I have a rather soft question. Let's assume that we consider the heat equation posed in $S^1$:
$$
\partial_t u=\partial_x^2u.
$$
It is well known that if we define the functionals
$$
H(t)=\int_{-\pi}^\pi u\log(u)-u+1dx,
$$
$$
E(t)=\int_{-\pi}^\pi u^2dx,
$$
they decay. As far as I know, in the literature, a functional similar to $H$ is called *entropy* while a functional like $E$ is called *energy*. I understand the physical reasons behind that.

The question is: are there some subtle mathematical differences between these two Lyapunov functionals that we emphasize by using different names entropy/energy? Or, on the other hand, it's just a reminder from the physical origin of both terms?