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