= Energy estimate
{wiki}
An energy estimate tests a differential equation against the solution or one of its derivatives to control a norm by initial data and forcing. Coercive terms represent dissipation, while <Young inequality> and the <Gronwall inequality> control lower-order terms.
For the difference $w$ of two smooth <heat equation> solutions with $D>0$, identical <PDE source term>, <initial conditions>, and either <periodic boundary conditions> or <Dirichlet boundary data>, <integration by parts> gives
$$
\frac{d}{dt}\int w^2\,dx=-2D\int w_x^2\,dx\leq0.
$$
The initially zero nonnegative <integral> stays zero, proving uniqueness. The same computation bounds changes in the <L2 norm> when the <initial conditions> differ. Specifying data and a solution class makes this argument meaningful.https://web.stanford.edu/class/math220b/handouts/heateqn.pdf[Stanford Math 220B heat-equation notes, section 2.8.3]{ref}
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