Heat equation energy identity
= Heat equation energy identity
{title2=$\frac d{dt}\|u\|_2^2=-2\|du\|_2^2$}
On a <closed manifold> with a <Riemannian metric>, a smooth solution of $\partial_tu+\Delta u=0$ for the <positive Laplace-Beltrami operator> satisfies the displayed identity by <integration by parts>. Zero initial data force the solution to vanish, proving uniqueness and the <semigroup property> for heat evolution.