ap.analysis of pdes – Definition of entropy and entropy flux of conservation laws: component-wise reasoning

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:

  1. 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 $u?$
  2. 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.