Theorems Propagation of Conormal Singularities for the Constant-Coefficient Wave Equation Attributions

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Let $n \in \mathbb{N}$, let $Y \subset \mathbb{R}^n$ be a smooth embedded submanifold, and let

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\begin{align*} N^*Y \setminus 0 := \{(y,\eta) \in T^*\mathbb{R}^n : y \in Y,\ \eta|_{T_yY}=0,\ \eta \neq 0\} \end{align*}

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denote the punctured conormal bundle of $Y$, viewed as a conic subset of $T^*\mathbb{R}^n \setminus 0$. Let $\mathcal{S}(\mathbb{R}^n)$ denote the [Schwartz space](/page/Schwartz%20Space) and let $\mathcal{S}'(\mathbb{R}^n)$ denote its [topological dual](/page/Topological%20Dual), the space of [tempered distributions](/page/Tempered%20Distributions). Let $u_0,u_1 \in \mathcal{E}'(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$ be compactly supported distributions satisfying

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\begin{align*} WF(u_0) \cup WF(u_1) \subset N^*Y \setminus 0. \end{align*}

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Let $|D_x| : \mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ denote the Fourier multiplier with symbol $|\xi|$. For each $t \in \mathbb{R}$, let $C(t) : \mathcal{E}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ and $S(t) : \mathcal{E}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ be the constant-coefficient wave propagators defined by $C(t)f=\cos(t|D_x|)f$ and $S(t)f=|D_x|^{-1}\sin(t|D_x|)f$, where the multiplier $|\xi|^{-1}\sin(t|\xi|)$ is extended smoothly at $\xi=0$ by the value $t$. Let

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\begin{align*} P := D_t^2 - \Delta_x : \mathcal{D}'(\mathbb{R}_t \times \mathbb{R}_x^n) \to \mathcal{D}'(\mathbb{R}_t \times \mathbb{R}_x^n) \end{align*}

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be the wave operator, where $D_t$ denotes distributional differentiation in the $t$-variable and $\Delta_x$ denotes the Euclidean Laplacian in the $x$-variables. Let $u \in \mathcal{D}'(\mathbb{R}_t \times \mathbb{R}_x^n)$ be the distributional Cauchy solution given by

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\begin{align*} u(t,\cdot) = C(t)u_0 + S(t)u_1. \end{align*}

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Equivalently, $Pu=0$ with Cauchy data

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\begin{align*} i_0^*u = u_0, \qquad i_0^*D_tu = u_1, \end{align*}

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where $i_0 : \mathbb{R}^n \to \mathbb{R}_t \times \mathbb{R}_x^n$ is the embedding $i_0(x)=(0,x)$ and the traces are understood through the standard wave Cauchy evolution on compactly supported distributions.

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Let

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\begin{align*} p : T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0 \to \mathbb{R} \end{align*}

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be the principal symbol of $P$, defined by

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\begin{align*} p(t,x,\tau,\xi)=\tau^2-|\xi|^2. \end{align*}

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Let $H_p$ denote its Hamilton vector field

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\begin{align*} H_p = 2\tau\,\partial_t - 2\sum_{j=1}^n \xi_j\,\partial_{x_j} \end{align*}

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on $T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0$. For $y \in Y$, $(y,\eta) \in N^*Y \setminus 0$, and $\sigma \in \{-1,1\}$, define the bicharacteristic map $\gamma_{y,\eta,\sigma}: \mathbb{R} \to T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0$ by

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\begin{align*} \gamma_{y,\eta,\sigma}(s) = (2\sigma |\eta|s, y - 2s\eta, \sigma |\eta|, \eta). \end{align*}

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Equivalently, $\gamma_{y,\eta,\sigma}$ is the integral curve of $H_p$ with initial point

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\begin{align*} \gamma_{y,\eta,\sigma}(0) = (0,y,\sigma |\eta|,\eta). \end{align*}

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Then

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\begin{align*} WF(u) \subset \bigcup_{(y,\eta) \in N^*Y \setminus 0}\bigcup_{\sigma \in \{-1,1\}} \gamma_{y,\eta,\sigma}(\mathbb{R}). \end{align*}

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Equivalently, if $(t,x,\tau,\xi) \in WF(u)$, then there exist a nonzero conormal covector $(y,\eta) \in N^*Y \setminus 0$, a sign $\sigma \in \{-1,1\}$, and a parameter $s \in \mathbb{R}$ such that $(t,x,\tau,\xi) = \gamma_{y,\eta,\sigma}(s)$.

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