TITLE: How define the entropy of heat equation?
QUESTION [1 upvotes]: Today, I report a paper about Ricci flow, I saw entropy. As I know, entropy is a physical  term.And I know it is used to describe how far the system from heat death.But I don't know the equation of entropy and the precise define.
For example, if there is a heat  equation and suitable initial condition,how to define the entropy ?
$$ 
\left\{
\begin{aligned}
&u_t  =\Delta u ~~~~\text{  in $~\Omega\times(0,T]$}\\
&u(x,0) =u_0(x)
\end{aligned}
\right.
$$

REPLY [0 votes]: Much too late, but there is a very nice script by Evans on the subject: https://math.berkeley.edu/~evans/entropy.and.PDE.pdf
The heat equation and, more generally parabolic equations, is treated on pp. 90 ff.
Very roughly, the entropy is defined as $\log(u)$, where $u$ is the solution to the heat equation.
Hope that helps!
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