Em Wave Equation Prerequisites — Examples

Given: The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; a speed $c>0$; the data of clause (i) $t_0=0$, $a=b=0$, $h\equiv 1$ on ${\mathbb{R}}^3\times [0,T]$; of clause (ii) $a=b=0$, $t_0=0$, $h(x,t)=h_0(x)$ with $h_0\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$ vanishing off ${\bar{B}}_1(0)$; and of clause (iii) $a=0$, $b(x)=x_1$, $h=0$, with $t_0<T$ arbitrary.

Wave Cauchy problem: for $c>0$, $t_0<T$, $a,b\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$ and $h$ of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times [t_0,T]$ in the extension sense, the function $u(x,t)={\mathrm{\partial }}_s{H}_a(x,s)+H_b(x,s)+c^2{\int }_{t_0}^t{H}_{h(\cdot ,v)}(x,t-v)dv,s=t-t_0,$ where $H_{\phi }(x,\sigma )=\sigma M_{c\sigma }\phi (x)$ and $M_r\phi (x)=\frac{1}{4\pi }{\int }_{S^2}\phi (x+rn)d\mathrm{\Omega }(n)$, is of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times [t_0,T]$ in the extension sense and satisfies $Wu=h$, $u(\cdot ,t_0)=a$ and ${\mathrm{\partial }}_{t}u(\cdot ,t_0)=b$ with $W=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$; every $C^2$ extension-sense solution with the same data equals it (clause (b)); and if $a$ vanishes off a compact $K_a$, $b$ off a compact $K_b$ and every $h(\cdot ,v)$ off one fixed compact $K_h$, with $K_a\cup K_b\cup K_h\subseteq {\bar{B}}_{\rho }(0)$, then $u(x,t)=0$ whenever $\mathrm{\mid }x\mathrm{\mid }>\rho +c(t-t_0)$ (clause (d)) (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support).

For $f\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$, $r\mapsto M_{r}f(x)$ is smooth for $r\ge 0$ with the normalisation of [F1] and no decay hypotheses; for each fixed $f$ the function $H_f(x,t)=t{M}_{ct}f(x)$ is the one used in [F1] (The spherical mean identity behind the wave equation).

The reflection $n\mapsto -n$ preserves the surface measure $d\mathrm{\Omega }$, ${\int }_{S^2}1d\mathrm{\Omega }=4\pi$, ${\int }_{S^2}nd\mathrm{\Omega }(n)=0$, and the chart surface measure on $S^2$ is the polar surface measure (Reflection invariance and vanishing first moment of the sphere measure, Agreement with the existing polar sphere measure, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Consequently $M_{r}1(x)=1$ and $M_r(x\mapsto x_1)(x)=x_1$ for every $r\ge 0$ and every $x\in {\mathbb{R}}^3$, since the sphere integral of the constant $1$ is $4\pi$ and the sphere integral of $n\mapsto x_1+r{n}_1$ is $4\pi x_1$.

Energy: for a $C^2$ function $u$ the energy density is $e=\frac{1}{2}(u_t^2+c^2\mathrm{\mid }Du{\mathrm{\mid }}^2)$ with $u_t={\mathrm{\partial }}_{t}u$ and $Du$ the spatial gradient; for measurable $\mathrm{\Omega }$ with ${\int }_{\mathrm{\Omega }}e<\mathrm{\infty }$ the total energy is $E_{\mathrm{\Omega }}(t)={\int }_{\mathrm{\Omega }}e(x,t)dx$, and when the integral is infinite the total energy is $+\mathrm{\infty }$ (Wave energy density, energy flux and total energy).

Energy uniqueness for the wave Cauchy problem: under ${\mathrm{A}\mathrm{C}}_{\omega }$, if a classical solution of ${\mathrm{\square }}_{c}u=0$ has Cauchy data $(u_0,u_1)$ with differentiable $u_0$ and the total energy has a finite initial value $E_{{\mathbb{R}}^3}(0)=\frac{1}{2}{\int }_{{\mathbb{R}}^3}(u_1^2+c^2\mathrm{\mid }D{u}_0{\mathrm{\mid }}^2)$ and is conserved in the sharp form $E_{{\mathbb{R}}^3}(t)=E_{{\mathbb{R}}^3}(0)$, then vanishing data force $u\equiv 0$ (Energy uniqueness for the wave Cauchy problem).

For nonnegative measurable $f\le g$ one has $\int f\le \int g$; Tonelli's theorem permits the iterated integral of a nonnegative measurable function on a product of $\sigma$-finite spaces to be computed in either order; the box $[-R,R{]}^3$ has Lebesgue measure $(2R{)}^3$ with ${\int }_{-R}^R{x}_1^{2}d{x}_1=2R^3\mathrm{/}3$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included, Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative).

The published Duhamel principle for the forced wave equation assumes a source continuous and compactly supported in $x$ for each fixed $s$, with all spatial derivatives through order $\lfloor n\mathrm{/}2\rfloor +1$ jointly continuous; its launched-solution construction applies to the admissible slices of the source (Duhamel's principle for the wave equation).

The Euclidean norm obeys the triangle inequality and hence $\mathrm{\mid }y\mathrm{\mid }\ge \mathrm{\mid }x\mathrm{\mid }-\mathrm{\mid }y-x\mathrm{\mid }$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

Every continuous function on a closed nondegenerate Euclidean rectangle is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable); under ${\mathrm{A}\mathrm{C}}_{\omega }$, a bounded Borel Riemann integrand on such a rectangle has the same Lebesgue integral (Borel Darboux integrands in finite dimension). In particular these apply to the continuous polynomial $x_1\mapsto x_1^2$ on $[-R,R]$.

Under ${\mathrm{A}\mathrm{C}}_{\omega }$, ${\lambda }_m\times {\lambda }_n$ equals ${\lambda }_{m+n}$ on Borel sets for $m,n\ge 1$ (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}). Applying this with $(m,n)=(1,1)$ and $(2,1)$ identifies the iterated product integral of the continuous nonnegative energy density on $[-R,R{]}^3$ with its Euclidean Lebesgue integral. The factor measures are sigma-finite because the finite-measure intervals $[-N,N]$ exhaust $\mathbb{R}$ by [F6].

The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).