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?



Mathematical difference between entropy and energy
First, note that entropy is well-defined only if $u$ is positive. Using integration by parts, it is in fact true on the flat torus or $\mathbb{R}^n$ (if $u$ is nonnegative and decays in space fast enough) that
$$
\frac{\partial}{\partial t}\frac{1}{p(p-1)}\log \int u^p 
= -\frac{\int u^{p-2}|\nabla u|^2}{\int u^p}\le 0
$$
for all $p > 1$. The case $p = 2$ is the usual $L^2$ energy inequality. If you normalize $u$ so that
$$
\int u = 1
$$
and take the limit $p \rightarrow 1$, you get the entropy inequality. So there are $L^p$ analogues of energy or entropy, depending on your point of view.


In information theory, if $u$ is a probability density, then
$$
-\int u\log u
$$
is called Shannon entropy and
$$
\frac{1}{1-p}\log\int u^p
$$
is called Rényi entropy.


