On the definition of hyperbolic conservation law There is a Wikipedia article and an article by A. Bressan that introduce the notion of hyperbolic conservation law.  I'm not used to this area, so I have some questions about the definition.  I will stick to the one dimensional case.
Let $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and
let $j : \mathbb{R} \to \mathbb{R}$.  A one dimensional conservation equation
is the equation
$$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
So here is my basic questions: What are the types (domain and codomain) of $y$ and $j$?  What is the quantification: i.e. is it, for any $y$ and $j$ of those types that satisfy the differential equation, we say that $y$ and $j$ form a conservation law. --?  Or do $y$ and $j$ need to have other dependencies as well.
Also, if you want to leave any other comments regarding one dimensional conservation laws -- please do.  I would love to learn more.
 A: You asked a couple of questions, so I'll try to take them in order.

What are the types (domain and codomain) of $y$  and $j$? 

Mathematically, I think you are on good ground to choose any domain and codomain for $y$.  The domain of $j$ must include the domain of $y$, of course, and it should not depend directly on $x_1$ or $x_2$.  The codomain needs to be consistent with $y$ to the extent that it makes sense to add a time derivative of $y$ with a gradient of $j$.  From a physics point of view, in most cases expect $y$ to be a function of space and time.  Beyond that you have no restrictions in general - although the specific physical scenario may imply additional requirements.

What is the quantification: i.e. is it, for any y  and j  of those types that satisfy the differential equation, we say that y  and j  form a conservation law. --? Or do y  and j  need to have other dependencies as well.

Yes, if they satisfy the equation, then they form a conservation law.  I don't think that there's more to it than that. I'd take the equation to define a conservation law.

Also, if you want to leave any other comments regarding one dimensional conservation laws -- please do. I would love to learn more.

Here are some assorted facts / concepts, which apply more generally than 1D:


*

*Every conservation law corresponds to a symmetry.  This is closely connected to Noether's theorem, which you can look-up.  A "symmetry" in this case is more general than the every-day examples like "left-right symmetry."  It means that some physical law is does not depend on some identifiable quantity.  Gauge invariance from electrodynamics is an example of this broader type of symmetry.  The action for EM does not change when you make a gauge transformation.

*Another similar but different concept is a constraint.  A constraint must be satisfied at each point in space and time. The conserved quantity is only conserved in a more global (I'd like to say "integral") sense since whatever stuff is represented by your $y$ is allowed to move to another place in space - It just cannot be created or destroyed.  Using EM again, $\nabla \cdot E = \rho/\epsilon_0$ is a constraint since it's satisfied everywhere and for all time.  Each constraint (trivially) implies a conserved quantity - In this case $\partial_t (\nabla \cdot E - \rho/\epsilon_0) = 0$.  Converted to the notation of your question, in this case $y = \nabla \cdot E - \rho/\epsilon_0$ and $j(y) = 0$.

