Wave energy estimate
= Wave energy estimate
{title2=$\|\partial u(t)\|_2\leq\|\partial u(0)\|_2+\int_0^t\|F(s)\|_2\,ds$}
If $\Box u=F$ with <d'Alembert operator> $\Box=-\partial_t^2+\Delta$, then $\|\partial u(t)\|_2\leq\|\partial u(0)\|_2+\int_0^t\|F(s)\|_2\,ds$, where $\partial u=(u_t,\nabla u)$. Test the <wave equation> against $u_t$, use <integration by parts>, and apply the <Cauchy-Schwarz inequality>. The same <energy estimate> applies to a finite system of <wave equations>.