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 \emph{entropy} while a functional like $E$ is called \emph{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?

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?