Theorems Finite Propagation Speed for Retarded Strictly Hyperbolic Parametrices Attributions

Attributions & Verification

Track contributions and verify content correctness

Statement

Let $Y$ be a smooth $n$-dimensional manifold, let $(O,\varphi)$ be a coordinate chart with $\varphi:O\to V\subset \mathbb{R}^n$, let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}$, and let $J\subset\mathbb{R}$ be an open interval with $0,t_0\in J$ and $t_0>0$. Write points of $J\times O$ in local coordinates as $(t,x)\in J\times V$, where $x=\varphi(y)$. Let $P$ be a scalar differential operator of order $m$ with smooth coefficients on $J\times O$. Assume that, in these coordinates, the principal symbol of $P$ factors as

paragraph admin

\begin{align*} p_m(t,x,\tau,\xi)=a(t,x)\prod_{j=1}^m(\tau-\lambda_j(t,x,\xi)), \end{align*}

latex_env admin

where $a:J\times V\to\mathbb{R}$ is smooth and nonvanishing, and for every $1\leq j\leq m$ the map

paragraph admin

\begin{align*} \lambda_j:J\times V\times \mathbb{R}^n_0\to\mathbb{R} \end{align*}

latex_env admin

is smooth, real-valued, positively homogeneous of degree $1$ in $\xi$, and the values $\lambda_1(t,x,\xi),\dots,\lambda_m(t,x,\xi)$ are pairwise distinct for every $(t,x,\xi)\in J\times V\times\mathbb{R}^n_0$. For $1\leq j\leq m$, define $v_j:J\times V\times \mathbb{R}^n_0\to\mathbb{R}^n$ by

paragraph admin

\begin{align*} v_j(t,x,\xi):=-\partial_\xi\lambda_j(t,x,\xi). \end{align*}

latex_env admin

Assume that the Cauchy problem for $P$ at $t=0$ is locally well posed in an energy space for which the traces

paragraph admin

\begin{align*} \gamma_k u:=\partial_t^k u(0,\cdot)\in\mathcal{D}'(O) \end{align*}

latex_env admin

exist for $0\leq k\leq m-1$. Assume also the following local domain-of-dependence property. If $t_*\in J$ with $t_*>0$, $x_*\in V$, $c>0$, and

paragraph admin

\begin{align*} K:=\{(s,y)\in[0,t_*]\times O:|\varphi(y)-x_*|\leq c(t_*-s)\} \end{align*}

latex_env admin

is compactly contained in $J\times O$, then every energy-class solution $w$ of $Pw=h$ satisfying

paragraph admin

\begin{align*} h=0\quad\text{in }\mathcal{D}'(\operatorname{int}_{J\times O}K) \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} \gamma_k w=0\quad\text{in }\mathcal{D}'(\{y\in O:|\varphi(y)-x_*|<ct_*\}) \end{align*}

latex_env admin

for all $0\leq k\leq m-1$ vanishes in the energy sense, and therefore in the distributional sense, on some open neighbourhood of $(t_*,\varphi^{-1}(x_*))$.

paragraph admin

Fix $c>0$ and $y_0\in O$, and set $x_0:=\varphi(y_0)$. Define

paragraph admin

\begin{align*} D_c(t_0,y_0):=\{(s,y)\in[0,t_0]\times O:|\varphi(y)-x_0|<c(t_0-s)\} \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} \overline D_c(t_0,y_0):=\{(s,y)\in[0,t_0]\times O:|\varphi(y)-x_0|\leq c(t_0-s)\}. \end{align*}

latex_env admin

Assume that $\overline D_c(t_0,y_0)$ is compactly contained in $J\times O$.

paragraph admin

Assume that every retarded bicharacteristic base curve relevant to the local retarded solution operator and ending at $(t_0,y_0)$ is, in the coordinate chart, the base projection of a smooth lifted Hamiltonian bicharacteristic and is represented by maps

paragraph admin

\begin{align*} x\in C^1([s,t_0];V),\qquad \xi\in C^1([s,t_0];\mathbb{R}^n_0) \end{align*}

latex_env admin

with $0\leq s\leq t_0$, $x(t_0)=x_0$, $x(s)=\varphi(z)$ for some $z\in O$, and such that for some branch $1\leq j\leq m$,

paragraph admin

\begin{align*} \frac{dx}{dr}(r)=-\partial_\xi\lambda_j(r,x(r),\xi(r)) \end{align*}

latex_env admin

for every $r\in[s,t_0]$. Suppose that every such curve is defined on all of $[s,t_0]$, remains in $V$, and satisfies

paragraph admin

\begin{align*} |\partial_\xi\lambda_j(r,x(r),\xi(r))|\leq c \end{align*}

latex_env admin

for every $r\in[s,t_0]$.

paragraph admin

Let $u$ be an energy-class solution of $Pu=f$ on $J\times O$ with Cauchy traces $\gamma_k u=g_k\in\mathcal{D}'(O)$ for $0\leq k\leq m-1$. If

paragraph admin

\begin{align*} f=0\quad\text{in }\mathcal{D}'(\operatorname{int}_{J\times O}\overline D_c(t_0,y_0)) \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} g_k=0\quad\text{in }\mathcal{D}'(\{y\in O:|\varphi(y)-x_0|<ct_0\}) \end{align*}

latex_env admin

for every $0\leq k\leq m-1$, then there exists an open neighbourhood $U\subset J\times O$ of $(t_0,y_0)$ such that $u=0$ in the energy sense, and therefore in $\mathcal{D}'(U)$.

paragraph admin

Independently of these vanishing hypotheses, assume that, microlocally near $(t_0,y_0)$, the local Cauchy solution operator admits a retarded representation modulo smoothing operators

paragraph admin

\begin{align*} u\equiv Ef+\sum_{k=0}^{m-1}E_k g_k. \end{align*}

latex_env admin

Here $E$ is a Fourier integral operator from distributions on $J\times O$ to distributions on $J\times O$ with closed conic canonical relation, in the twisted operator convention,

paragraph admin

\begin{align*} C_E\subset (T^*(J\times O)\setminus 0)\times (T^*(J\times O)\setminus 0), \end{align*}

latex_env admin

and each $E_k$ is a Fourier integral operator from distributions on $O$ to distributions on $J\times O$ with closed conic canonical relation, in the same convention,

paragraph admin

\begin{align*} C_k\subset (T^*(J\times O)\setminus 0)\times (T^*O\setminus 0). \end{align*}

latex_env admin

Assume that after insertion of properly supported pseudodifferential cutoffs supported in the microlocal coordinate neighbourhood where the representation is valid, the localized operators are properly supported Fourier integral operators with canonical relations contained in $C_E$ and $C_k$, respectively, so that the wave front mapping theorem for properly supported Fourier integral operators applies to these localized operators. Assume these canonical relations are retarded and time-oriented as follows. If

paragraph admin

\begin{align*} (((t_0,y_0),\eta),((s,z),\zeta))\in C_E, \end{align*}

latex_env admin

then $0\leq s\leq t_0$ and this element is generated by a smooth lifted bicharacteristic over one branch $\tau=\lambda_j(t,x,\xi)$ whose base projection is a $t$-parametrized curve from $(s,\varphi(z))$ to $(t_0,x_0)$ satisfying the velocity equation above. If

paragraph admin

\begin{align*} (((t_0,y_0),\eta),(z,\zeta))\in C_k, \end{align*}

latex_env admin

then this element is generated by a smooth lifted bicharacteristic over one branch $\tau=\lambda_j(t,x,\xi)$ whose base projection is a $t$-parametrized curve from $(0,\varphi(z))$ to $(t_0,x_0)$ satisfying the same velocity equation. No advanced component, no component based at a time $s>t_0$, and no branch outside these $t$-parametrized bicharacteristic flows occurs in the microlocal representation near $(t_0,y_0)$.

paragraph admin

Then, for every nonzero covector $\eta\in T^*_{(t_0,y_0)}(J\times O)$, if

paragraph admin

\begin{align*} ((t_0,y_0),\eta)\in\operatorname{WF}(u), \end{align*}

latex_env admin

then at least one of the following alternatives holds.

paragraph admin

First, there exist $(s,z)\in\overline D_c(t_0,y_0)$ and a nonzero covector $\zeta\in T^*_{(s,z)}(J\times O)$ such that

paragraph admin

\begin{align*} ((s,z),\zeta)\in\operatorname{WF}(f) \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} (((t_0,y_0),\eta),((s,z),\zeta))\in C_E. \end{align*}

latex_env admin

Second, there exist an integer $k$ with $0\leq k\leq m-1$, a point $z\in O$ with

paragraph admin

\begin{align*} |\varphi(z)-x_0|\leq ct_0, \end{align*}

latex_env admin

and a nonzero covector $\zeta\in T^*_zO$ such that

paragraph admin

\begin{align*} (z,\zeta)\in\operatorname{WF}(g_k) \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} (((t_0,y_0),\eta),(z,\zeta))\in C_k. \end{align*}

latex_env admin

Equivalently, under the stated retarded time-oriented parametrix hypotheses, source and Cauchy-data canonical-relation branches whose base projections begin strictly outside the closed local backward cone $\overline D_c(t_0,y_0)$, respectively strictly outside the closed initial ball

paragraph admin

\begin{align*} \{z\in O:|\varphi(z)-x_0|\leq ct_0\}, \end{align*}

latex_env admin

do not contribute to $\operatorname{WF}(u)$ over $(t_0,y_0)$.

paragraph admin

Verification Progress

59 Total Blocks

0 Verified

0% verified

Contributors

admin 59 blocks (0 verified)

Who Can Verify

Areas: Analysis

Viktor Miykov Admin

Max Vassiliev Global Reviewer

Horia Neagu Global Reviewer

강현욱 Global Reviewer

Demo Testing Global Reviewer

Archie Pennycook Global Reviewer

Quick Actions

Edit Theorem

Raw Attribution Data

Loading attribution data...

What brings you to Androma?

Start with a route through the knowledge graph.

Attributions & Verification

Statement

Verification Progress

Contributors

Who Can Verify

Quick Actions

Sign in to Androma

Check your inbox

One last step

Attributions & Verification

Statement

Verification Progress

Contributors

Who Can Verify

Quick Actions

Raw Attribution Data