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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:

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

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

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  1. 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.

  2. 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.

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