Why is the definition of entropy solution necessary to prove uniqueness for hyperbolic conservation laws? I'm aware that there are a lot of counterexamples to show that distributional solutions for hyperbolic (scalar) conservation laws are not unique.
However, I'd like to ask:


*

*Conceptually, at which point of a proof of uniqueness is the definition of distributional solution not enough to go on?


*Why is the definition of entropy solution useful in the proof of uniqueness for hyperbolic conservation laws?

 A: *

*Unless an evolution PDEs be linear, a uniqueness proof is always nonlinear in essence: you prove that some distance $d(u(t),v(t))$ between two solutions is bounded in terms of $d(u(0),v(0))$. To carry out the proof, you need to be able to compute the time derivative of $d(u,v)$, and for this you need the chain rule. It turns out that the solutions of hyperbolic conservation laws display sharp discontinuities, and therefore the chain rule does not apply.

*The entropy condition is a differential inequality that involves such a nonlinear quantity, typically $d(u,k)$ where $k$ is a constant state. This immediately yiedls uniqueness if the initial data is constant. When it is not, you can still conclude uniqueness if the solution is Lipschitz continuous, by a so-called weak-strong argument (similar to that for the Navier-Stokes equation). If the equation is scalar, Kruzkhov used the fact that the distance $|v-u|$ is a convex entropy with respect to both arguments and proved uniqueness of weak entropy solutions. For systems, the situation is much more complicated, and totally open in several space dimensions.
