanisms that
  are neglected. They have the effect to prevent characteristics getting
  out from shocks (discontinuity curves).

This physical intuition is obtained by a trick which is current in the study of hyperbolicity: one introduces a small dissipative term in the equation, and analyze the behavior of the limit of a sequence of solutions to these new equations.
