Wave energy estimate ($\Vert \partial u(t){\Vert }_2\le \Vert \partial u(0){\Vert }_2+{\int }_0^t\Vert F(s){\Vert }_{2}ds$)

If $\square u=F$ with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, then $\Vert \partial u(t){\Vert }_2\le \Vert \partial u(0){\Vert }_2+{\int }_0^t\Vert F(s){\Vert }_{2}ds$, 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.

For the unit-speed wave equation, the integral of $(u_t^2+|\nabla u{|}^2)/2$ over $B_r(x_0)$ at time $T\ge 0$ is at most its initial integral over $B_{r+T}(x_0)$. On a shrinking ball the outward energy flux minus the loss from its moving boundary is $-((u_t-{\partial }_{n}u{)}^2+|{\nabla }_{\tan }u{|}^2)/2$, hence nonpositive. This estimate establishes the domain of dependence and finite propagation speed without any assumptions at spatial infinity.