Let $(M,g)$ be a closed smooth Riemannian manifold, let $(N,h)$ be a smooth Riemannian manifold, and let $u: M \times [0,T) \to N$ be a smooth solution of the harmonic map heat flow $\partial_t u = \tau_g(u)$, where $\tau_g(u)$ denotes the tension field of the time-slice map $u_t: M \to N$, $x \mapsto u(x,t)$. Define the Dirichlet energy by

\begin{align*} E[u(t)] := \frac{1}{2}\int_M |du_t|_{g^* \otimes u_t^*h}^2\,d\operatorname{vol}_g. \end{align*}

Then, for every $t \in (0,T)$,

\begin{align*} \frac{d}{dt}E[u(t)] = -\int_M |\partial_t u(\cdot,t)|_h^2\,d\operatorname{vol}_g = -\int_M |\tau_g(u_t)|_h^2\,d\operatorname{vol}_g. \end{align*}

In particular, the function $t \mapsto E[u(t)]$ is nonincreasing on $[0,T)$.