Consider the conservation law

$$

partial_t u(x,t) + text{div } g(u(x,t)) =0, \

u in Usubseteq mathbb R^m, xin Xsubseteq mathbb R^n, G subseteq mathbb M^{ntimes m}(mathbb R).

$$

We usually equip the system with an **entropy** (which is a scalar) $eta$ with associate flux $Q in mathbb R^n$ related by

$$

DQ_{alpha} =Deta cdot DG_alpha, alpha = 1,2,ldots, n. (*)

$$

Here are my concerns about this definition. In thermodynamics, the entropy flow is simply heat flux divided by temperature. **It is NOT defined by the equation stated above via some other quantity $u.$**

Therefore, I am puzzled by the following questions:

- Is the thermodynamic entropy/entropy flux an entropy in the sense of the definition above? In other words, does the tranditional physical entropy arise from some quantity $u$ via the above equation? If so, what is the physical meaning of $u?$
- Are there other nonequivalent definitions of “mathematical” entropy? After all $(*)$ is not necessary for $eta(u)$ to be conserved for smooth solutions. To make sure $eta(u)$ is conserved for smooth solutions, we only need $text{div }Q = Deta cdot text{div }G(U)$ which is of course a much weaker condition.
**That leaves the possibility of defining other mathematical objects that looks somewhat like entropy.**