Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of $\Psi(r):=\int_0^r b(s)\;ds$ where $b:\mathbb{R}\to\mathbb{R}$ is $b(s) = s^{\frac 1 m}$?
This is associated to the porous medium equation $$u_t = \Delta u^m= \Delta b^{-1}(u).$$
Googling this is very messy.