2.1 Notation

We use C to denote universal positive constants. We write $A\lesssim B$ , $B\gtrsim A$ , or $A=O(B)$ if $|A|\leq CB$ for some $C>1$ . The values of all constants in this paper may vary from line to line. We write $A\sim B$ if $A\lesssim B$ and $B\lesssim A$ . We use $C_{v}$ , $\gtrsim _v$ , or $\lesssim _v$ if we want to emphasize that the constant depends on a parameter v. Moreover, if we make a statement such as ‘ $|A|\leq CB$ for all v in a certain set V’, then it implies that the constant C can be chosen to be uniform for all $v\in V$ .

Unless specified otherwise, we always assume that the Latin indices $i,j,k$ take values in $\{1,2,3\}$ and the Greek indices $\alpha ,\beta $ take values in $\{0,1,2,3\}$ . We use subscript to denote partial derivatives unless specified otherwise. For example, $u_{\alpha \beta }=\partial _\alpha \partial _\beta u$ , $q_r=\partial _rq$ , $A_q=\partial _qA$ , etc. For a fixed integer $k\geq 0$ , we use $\partial ^k$ to denote either a specific partial derivative of order k, or the collection of partial derivatives of order k.

Given a function $f=f(\omega )$ defined on $\mathbb {S}^2$ , in order to define angular derivatives $\partial _\omega $ , we first extend f to $\mathbb {R}^3\setminus 0$ by setting $f(x):=f(x/|x|)$ and then set $\partial _{\omega _i}f:=\partial _{x_i}f|_{\mathbb {S}^2}$ . To prevent confusion, we will only use $\partial _\omega $ to denote angular derivatives under the coordinate system $(s,q,\omega )$ , and we will never use it under the coordinate system $(t,r,\omega )$ . For a fixed integer $k\geq 0$ , we will use $\partial _\omega ^k$ to denote either a specific angular derivative of order k, or the collection of all angular derivatives of order k.

2.2 Dependence on $\varepsilon $

In this paper, the parameter $\varepsilon $ always denotes the size of initial data in (1.2). We always assume that $\varepsilon \ll 1$ which means $0<\varepsilon <\varepsilon _0$ for some sufficiently small constant $0<\varepsilon _0<1$ . We say that $\varepsilon \ll 1$ depends on v, or $\varepsilon \ll _v1$ , if we want to emphasize that this constant $\varepsilon _0$ depends on a parameter v.

The convention in the previous subsection allows us to write a constant depending on v as C without the subscript v. However, we make a special exception for $\varepsilon $ . In this paper, if a constant depends on $\varepsilon $ , then we must write it as $C_{\varepsilon }$ . In other words, any constant without the subscript $\varepsilon $ must be independent of the choice of $\varepsilon $ . Similarly for $\lesssim _\varepsilon $ , $\gtrsim _\varepsilon $ , or $\ll _\varepsilon $ .

Note that most statements in this paper follow this format: For each integer $N\geq 1$ , we have a certain property $(P_{N,\varepsilon })$ as long as $\varepsilon \ll _N1$ . For simplicity, later we will only say that we have $(P_{N,\varepsilon })$ for $\varepsilon \ll 1$ without explicitly stating the dependence of $\varepsilon $ on N. For example, we can say

$$ \begin{align*} (P_{N,\varepsilon}):\ t^{-1+C_N\varepsilon}\leq 1,\qquad \forall t\geq 1 \end{align*} $$

as long as $\varepsilon \ll 1$ . Note that $(P_{N,\varepsilon })$ holds for all $\varepsilon \in (0,\frac {1}{1+C_N})$ .

2.3 Commuting vector fields

Let Z be any of the following vector fields:

(2.1) $$ \begin{align}\begin{aligned} \partial_\alpha,\ \alpha=0,1,2,3;\ S=t\partial t+r\partial_r=t\partial_t+\sum_{i=1}^3x_i\partial_i;\\ \Omega_{ij}=x_i\partial_j-x_j\partial_i,\ 1\leq i<j\leq 3;\ \Omega_{0i}=x_i\partial_t+t\partial_i,\ i=1,2,3. \end{aligned}\end{align} $$

We write these vector fields as $Z_1,Z_2,\dots ,Z_{11}$ , respectively. For any multiindex $I=(i_1,\dots ,i_m)$ with length $m=|I|$ such that $1\leq i_*\leq 11$ , we set $Z^I=Z_{i_1}Z_{i_2}\cdots Z_{i_m}$ . Then we have Leibniz’s rule

(2.2) $$ \begin{align}Z^I(fg)=\sum_{|J|+|K|=|I|}C^I_{JK}Z^Jf\cdot Z^Kg,\qquad\text{where }C_{JK}^I\text{ are constants.}\end{align} $$

We have the following commutation properties:

(2.3) $$ \begin{align}[S,\square]=-2\square,\qquad[Z,\square]=0\text{ for other }Z;\end{align} $$

(2.4) $$ \begin{align}[Z,\widetilde{ Z }]=\sum_{|I|=1} C_{Z,\widetilde{ Z },I}Z^I,\qquad\text{where }C_{Z,\widetilde{ Z },I}\text{ are constants};\end{align} $$

(2.5) $$ \begin{align}[Z,\partial_\alpha]=\sum_\beta C_{Z,\alpha\beta}\partial_\beta,\hspace{2em} \text{where }C_{Z,\alpha\beta}\text{ are constants}.\end{align} $$

We now recall several useful estimates involving the commuting vector fields. Their proofs are standard and can be found in, for example, [Reference Sogge67, Reference Hörmander26, Reference Lindblad45]. We start with the following pointwise estimates.

Lemma 2.1. For any function $\phi =\phi (t,x)$ , we have

(2.6) $$ \begin{align}|\partial^k\phi|\leq C\langle t-r \rangle^{-k}\sum_{|I|\leq k}|Z^I\phi|,\qquad\forall k\geq 0;\end{align} $$

(2.7) $$ \begin{align}\sum_{i=1}^3|(\partial_i+\omega_i\partial_t)\phi|\leq C\langle t+r \rangle^{-1}|Z \phi|.\end{align} $$

For each $x\in \mathbb {R}^3$ , we set $r=|x|$ and $\omega =(\omega _i)_{i=1,2,3}=x/|x|$ . For each $s\in \mathbb {R}$ , we define the Japanese bracket $\langle s \rangle :=\sqrt {1+|s|^2}$ . We also define $|Z\phi |:=\sum _{|I|=1}|Z^I\phi |$ .

In addition, we have the Klainerman-Sobolev inequality.

Proposition 2.2. For $\phi \in C^\infty (\mathbb {R}^{1+3})$ which vanishes for large $|x|$ , we have

(2.8) $$ \begin{align}\langle t+|x| \rangle\langle t-|x| \rangle^{1/2}|\phi(t,x)|\leq C\sum_{|I|\leq 2}\left\lVert Z^I\phi(t,\cdot)\right\rVert_{L^2(\mathbb{R}^3)}.\end{align} $$

Moreover, we state a corollary of [Reference Yu74, Lemma 2.6]. It can be viewed as the estimates for Taylor’s series adapted to the commuting vector fields. Note that the original lemma is stronger but unnecessary for this paper.

Lemma 2.3. Fix two positive integer $N,l_0$ . Assume that $\phi =\phi (t,x)$ is an $\mathbb {R}^k$ -valued function satisfying $\sum _{|J|\leq N}|Z^J\phi |\leq 1$ at a certain point $(t_0,x_0)$ . Also assume that $f\in C^\infty (\mathbb {R}^k)$ and that $f(s)=O(|s|^{l_0})$ whenever s is near the origin. Then, at $(t_0,x_0)$ we have

$$ \begin{align*} \sum_{|J|\leq N}|Z^J(f(\phi))|\lesssim_{f,N} \sum_{|J|\leq N}|Z^J\phi|^{l_0}. \end{align*} $$

The implicit constant is independent of the choice of $(t_0,x_0)$ .

We also recall [Reference Yu74, Lemma 2.1] on the commutator $[\partial _t-\partial _r,Z^I]$ .

Lemma 2.4. For each multiindex I, we have

$$ \begin{align*} \ [\partial_t-\partial_r,Z^I]F&=\sum_{|J|<|I|}\left( f_0\cdot Z^J(F_t-F_r)+\sum_{i=1,2,3}f_0\cdot (\partial_i+\omega_i\partial_t)Z^JF \right). \end{align*} $$

Here, $f_0$ denotes an arbitrary finite sum of terms of the form $C\cdot \prod _{j=1}^p Z^{I_j}\omega _{i_j}$ with $i_*\in \{1,2,3\}$ , and we allow $f_0$ to vary from line to line.

2.4 A function space

To simplify our computations, we make the following definition. Fix a small parameter $\delta \in (0,1)$ . For each $\varepsilon \in (0,1)$ , we suppose that there is a nonempty open set $\mathcal {D}_\varepsilon \subset \{(t,x):\ t\geq e^{\delta /\varepsilon },\ |x|/t\in [1/2,2]\}$ .

Definition 2.5. Fix $n,m,p\in \mathbb {R}$ . Suppose that we have a function $F=F(t,x;\varepsilon )$ that is defined for all $(t,x)\in \mathcal {D}_\varepsilon $ as long as $\varepsilon \ll 1$ . We say that $F\in \varepsilon ^nS^{m,p}_{\mathcal {D}}$ if for each integer $N\geq 1$ , there exists an $\varepsilon _N\in (0,1)$ depending on $\{\mathcal {D}_*\},\delta ,N$ , such that for all $\varepsilon \in (0,\varepsilon _N)$ , we have $F(\cdot ;\varepsilon )\in C^N(\mathcal {D}_\varepsilon )$ and

(2.9) $$ \begin{align}\begin{aligned} \sum_{|I|\leq N}|Z^IF(t,x;\varepsilon)|\lesssim_N \varepsilon^nt^{m+C_N\varepsilon}\langle r-t \rangle^{p},\qquad \forall (t,x)\in\mathcal{D}_\varepsilon. \end{aligned}\end{align} $$

The constants in (2.9) depend on $\{\mathcal {D}_*\}$ , N, and $\delta $ , but they are independent of $\varepsilon \in (0,\varepsilon _N)$ .

Moreover, we say that $F\in \varepsilon ^nS^{m,p}_{\ln ,\mathcal {D}}$ if we add a factor $(1+\ln \langle r-t \rangle )$ on the right-hand side of (2.9). Such a definition is useful in the proof of Proposition 3.3.

Finally, we will omit the subscript $\mathcal {D}$ for simplicity if it is clear what $\mathcal {D}_\varepsilon $ is according to the context. That is, we can write $\varepsilon ^n S^{m,p}$ and $\varepsilon ^n S^{m,p}_{\ln }$ . If we have $n=0$ , then we will also omit $\varepsilon ^0$ and write $S^{m,p}$ and $S^{m,p}_{\ln }$ .

We summarize some useful properties in the next lemma. This lemma is essentially the same as [Reference Yu74, Lemma 2.8], so we omit the proof here.

We also present a key lemma that will be applied repeatedly in this paper.

Lemma 2.7. Let $\rho =\rho (t,x;\varepsilon )\in S^{0,1}$ such that $\langle r-t \rangle \lesssim \langle \rho (t,x;\varepsilon ) \rangle t^{C\varepsilon }$ in $\mathcal {D}_\varepsilon $ for $\varepsilon \ll 1$ . Let $\widetilde { \mathcal {D} }_\varepsilon \subset [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ be an open set such that

$$ \begin{align*} \widetilde{ \mathcal{D} }_\varepsilon\supset\{(\varepsilon\ln t-\delta,\rho(t,x;\varepsilon),x/|x|):\ (t,x)\in\mathcal{D}_\varepsilon\}. \end{align*} $$

Suppose that $G=G(s,q,\omega ;\varepsilon )$ is a function such that for each integer $N\geq 0$ , as long as $\varepsilon \ll _N1$ , we have $G(\cdot ;\varepsilon )\in C^N(\widetilde { D }_\varepsilon )$ and

(2.10) $$ \begin{align}\begin{aligned} \sum_{a+b+c\leq N}\langle q \rangle^{a}\varepsilon^{b}|\partial_q^a\partial_s^b\partial_\omega^cG(s,q,\omega;\varepsilon)|\lesssim_N \varepsilon^n\langle q \rangle^{p}e^{C_Ns+m\cdot \frac{s+\delta}{\varepsilon}},\qquad (s,q,\omega)\in\widetilde{ \mathcal{D} }_\varepsilon. \end{aligned}\end{align} $$

Then we have $G(\varepsilon \ln t-\delta ,\rho (t,x;\varepsilon ),x/|x|;\varepsilon )\in \varepsilon ^nS^{m,p}$ .

Proof. We first notice that

$$ \begin{align*} \langle r-t \rangle t^{-C\varepsilon}\lesssim \langle \rho \rangle\lesssim t^{C\varepsilon}\langle r-t \rangle,\qquad \forall (t,x)\in\mathcal{D}\cap\{t\geq e^{\delta/\varepsilon}\}. \end{align*} $$

The first half is in the assumption of this lemma, and the second half follows from $\rho \in S^{0,1}$ .

Fix an integer $N\geq 0$ . Choose $\varepsilon \ll _N1$ so that $G(s,q,\omega ;\varepsilon )\in C^{N}(\widetilde { \mathcal {D} }_\varepsilon )$ , $\rho (t,x;\varepsilon )\in C^N(\mathcal {D}_\varepsilon )$ , and that both (2.9) (with F replaced by $\rho $ and $(n,m,p)=(0,0,1)$ ) and (2.10) hold for this N. We need to check (2.9) with F replaced by $G(t,x;\varepsilon ):=G(\varepsilon \ln t-\delta ,\rho (t,x;\varepsilon ),x/|x|;\varepsilon )$ and $m=0$ . Fix a multiindex I with $|I|\leq N$ . If $|I|=0$ , then (2.10) implies that

$$ \begin{align*} |G|\lesssim \varepsilon^n\langle \rho \rangle^p t^{m+C\varepsilon}\lesssim \varepsilon^n\langle r-t \rangle^p t^{m+C\varepsilon}\qquad \forall (t,x)\in \mathcal{D}_\varepsilon, \end{align*} $$

so we are done. If $|I|>0$ , we write $Z^I(G(t,x;\varepsilon ))$ as a linear combination (with real constant coefficients independent of $\varepsilon $ ) of terms of the form

$$ \begin{align*} \partial_q^a\partial_s^b\partial_\omega^cG(\varepsilon\ln t-\delta,\rho(t,x;\varepsilon),x/|x|;\varepsilon)\cdot\prod_{j=1}^bZ^{I_j}(\varepsilon\ln t-\delta)\cdot\prod_{j=1}^aZ^{J_j}\rho\cdot\prod_{j=1}^cZ^{K_j}\omega. \end{align*} $$

Here, $a+b+c\leq N$ , each of $I_*,J_*,K_*$ is nonzero, and the sum of these multiindices is $|I|$ . Since $|I_*|>0$ , we have $Z^{I_*}(\varepsilon \ln t-\delta )=O(\varepsilon )$ . Since $\rho \in S^{0,1}$ and $\omega \in S^{0,0}$ , we have $Z^{J_*}\rho =O(t^{C\varepsilon }\langle r-t \rangle ) $ and $Z^{K_*}\omega =O(1)$ (we can show that there is no $t^{C\varepsilon }$ factor here). We also apply (2.10) to control the derivatives of G. In summary, each term of the form above is controlled by

$$ \begin{align*} \varepsilon^{n-b}\langle \rho \rangle^{p-a}t^{m+C\varepsilon}\cdot \varepsilon^b\cdot \langle r-t \rangle^{a}t^{C\varepsilon}\cdot 1\lesssim\varepsilon^n\langle r-t \rangle^{p}t^{m+C\varepsilon}. \end{align*} $$

Thus, we have $Z^IG=O(\varepsilon ^n\langle r-t \rangle ^{p}t^{m+C\varepsilon })$ in $\mathcal {D}_\varepsilon $ .

3.1 Global existence

Fix $u_0,u_1\in C_c^\infty (\mathbb {R}^3)$ . Choose $R>0$ such that $\operatorname {\mathrm {supp}}(u_0,u_1)\subset \{|x|\leq R\}$ , and we remark that the meaning of R will remain unchanged in the rest of this paper. For $\varepsilon \ll 1$ , the Cauchy problem (1.1) and (1.2) admits a unique global $C^\infty $ solution $u=u(t,x;\varepsilon )$ for all $t\geq 0$ . Sometimes we omit $\varepsilon $ and write $u(t,x)$ for simplicity. Here, we first apply the global existence result for (1.1) in Lindblad [Reference Lindblad45] to get a global $C^{14}$ solution. Then, we apply [Reference Sogge67, Theorems I.4.1 and I.4.2] to show that this $C^{14}$ solution is indeed the unique $C^\infty $ solution.

Moreover, for each nonnegative integer N, we have the following pointwise bounds as long as $\varepsilon \ll _N1$ :

(3.1) $$ \begin{align}\begin{aligned} \sum_{|I|\leq N}|Z^Iu(t,x;\varepsilon)|\lesssim_N \varepsilon \langle t \rangle^{-1+C_N\varepsilon},\qquad \forall t\geq 0,\ x\in\mathbb{R}^3. \end{aligned}\end{align} $$

There is a better bound for $\partial u$ : $|\partial u|\lesssim \varepsilon \langle t \rangle ^{-1}$ . By the finite speed of propagation, we also have $u\equiv 0$ for $r-t\geq R$ .

3.2 The optical function

Fix a small parameter $\delta \in (0,1)$ . Define

(3.2) $$ \begin{align}\begin{aligned} \Omega=\Omega_{\delta,R,\varepsilon}:=\{(t,x)\in(e^{\delta/\varepsilon},\infty)\times\mathbb{R}^3:\ |x|>\frac{1}{2}(t+e^{\delta/\varepsilon})+2R\}. \end{aligned}\end{align} $$

Heuristically, one can view $\Omega $ as $\{t>e^{\delta /\varepsilon },\ |x|>t/2\}$ . Here, we use (3.2) for technical reasons; see the discussion below (3.3). From now on, if we use the notation $\varepsilon ^nS^{m,p}$ or $\varepsilon ^nS^{m,p}_{\ln }$ , we always take $\mathcal {D}_{\varepsilon }=\Omega _{\delta ,R,\varepsilon }\cap \{r-t<2R\}$ in Definition 2.5. Thus, by (3.1), we have $u\in \varepsilon S^{-1,0}$ .

In [Reference Yu75, Propositions 3.1 and 4.1], we applied the method of characteristics and proved that the eikonal equation

(3.3) $$ \begin{align}\begin{aligned} g^{\alpha\beta}(u)\partial_\alpha q\partial_\beta q=0\quad\text{in }\Omega;\qquad q=|x|-t\quad \text{on }\partial\Omega \end{aligned}\end{align} $$

admits a unique global $C^2$ solution $q=q(t,x;\varepsilon )$ in $\Omega $ . Because $\partial \Omega $ is not a smooth surface, it seems that we need to carefully choose the boundary condition near the ‘corner’ of $\partial \Omega $

$$ \begin{align*} \{t=e^{\delta/\varepsilon},|x|=e^{\delta/\varepsilon}+2R\}\subset\partial\Omega \end{align*} $$

to make q a $C^2$ solution. However, it is not necessary here because of our choice (3.2) of $\Omega $ . We have $u|_{r-t\geq R}\equiv 0$ and $q|_{r-t\geq R}=r-t$ , so we only need to consider those characteristics emanating from a smooth subset of $\partial \Omega $ (i.e., $\partial \Omega \cap \{t>e^{\delta /\varepsilon }\}$ ). Besides, we have $q\in S^{0,1}$ and $\langle r-t \rangle \lesssim \langle q \rangle t^{C\varepsilon }$ in $\Omega $ , so we can take $\rho =q$ in Lemma 2.7. To see this, we apply [Reference Yu75, Lemma 3.8] to obtain $|q-(r-t)|\lesssim t^{C\varepsilon }$ and $t^{-C\varepsilon }\lesssim \langle q \rangle /\langle r-t \rangle \lesssim t^{C\varepsilon }$ in $\Omega $ . Next, we apply [Reference Yu75, Proposition 4.1] to conclude that for each fixed integer $N\geq 2$ , the $C^2$ solution q is $C^N$ as long as $\varepsilon \ll _N1$ . We also have $\sum _{|I|\leq N}|Z^Iq|\lesssim \langle q \rangle t^{C\varepsilon }\lesssim \langle r-t \rangle t^{C\varepsilon }$ in $\Omega $ .

Next, if we set

(3.4) $$ \begin{align}\begin{aligned} \Omega'&:=\{(s,q,\omega)\in[0,\infty)\times\mathbb{R}\times\mathbb{S}^2:\ q>\frac{1}{2}(1-e^{s/\varepsilon})e^{\delta/\varepsilon}+2R\}, \end{aligned}\end{align} $$

then the map

$$ \begin{align*} \Omega\to\Omega':\ (t,x)\mapsto (\varepsilon\ln t-\delta,q(t,x;\varepsilon),x/|x|) \end{align*} $$

is bijective and has an inverse map. As a result, any function defined for all $(t,x)\in \Omega $ induces a function of $(s,q,\omega )$ . Again, for each fixed integer $N\geq 2$ , both the map $\Omega \to \Omega '$ and its inverse are $C^N$ as long as $\varepsilon \ll _N1$ .

3.3 The geometric reduced system

In $\Omega $ , we set $(\mu ,U)(t,x;\varepsilon ):=(q_t-q_r,\varepsilon ^{-1}ru)(t,x;\varepsilon )$ and obtain an induced function $(\mu ,U)(s,q,\omega ;\varepsilon )$ in $\Omega '$ . In [Reference Yu75, Section 5], we proved that $(\mu ,U)$ is an approximate solution to the geometric reduced system (1.14). In addition, the following three limits exist for all $(q,\omega )\in \mathbb {R}\times \mathbb {S}^2$ :

(3.5) $$ \begin{align}\begin{aligned} \left\{\begin{array}{l} \displaystyle A(q,\omega;\varepsilon):=-\frac{1}{2}\lim_{s\to\infty}(\mu U_q)(s,q,\omega;\varepsilon),\\ \displaystyle A_1(q,\omega;\varepsilon):=\lim_{s\to\infty}\exp(\frac{1}{2}G(\omega)A(q,\omega;\varepsilon)s)\mu(s,q,\omega;\varepsilon),\\ \displaystyle A_2(q,\omega;\varepsilon):=\lim_{s\to\infty}\exp(-\frac{1}{2}G(\omega)A(q,\omega;\varepsilon)s)U_q(s,q,\omega;\varepsilon). \end{array}\right. \end{aligned}\end{align} $$

Note that $A_1A_2\equiv -2A$ . Since $u\equiv 0$ whenever $r\geq t+R$ , we have $(A,A_2)\equiv 0$ and $A_1\equiv -2$ for $q\geq R$ . By setting

(3.6) $$ \begin{align}\begin{aligned} \left\{\begin{array}{l} \displaystyle\widetilde{ \mu }(s,q,\omega;\varepsilon):=A_1\exp(-\frac{1}{2}G(\omega)As),\\[1em] \displaystyle \widetilde{ U }_q(s,q,\omega;\varepsilon):=A_2\exp(\frac{1}{2}G(\omega)As),\\[1em]\displaystyle \lim_{q\to\infty}\widetilde{ U }(s,q,\omega;\varepsilon)=0, \end{array}\right. \end{aligned}\end{align} $$

we obtain an exact solution $(\widetilde { \mu },\widetilde { U })$ to the reduced system (1.14). We refer our readers to [Reference Yu75, Proposition 5.1] for more details. Since $A,A_1,A_2$ are defined for all $(q,\omega )\in \mathbb {R}\times \mathbb {S}^2$ , we emphasize that $(\widetilde { \mu },\widetilde { U })$ is defined for all $(s,q,\omega )\in \mathbb {R}\times \mathbb {R}\times \mathbb {S}^2$ . However, $(\mu ,U)$ is only defined in $\Omega '$ .

In [Reference Yu75, Proposition 5.1], we also proved several bounds for $A,A_1,A_2$ in the coordinate set $(q,\omega )$ :

(3.7) $$ \begin{align}\begin{aligned} \partial_q^a\partial_\omega^c(A,A_2)=O_{a,c}(\langle q \rangle^{-1-a+C_{a,c}\varepsilon}),\quad \partial_q^a\partial_\omega^cA_1=O_{a,c}(\langle q \rangle^{-a+C_{a,c}\varepsilon}),\qquad \forall a,c\geq 0. \end{aligned}\end{align} $$

Here, we remind our readers of the convention introduced in Section 2.2. First, the estimates (3.7) should be understood in the following way: for each integer $N\geq 0$ , as long as $\varepsilon \ll _N1$ , the functions $A,A_1,A_2$ are $C^N$ and satisfy (3.7) for all $a+c\leq N$ . Moreover, we emphasize that the constants in (3.7) can be chosen to be uniform for all $\varepsilon \ll _N1$ .

In [Reference Yu75, Lemma 5.8], we proved that

(3.8) $$ \begin{align}\begin{aligned} |A_1+2|\leq \langle q \rangle^{-1+C\varepsilon},\qquad -3\leq A_1\leq -1<0. \end{aligned}\end{align} $$

Finally, we recall from [Reference Yu75, Proposition 5.1, (5.11), and (5.12)] that

(3.9) $$ \begin{align}\begin{aligned} \partial_q^a\partial_s^b\partial_\omega^c(\mu-\widetilde{ \mu })&=O_{a,b,c}(\varepsilon^{-b}\langle q \rangle^{-a}t^{-1+C_{a,b,c}\varepsilon}),\qquad &\forall a,b,c\geq 0,\ (s,q,\omega)\in\Omega';\\ \partial_q^a\partial_s^b\partial_\omega^c(U-\widetilde{ U })&=O_{a,b,c}(\varepsilon^{-b}\langle q \rangle^{1-a}t^{-1+C_{a,b,c}\varepsilon}),\qquad& \forall a,b,c\geq 0,\ (s,q,\omega)\in\Omega'; \end{aligned}\end{align} $$

(3.10) $$ \begin{align}\begin{aligned} \partial_q^a\partial_s^b\partial_\omega^c(\widetilde{ \mu },\widetilde{ U })&=O_{a,b,c}(\varepsilon^{-b}\langle q \rangle^{-a}t^{C_{a,b,c}\varepsilon}),\qquad &\forall a,b,c\geq 0,\ (s,q,\omega)\in\Omega'. \end{aligned}\end{align} $$

All the functions above are defined for $(s,q,\omega )\in \Omega '\subset [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ where $\Omega '$ is defined by (3.4). Besides, we set $t=e^{\frac {s+\delta }{\varepsilon }}$ . Again, all the estimates above should be understood in the following way: for each integer $N\geq 0$ , we can choose $\varepsilon \ll _N1$ so that these estimates hold for all $a+b+c\leq N$ .

As discussed in Section 3.2, every function of $(s,q,\omega )$ induces a function of $(t,x)$ via the map $(t,x)\mapsto (\varepsilon \ln t-\delta ,q(t,x;\varepsilon ),x/|x|)$ . Below (3.3), we mentioned that $q\in S^{0,1}$ and $\langle r-t \rangle \lesssim \langle q \rangle t^{C\varepsilon }$ . Thus, we can apply Lemma 2.7 with $\rho =q$ to obtain the following lemma.

Lemma 3.1. We have $u-\varepsilon r^{-1}\widetilde { U }\in \varepsilon S^{-2,1}$ , and $\partial _\alpha u-\varepsilon \widehat { \omega }_\alpha r^{-1}A\in \varepsilon S^{-2,0}$ for $\alpha =0,1,2,3$ . Here, $\widehat { \omega }_0=-1$ and $\widehat { \omega }_j=\omega _j$ for $j=1,2,3$ . That is, for each fixed integer $N\geq 0$ , as long as $\varepsilon \ll _N1$ , we have

$$ \begin{align*} \sum_{|I|\leq N}|Z^I(u(t,x)-\varepsilon r^{-1}\widetilde{ U }(\varepsilon\ln t-\delta,q(t,x),x/|x|))|&\lesssim \varepsilon t^{-2+C\varepsilon}\langle r-t \rangle,\\ \sum_{|I|\leq N}|Z^I(\partial_\alpha u(t,x)-\varepsilon \widehat{ \omega }_\alpha r^{-1}A(q(t,x),x/|x|))|&\lesssim \varepsilon t^{-2+C\varepsilon} \end{align*} $$

for all $(t,x)\in \Omega $ . Note that the estimates hold trivially when $r-t\geq R$ since $u,\widetilde { U },A$ vanish there.

Proof. We apply Lemma 2.7 with $\rho =q$ , $G=U-\widetilde { U }$ , and $(n,m,p)=(0,-1,1)$ . Thus, we have $U-\widetilde { U }\in \ S^{-1,1}$ . By Lemma 2.6, we have $\varepsilon r^{-1}\in \varepsilon S^{-1,0}$ and thus $u-\varepsilon r^{-1}\widetilde { U }=\varepsilon r^{-1}(U-\widetilde { U })\in \varepsilon S^{-1,0}\cdot S^{-1,1}\subset \varepsilon S^{-2,1}$ .

Next, by part (d) of Lemma 2.6, we have $\partial _{\alpha }(u-\varepsilon r^{-1}\widetilde { U })\in \varepsilon S^{-2,0}$ . It suffices to prove that $\partial _{\alpha }(\varepsilon r^{-1}\widetilde { U })-\varepsilon \widehat { \omega }_\alpha r^{-1}A\in \varepsilon S^{-2,0}$ . By the chain rule and since $A=-\frac {1}{2}\widetilde { \mu }\widetilde { U }_q$ , we have

$$ \begin{align*} &\partial_\alpha(\varepsilon r^{-1}\widetilde{ U })-\varepsilon r^{-1}\widehat{ \omega }_\alpha A\\& \quad =\varepsilon r^{-1}(\widetilde{ U }_qq_\alpha-\widehat{ \omega }_\alpha A)-\varepsilon r^{-2}(\partial r)\cdot \widetilde{ U }+\varepsilon r^{-1}\widetilde{ U }_s\cdot\partial(\varepsilon\ln t)+\varepsilon r^{-1}\widetilde{ U }_{\omega}\cdot\partial\omega\\& \quad =\varepsilon r^{-1}(q_\alpha+\frac{1}{2}\widehat{ \omega }_\alpha \widetilde{ \mu })\widetilde{ U }_q+\varepsilon S^{-2,0}. \end{align*} $$

To obtain the last estimate, we first apply (3.10) and Lemma 2.7 to obtain $\widetilde { U },\widetilde { U }_\omega \in S^{0,0}$ and $\widetilde { U }_s\in \varepsilon ^{-1}S^{0,0}$ . By Lemma 2.6, we have $\varepsilon r^{-2}\partial r,\varepsilon r^{-1}\partial \omega \in \varepsilon S^{-2,0}$ and $\varepsilon r^{-1}\partial (\varepsilon \ln t)\in \varepsilon ^2S^{-2,0}$ .

Finally, we need to show $\varepsilon r^{-1}(q_\alpha +\frac {1}{2}\widehat { \omega }_\alpha \widetilde { \mu })\widetilde { U }_q\in \varepsilon S^{2,0}$ . We have $\varepsilon r^{-1}\in \varepsilon S^{-1,0}$ and $\widetilde { U }_q\in S^{0,-1}$ by (3.10) and Lemma 2.7. Moreover, we have

$$ \begin{align*} q_t&=\frac{1}{2}\widetilde{ \mu }+\frac{1}{2}(\mu-\widetilde{ \mu })+\frac{1}{2}(q_t+q_r),\\ q_i&=-\frac{1}{2}\widetilde{ \mu }\omega_i-\frac{1}{2}(\mu-\widetilde{ \mu })\omega_i+(q_i-\omega_iq_r). \end{align*} $$

By (3.9) and Lemma 2.7, we have $\mu -\widetilde { \mu }\in S^{-1,0}$ . Since $q\in S^{0,1}$ , we apply part (d) of Lemma 2.6 to obtain $q_t+q_r,q_i-\omega _iq_r\in S^{-1,1}$ . As a result, we have $q_\alpha +\frac {1}{2}\widetilde { \mu }\widehat { \omega }_\alpha \in S^{-1,1}$ . In summary, we have

$$ \begin{align*} \varepsilon r^{-1}(q_\alpha+\frac{1}{2}\widehat{ \omega }_\alpha \widetilde{ \mu })\widetilde{ U }_q\in \varepsilon S^{-1,0}\cdot S^{-1,1}\cdot S^{0,-1}\subset \varepsilon S^{-2,0}.\\[-41pt] \end{align*} $$

3.4 Approximation

In [Reference Yu75, Section 7], we used the limits in (3.5) to generate an approximate solution to (1.1) that is sufficiently close to the exact solution u in some sense. The construction is as follows. Recall from (3.6) that we have an exact solution $(\widetilde { \mu },\widetilde { U })(s,q,\omega ;\varepsilon )$ to the geometric reduced system (1.14). Now, we define a new function $\widetilde { q }=\widetilde { q }(t,x;\varepsilon )$ in $\Omega $ by solving

(3.11) $$ \begin{align}\begin{aligned} \widetilde{ q }_t-\widetilde{ q }_r=\widetilde{ \mu }(\varepsilon\ln t-\delta,\widetilde{ q }(t,x;\varepsilon),x/|x|;\varepsilon)\quad\text{in }\Omega;\qquad \widetilde{ q }=|x|-t\text{ for }|x|-t\geq 2R. \end{aligned}\end{align} $$

We then define

(3.12) $$ \begin{align}\begin{aligned} \widetilde{ u }(t,x;\varepsilon)&=\varepsilon |x|^{-1}\widetilde{ U }(\varepsilon\ln t-\delta,\widetilde{ q }(t,x;\varepsilon),x/|x|;\varepsilon),\qquad \forall (t,x)\in\Omega. \end{aligned}\end{align} $$

Once again, both $\widetilde { q }$ and $\widetilde { u }$ are $C^N$ as long as $\varepsilon \ll _N1$ . In [Reference Yu75, Proposition 7.1 and Lemma 7.7], we proved that this $\widetilde { u }$ is an approximate solution to (1.1), that for each integer $N\geq 0$ , as long as $\varepsilon \ll _N1$ ,

(3.13) $$ \begin{align}\begin{aligned} \sum_{|I|\leq N}|Z^I\widetilde{ u }(t,x;\varepsilon)|\lesssim \varepsilon t^{-1+C\varepsilon},\qquad \forall (t,x)\in\Omega, \end{aligned}\end{align} $$

and that for each $\gamma \in (0,1)$ and each integer $N\geq 0$ , as long as $\varepsilon \ll _{\gamma ,N}1$ ,

(3.14) $$ \begin{align}\begin{aligned} \sum_{|I|\leq N}|Z^I(u-\widetilde{ u })(t,x;\varepsilon)|\lesssim_{\gamma,N}\varepsilon t^{-2+C_N\varepsilon}\langle r-t \rangle,\qquad \forall (t,x)\in\Omega\cap\{|r-t|\lesssim t^\gamma\}. \end{aligned}\end{align} $$

We ask our readers to compare (3.14) with Lemma 3.1. First, the estimates (3.14) hold in $\Omega \cap \{|r-t|\lesssim t^\gamma \}$ , and the implicit constants depend on $\gamma $ . From this point of view, the estimates in Lemma 3.1 are better since they hold in $\Omega $ . However, we can construct $\widetilde { u }$ from the functions $A,A_1,A_2$ without knowing what the exact solution u to (1.1) and the exact optical function q are. Recall that we construct an approximate optical function $\widetilde { q }$ by solving a transport equation which does not depend on u and q. However, to apply Lemma 3.1, we need the exact optical function q which is determined by the eikonal equation (3.3) involving u.

Here, we recall an alternative way to define $\widetilde { u }$ from [Reference Yu75, Section 7.2] which will simplify our computations. DefineFootnote ¹³

(3.15) $$ \begin{align}\begin{aligned} F(q,\omega;\varepsilon):=2R-\int_{2R}^q \frac{2}{A_1(p,\omega;\varepsilon)}\ dp,\qquad\forall (q,\omega)\in\mathbb{R}\times\mathbb{S}^2. \end{aligned}\end{align} $$

Recall that $A_1\in [-3,-1]$ everywhere, so F is well defined. In [Reference Yu75, Section 7.2], we proved that $\langle F(q,\omega ;\varepsilon ) \rangle \sim \langle q \rangle $ for all $(q,\omega )\in \mathbb {R}\times \mathbb {S}^2$ . For each fixed $\omega \in \mathbb {S}^2$ , the map $q\mapsto F(q,\omega ;\varepsilon )$ has an inverse $\widehat { F }(q,\omega ;\varepsilon )$ . Set

(3.16) $$ \begin{align}\begin{aligned} \left\{\begin{array}{l} \displaystyle \widehat{ A }(q,\omega;\varepsilon):=A(\widehat{ F }(q,\omega;\varepsilon),\omega;\varepsilon),\\ \displaystyle \widehat{ \mu }(s,q,\omega;\varepsilon):=-2\exp(-\frac{1}{2}G(\omega)\widehat{ A }(q,\omega;\varepsilon)s),\\ \displaystyle \widehat{ U }(s,q,\omega;\varepsilon):=-\int_q^\infty \widehat{ A }(p,\omega;\varepsilon)\exp(\frac{1}{2}G(\omega)\widehat{ A }(p,\omega;\varepsilon)s) \ dp.\end{array}\right. \end{aligned}\end{align} $$

This $\widehat { A }$ is called the scattering data for the asymptotic completeness problem. Note that $(\widehat { \mu },\widehat { U })$ is again a solution to the geometric reduced system (1.14).

We summarize several estimates for $\widehat { A },\widehat { \mu },\widehat { U }$ in the next lemma. We again remind our readers of the convention in Section 2.2.

Lemma 3.2. For all $a,c\geq 0$ , we have

(3.17) $$ \begin{align}\begin{aligned} \partial_q^a\partial_\omega^c\widehat{ A }=O_{a,c}(\langle q \rangle^{-1-a+C_{a,c}\varepsilon}),\qquad \forall (q,\omega)\in\mathbb{R}\times\mathbb{S}^2. \end{aligned}\end{align} $$

For all $a,b,c\geq 0$ , we have

(3.18) $$ \begin{align}\begin{aligned} \partial_q^a\partial_s^b\partial_\omega^c(\widehat{ \mu }+2)&=O_{a,b,c}(\langle q \rangle^{-\max\{1,b\}-a+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s});\\ \partial_q^a\partial_s^b \partial_\omega^c\widehat{ U }_q&=O_{a,b,c}(\langle q \rangle^{-1-a-b+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s});\\ \partial_s^b \partial_\omega^c\widehat{ U }&=O_{b,c}((\varepsilon^{-1}\langle q \rangle^{C_{b,c}\varepsilon}1_{b=0}+1_{b>0})e^{C_{b,c}s}) \end{aligned}\end{align} $$

for all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ . Finally, for all $c\geq 0$ , $s\geq 0$ , $\omega \in \mathbb {S}^2$ , and $q>C_0e^{C_0s}(e^{\delta /\varepsilon }-e^{(\delta +s)/\varepsilon }-1)$ for some constant $C_0>1$ , we have

(3.19) $$ \begin{align}\begin{aligned} \partial_\omega^c\widehat{ U }&=O_{c,C_0}(e^{C_{c,C_0}s}). \end{aligned}\end{align} $$

Proof. The estimates (3.17) have been proved in [Reference Yu75, Proposition 7.3]. We now prove (3.18) for all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ . Since $|e^x-1|\leq |x|e^{|x|}$ , we have $|\widehat { \mu }+2|\lesssim |G(\omega )\widehat { A }s|e^{Cs}\lesssim \langle q \rangle ^{-1+C\varepsilon }e^{Cs}$ . Next, for $a+c>0$ , we write $\partial _q^a\partial _\omega ^c\widehat { \mu }$ as a linear combination (with real constant coefficients depending only on a and c) of terms of the form

$$ \begin{align*} \exp(-\frac{1}{2}G \widehat{ A }s)\cdot\prod_{j=1}^k \partial_q^{a_j}\partial_\omega^{c_j}(-\frac{1}{2}G \widehat{ A }s),\qquad \sum a_*=a,\ \sum c_*=c,\ a_j+c_j>0\text{ for each }j. \end{align*} $$

The estimates for $\widehat { A }$ imply that

$$ \begin{align*} |\exp(-\frac{1}{2}G \widehat{ A }s)\cdot\prod_{j=1}^k \partial_q^{a_j}\partial_\omega^{c_j}(-\frac{1}{2}G \widehat{ A }s)|\lesssim e^{Cs}\cdot \langle q \rangle^{-1-\sum a_j+C\varepsilon}s^k\lesssim \langle q \rangle^{-1-a+C\varepsilon}e^{Cs}. \end{align*} $$

In summary, we have $|\partial _q^a\partial _\omega ^c\widehat { \mu }|\lesssim _{a,c} \langle q \rangle ^{-1-a+C_{a,c}\varepsilon }e^{C_{a,c}s}$ . That is, we obtain the first estimate in (3.18) with $b=0$ .

Now, by (3.16), we notice that

(3.20) $$ \begin{align}\begin{aligned} \partial_q^a\partial_s^b\partial_\omega^c\widehat{ U }_q&=\partial_q^a\partial_\omega^c((\frac{1}{2}G\widehat{ A })^b\widehat{ A }\cdot\exp(\frac{1}{2}G\widehat{ A }s)),\qquad a,b,c\geq 0;\\ \partial_q^a\partial_s^b\partial_\omega^c\widehat{ \mu }&=\partial_q^a\partial_\omega^c((-\frac{1}{2}G\widehat{ A })^b\cdot\exp(-\frac{1}{2}G\widehat{ A }s)),\qquad a,c\geq 0,b>0. \end{aligned}\end{align} $$

Both of these rows are of the form

$$ \begin{align*} \partial_q^a\partial_\omega^c[F_1(q,\omega)\cdot \exp(F_2(q,\omega)s)]. \end{align*} $$

Here, for all $a,c\geq 0$ and $j=1,2$ , we have $\partial _q^a\partial _\omega ^cF_j=O_{a,c}(\langle q \rangle ^{-b_j+C_{a,c}\varepsilon })$ for some fixed constants $b_1>0$ and $b_2=1$ . Following the same proof in the previous paragraph for $\widehat { \mu }$ , we obtain that for all $a,c\geq 0$ and $(s,q,\omega )$ ,

$$ \begin{align*} \partial_q^a\partial_\omega^c(\exp(F_2(q,\omega)s)-1)=O_{a,c}(\langle q \rangle^{-1-a+C_{a,c}\varepsilon}e^{C_{a,c}s}). \end{align*} $$

Thus, by Leibniz’s rule, for all $a,c\geq 0$ and $(s,q,\omega )$ , we have

$$ \begin{align*} &\partial_q^a\partial_\omega^c[F_1(q,\omega)\cdot \exp(F_2(q,\omega)s)]=\partial_q^a\partial_\omega^cF_1+\partial_q^a\partial_\omega^c[F_1 \cdot (\exp(F_2 s)-1)]\\&\quad =O_{a,c}(\langle q \rangle^{-b_1-a+C_{a,c}\varepsilon}e^{C_{a,c}s}+\langle q \rangle^{-b_1-1-a+C_{a,c}\varepsilon}e^{C_{a,c}s})=O_{a,c}(\langle q \rangle^{-b_1-a+C_{a,c}\varepsilon}e^{C_{a,c}s}). \end{align*} $$

To estimate the derivatives of $\widehat { U }_q$ , we use (3.20) to set $F_1=(\frac {1}{2}G\widehat { A })^b\widehat { A }$ and $F_2=\frac {1}{2}G\widehat { A }$ . In this case, we have $b_1=b+1$ , so we obtain the second row of (3.18). To estimate the derivatives of $\widehat { \mu }$ , we use (3.20) to set $F_1=(\frac {1}{2}G\widehat { A })^b$ and $F_2=-\frac {1}{2}G\widehat { A }$ . In this case, we have $b_1=b$ , so we obtain the first row of (3.18) with $b>0$ .

To end the proof of (3.18), we prove the bounds for $\partial _s^b\partial _\omega ^c\widehat { U }$ . Note that $\widehat { U }\equiv 0$ for $q\geq R$ and that for $q<R$ ,

$$ \begin{align*} |\partial_s^b\partial_\omega^c\widehat{ U }(s,q,\omega)|&\lesssim \int_{q}^R \langle \rho \rangle^{-1-b+C\varepsilon}e^{Cs}\ d\rho. \end{align*} $$

When $b=0$ , the right side is $O(\varepsilon ^{-1}\langle q \rangle ^{C\varepsilon }e^{Cs})$ . If $b\geq 1$ , the right side is $O(e^{Cs})$ .

Finally, we check (3.19). Recall from [Reference Yu75, Proposition 7.3] that $\partial _\omega ^c\widehat { U }=O(e^{Cs})$ for $(s,q,\omega )\in \Omega '$ . That is, $(s,\omega )\in [0,\infty )\times \mathbb {S}^2$ and $q>q_{(s,\varepsilon )}:=\frac {1}{2}(e^{\delta /\varepsilon }-e^{(s+\delta )/\varepsilon })+2R$ . Since $e^{\delta /\varepsilon }-e^{(\delta +s)/\varepsilon }-1=2q_{(s,\varepsilon )}-4R-1<0$ and since $q_{(s,\varepsilon )}\leq 2R$ , we have

$$ \begin{align*} \langle q \rangle/\langle q_{(s,\varepsilon)} \rangle+\langle q_{(s,\varepsilon)} \rangle/\langle q \rangle\lesssim_{R,C_0}e^{C_0s},\qquad \text{whenever } C_0e^{C_0s}(e^{\delta/\varepsilon}-e^{(\delta+s)/\varepsilon}-1)<q\leq q_{(s,\varepsilon)}. \end{align*} $$

Since $\partial _\omega ^c\widehat { U }_q=O(\langle q \rangle ^{-1+C\varepsilon }e^{Cs})$ by (3.18), by the mean value theorem, we have

$$ \begin{align*} \partial_\omega^c\widehat{ U }(s,q,\omega;\varepsilon)&=\partial_\omega^c\widehat{ U }(s,q_{(s,\varepsilon)},\omega;\varepsilon)+O(|q-q_{(s,\varepsilon)}|\cdot \langle q_{(s,\varepsilon)} \rangle^{-1+C\varepsilon}e^{Cs})\\ &=O(e^{Cs})+O(\langle q_{(s,\varepsilon)} \rangle\cdot \langle q_{(s,\varepsilon)} \rangle^{-1+C\varepsilon}\cdot e^{Cs})=O(e^{Cs}) \end{align*} $$

for all $C_0e^{C_0s}(e^{\delta /\varepsilon }-e^{(\delta +s)/\varepsilon }-1)<q\leq q_{(s,\varepsilon )}$ . This finishes the proof.

It follows from [Reference Yu75, Lemma 7.2] that $\widehat { q }(t,x;\varepsilon ):=F(\widetilde { q }(t,x;\varepsilon ),x/|x|;\varepsilon )$ solves

(3.21) $$ \begin{align}\begin{aligned} \widehat{ q }_t-\widehat{ q }_r=\widehat{ \mu }(\varepsilon\ln t-\delta,\widehat{ q }(t,x;\varepsilon),x/|x|;\varepsilon),\qquad\text{in }\Omega, \end{aligned}\end{align} $$

that $\widehat { q }=r-t$ for $r-t\geq R$ , and that

(3.22) $$ \begin{align}\begin{aligned} \widetilde{ u }(t,x;\varepsilon)=\widehat{ u }(t,x;\varepsilon):=\varepsilon |x|^{-1}\widehat{ U }(\varepsilon\ln t-\delta,\widehat{ q }(t,x;\varepsilon),x/|x|;\varepsilon),\qquad\text{in }\Omega. \end{aligned}\end{align} $$

In the rest of this subsection, every function of $(s,q,\omega )$ induces a function of $(t,x)$ via the map $(t,x)\mapsto (\varepsilon \ln t-\delta ,\widehat { q }(t,x;\varepsilon ),x/|x|)$ . Note that this is different from the setting in Section 3.3. By [Reference Yu75, Lemmas 7.4 and 7.7], we have $\widehat { q }\in S^{0,1}$ and $\langle r-t \rangle \lesssim \langle \widehat { q } \rangle t^{C\varepsilon }$ . Thus, we can apply Lemma 2.7 with $\rho =\widehat { q }$ . For convenience, we list some results from (3.18) and (3.19):

(3.23) $$ \begin{align}\begin{aligned} \widehat{ U },\varepsilon\widehat{ U }_s\in S^{0,0};\ \widehat{ u }=\varepsilon r^{-1}\widehat{ U }\in \varepsilon S^{-1,0};\ \widehat{ \mu }+2\in S^{0,-1}; \end{aligned}\end{align} $$

All the functions here are of $(t,x)$ via the map $(t,x)\mapsto (\varepsilon \ln t-\delta ,\widehat { q }(t,x;\varepsilon ),x/|x|)=(s,q,\omega )$ . We will use them in the next subsection. Note that $(t,x)\in \Omega $ does not imply that $(\varepsilon \ln t-\delta ,\widehat { q },x/|x|)\in \Omega '$ . Instead, by [Reference Yu75, Lemma 7.4], we have $\langle \widehat { q } \rangle /\langle r-t \rangle +\langle r-t \rangle /\langle \widehat { q } \rangle \lesssim t^{C\varepsilon }$ in $\Omega $ . That is, in $\Omega \cap \{r-t\leq 2R\}$ , we have $\widehat { q }\leq 2R$ and

$$ \begin{align*} 0<(2R+1-\widehat{ q })\sim\langle \widehat{ q } \rangle\lesssim\langle r-t \rangle t^{C\varepsilon}\lesssim (t-e^{\delta/\varepsilon}+1)t^{C\varepsilon}. \end{align*} $$

We hope that the estimate (3.19) holds for such a $\widehat { q }$ , so we assume $q>C_0e^{C_0s}(e^{\delta /\varepsilon }-e^{(\delta +s)/\varepsilon }-1)$ in Lemma 3.2.

3.5 Further computations

In the next proposition, we present asymptotic formulas for $Z^I\widehat { u }$ and $(\partial _t-\partial _r)Z^I\widehat { u }$ in $\Omega $ . We remind our readers of the definitions of $\varepsilon ^*S^{*,*}$ and $\varepsilon ^*S^{*,*}_{\ln }$ in Definition 2.5.

Proposition 3.3. For each multiindex I, there exist functions $U_I=U_I(s,q,\omega ;\varepsilon )$ and $A_I=A_I(q,\omega ;\varepsilon )$ defined for all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ and $\varepsilon \ll _I1$ , such that

(3.24) $$ \begin{align}\begin{aligned} r Z^I\widehat{ u }&=\varepsilon U_I(\varepsilon\ln t-\delta,|x|-t,x/|x|;\varepsilon)+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\ r(\partial_t-\partial_r)Z^I\widehat{ u }&=\varepsilon A_I(|x|-t,x/|x|;\varepsilon)+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{aligned} \end{align} $$

Here, $U_I$ and $A_I$ are defined inductively on $|I|$ . First, we have $U_0=\widehat { U }$ and $A_0=-2\widehat { A }$ . Moreover, for each $|I|>0$ , by writing $Z^I=ZZ^{I'}$ with $|I'|=|I|-1$ , we define $U_I$ and $A_I$ inductively by

(3.25) $$ \begin{align} U_I(s,q,\omega;\varepsilon)&=\left\{ \begin{array}{ll} \varepsilon\partial_sU_{I'}+q\partial_qU_{I'}-U_{I'}, & Z^I=SZ^{I'}; \\ (\omega_i\partial_{\omega_j}-\omega_j\partial_{\omega_i})U_{I'}, & Z^I=\Omega_{ij}Z^{I'},\ 1\leq i<j\leq 3;\\ \varepsilon\omega_i\partial_sU_{I'}-q\omega_i\partial_qU_{I'}+\partial_{\omega_i}U_{I'}-\omega_iU_{I'}, & Z^I=\Omega_{0i}Z^{I'},\ 1\leq i\leq 3;\\ 0, & Z^I=\partial Z^{I'}; \end{array}\right. \end{align} $$

(3.26) $$ \begin{align}A_I(q,\omega;\varepsilon)&=\left\{ \begin{array}{ll} q\partial_qA_{I'}, & Z^I=SZ^{I'}; \\ (\omega_i\partial_{\omega_j}-\omega_j\partial_{\omega_i})A_{I'}, & Z^I=\Omega_{ij}Z^{I'},\ 1\leq i<j\leq 3;\\ -q\omega_i\partial_qA_{I'}+\partial_{\omega_i}A_{I'}-2\omega_iA_{I'}, & Z^I=\Omega_{0i}Z^{I'},\ 1\leq i\leq 3;\\ 0, & Z^I=\partial Z^{I'}; \end{array}\right.\end{align} $$

Note that $\widehat { U }\equiv \widehat { A }\equiv 0$ for $q\geq R$ , so by (3.25) and (3.26), we have $U_I\equiv A_I\equiv 0$ for all $q\geq R$ . For all $a,b,c\geq 0$ , we have

(3.27) $$ \begin{align}\begin{aligned} \partial_q^a\partial_\omega^cA_I&=O_{a,c}(\langle q \rangle^{-1-a+C_{a,c}\varepsilon});\\ \partial_q^a\partial_s^b \partial_\omega^c\partial_qU_I&=O_{a,b,c}(\langle q \rangle^{-1-a-b+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s});\\ \partial_s^b \partial_\omega^cU_I&=O_{b,c}((\varepsilon^{-1}\langle q \rangle^{C_{b,c}\varepsilon}1_{b=0}+1_{b>0})e^{C_{b,c}s}) \end{aligned}\end{align} $$

for all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ . For all $c\geq 0$ , we have

(3.28) $$ \begin{align}\begin{aligned} \partial_\omega^cU_I=O_{c,C_0}(e^{C_{c,C_0}s}) \end{aligned}\end{align} $$

for all $s>0$ , $\omega \in \mathbb {S}^2$ , and $q>C_0e^{C_0s}(e^{\delta /\varepsilon }-e^{(\delta +s)/\varepsilon }-1)$ for some constant $C_0>1$ . Finally, for all $a,b,c\geq 0$ , we have

(3.29) $$ \begin{align}\begin{aligned} \partial_q^a\partial_s^b\partial_\omega^c(2\partial_qU_I+A_I)(s,q,\omega;\varepsilon) &=O_{a,b,c}(\langle q \rangle^{-\max\{1,b\}-1-a+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s}) \end{aligned}\end{align} $$

for all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ .

Remark 3.3.1. Since $\widehat { u }=\varepsilon r^{-1}\widehat { U }(s,\widehat { q },\omega )$ is defined explicitly by (3.22), one can compute $Z^I\widehat { u }$ by the chain rule and Leibniz’s rule. This has been done in, for example, the proof of Lemma 2.7. Thus, we can obtain an identity for $Z^I\widehat { u }$ . Similarly for $(\partial _t-\partial _r)Z^I\widehat { u }$ . In this paper, we prefer the asymptotic formulas (3.24) over these identities because (3.24) are much simpler.

Let us compute $Z\widehat { u }$ as an example. It suffices to compute $Z(r\widehat { u })=Z(\varepsilon \widehat { U }(\varepsilon \ln t-\delta ,\widehat { q },\omega ))$ because of the commutator $[Z,r]=Zr$ . By the chain rule, we obtain the following term in the expansion of $Z(r\widehat { u })$ :

(3.30) $$ \begin{align}\begin{aligned} \varepsilon\widehat{ U }_q(\varepsilon\ln t-\delta,\widehat{ q },\omega)\cdot Z\widehat{ q }. \end{aligned}\end{align} $$

Recall that $\widehat { q }$ is a solution to the transport equation (3.21). There is no explicit formula for $Z\widehat { q }$ , so we cannot further simplify (3.30). In Proposition 3.3, we use an alternative way. We first show that $Z(r\widehat { u })$ equals $Z(\varepsilon \widehat { U }(\varepsilon \ln t-\delta ,r-t,\omega ))$ up to a remainder term. This remainder term is good because we expect $\widehat { q }\approx r-t$ to some extent; see Lemma 3.4 below. By the chain rule, we replace (3.30) with

$$ \begin{align*} \varepsilon\widehat{ U }_q(\varepsilon\ln t-\delta,r-t,\omega)\cdot Z(r-t). \end{align*} $$

We can compute this expression explicitly because we have $S(r-t)=r-t$ , $\Omega _{0i}(r-t)=(t-r)\omega _i$ , and $Z(r-t)=0$ for other Z. As a result, we obtain a simpler asymptotic formula for $Z\widehat { u }$ .

The key step in the proof of Proposition 3.3 is to show the following estimates:

(3.31) $$ \begin{align}\begin{aligned} r\widehat{ u }-\varepsilon\widehat{ U }(\varepsilon\ln t-\delta,r-t,\omega;\varepsilon)&\in\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\ (\partial_t-\partial_r)(r\widehat{ u })+2\varepsilon\widehat{ A }(r-t,\omega;\varepsilon)&\in\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{aligned}\end{align} $$

Recall that $r=|x|$ and $\omega =x/|x|$ . Proving these estimates requires several estimates for the approximate optical function $\widehat { q }$ ; see Section 3.5.1.

In Section 3.5.2, we extend (3.31) by showing that for each multiindex I, we have

(3.32) $$ \begin{align}\begin{aligned} rZ^I\widehat{ u }&=\varepsilon U_I(\varepsilon\ln t-\delta,r-t,\omega;\varepsilon)+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\(\partial_t-\partial_r)(rZ^I\widehat{ u })&=\varepsilon A_I(r-t,\omega;\varepsilon)+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{aligned} \end{align} $$

Here, the $U_I$ and $A_I$ are defined in the statement of Proposition 3.3: we set $(U_0,A_0)=(\widehat { U },-2\widehat { A })$ and define $(U_I,A_I)$ inductively by (3.25) and (3.26) for $|I|>0$ . We can induct on $|I|$ to check (3.27) and (3.28); the details of the proofs are omitted. We also notice that $[\partial _t-\partial _r,r]Z^I\widehat { u }=-Z^I\widehat { u }=O(\varepsilon t^{-1+C\varepsilon })$ by (3.13), so we obtain (3.24).

Finally, we prove (3.29). By setting $B_I=2\partial _qU_I+A_I$ for each I, we note that the $B_I$ ’s satisfy the induction formulas

$$ \begin{align*} B_I=\left\{ \begin{array}{ll} \varepsilon\partial_sb:{I'}+q\partial_qB_{I'}, & Z^I=SZ^{I'}; \\ (\omega_i\partial_{\omega_j}-\omega_j\partial_{\omega_i})B_{I'}, & Z^I=\Omega_{ij}Z^{I'},\ 1\leq i<j\leq 3;\\ \varepsilon\omega_i\partial_sb:{I'}-q\omega_i\partial_qB_{I'}+\partial_{\omega_i}B_{I'}-2\omega_iB_{I'}, & Z^I=\Omega_{0i}Z^{I'},\ 1\leq i\leq 3;\\ 0, & Z^I=\partial Z^{I'}. \end{array}\right. \end{align*} $$

We also have $B_0=2\widehat { A }(\exp (\frac {1}{2}G(\omega )\widehat { A }s)-1)$ . Since $\exp (\frac {1}{2}G(\omega )\widehat { A }s)-1$ has essentially the same form as $\widehat { \mu }+2$ , by following the proof of (3.18), we obtain for all $a,b,c\geq 0$ and all $(s,q,\omega )\in [0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ ,

$$ \begin{align*} \partial_q^a\partial_s^b\partial_\omega^c (\exp(\frac{1}{2}G(\omega)\widehat{ A }s)-1)=O_{a,b,c}(\langle q \rangle^{-\max\{1,b\}-a+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s}). \end{align*} $$

It follows from Leibniz’s rule that

$$ \begin{align*} \partial_q^a\partial_s^b\partial_\omega^cB_0&=O_{a,b,c}(\langle q \rangle^{-\max\{1,b\}-1-a+C_{a,b,c}\varepsilon}e^{C_{a,b,c}s}). \end{align*} $$

Now, (3.29) follows by induction on $|I|$ . We again omit the details of the proof.

3.5.1 Proof of (3.31)

By the definition of $\widehat { u }$ , in $\Omega $ we have

$$ \begin{align*} r\widehat{ u }&=\varepsilon\widehat{ U },\\ (\partial_t-\partial_r)(r\widehat{ u })&=(\partial_t-\partial_r)(\varepsilon\widehat{ U })=\varepsilon^2t^{-1}\widehat{ U }_s+\varepsilon\widehat{ U }_q\widehat{ \mu }=\varepsilon^2t^{-1}\widehat{ U }_s-2\varepsilon\widehat{ A }. \end{align*} $$

Here, $\widehat { U },\widehat { U }_s,\widehat { A }$ are functions of $(t,x)$ via the map $(t,x)\mapsto (\varepsilon \ln t-\delta ,\widehat { q }(t,x),x/|x|)$ . By (3.23) at the end of Section 3.4, we have $\varepsilon \widehat { U }_s\in S^{0,0}$ . Thus, we have

$$ \begin{align*} (\partial_t-\partial_r)(r\widehat{ u })&=-2\varepsilon\widehat{ A }+\varepsilon S^{-1,0}. \end{align*} $$

It remains to show that

(3.33) $$ \begin{align}\begin{aligned} \varepsilon \widehat{ U }(\varepsilon\ln t-\delta,\widehat{ q },\omega;\varepsilon)-\varepsilon \widehat{ U }(\varepsilon\ln t-\delta,r-t,\omega;\varepsilon)&\in \varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\\varepsilon \widehat{ A }(\widehat{ q },\omega;\varepsilon)-\varepsilon \widehat{ A }(r-t,\omega;\varepsilon)&\in \varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{aligned}\end{align} $$

We first prove a lemma for $\widehat { q }$ . This lemma both relies on and improves [Reference Yu75, Lemma 7.7].

Lemma 3.4. We have $\widehat { q }-r+t\in S^{0,0}_{\ln }$ . That is, for each I, we have $Z^I(\widehat { q }-r+t)=O((1+\ln \langle r-t \rangle )t^{C\varepsilon })$ in $\Omega $ .

Proof. For $r-t\geq R$ , we have $q=r-t$ . It remains to prove $Z^I(\widehat { q }-r+t)=O((1+\ln \langle r-t \rangle )t^{C\varepsilon })$ for each I in $\Omega \cap \{r-t<2R\}$ . Before we start the proof, we recall from [Reference Yu75, Lemma 7.4] that $t^{-C\varepsilon }\lesssim \langle \widehat { q } \rangle /\langle r-t \rangle \lesssim t^{C\varepsilon }$ in $\Omega $ . Thus, every power of $\langle \widehat { q } \rangle $ can be controlled by $\langle r-t \rangle t^{C\varepsilon }$ and vice versa.

We first apply Lemma 2.4 to compute $(\partial _t-\partial _r)Z^I(\widehat { q }-r+t)$ for each multiindex I. For each integer $N\geq 0$ , in $\Omega $ we have

(3.34) $$ \begin{align}\begin{aligned} &\sum_{|I|\leq N}|(\partial_t-\partial_r)Z^I(\widehat{ q }-r+t)|\\&\quad\lesssim \sum_{|I|\leq N}|Z^I(\widehat{ \mu }+2)|+\sum_{|J|<N}\left( |Z^J(\widehat{ \mu }+2)|+\sum_{j=1}^3|(\partial_j+\omega_j\partial_t)Z^J(\widehat{ q }-r+t)| \right)\\&\quad\lesssim \sum_{|I|\leq N}|Z^I(\widehat{ \mu }+2)|+\langle r+t \rangle^{-1}\sum_{|I|\leq N}|Z^I(\widehat{ q }-r+t)|. \end{aligned}\end{align} $$

Here, we recall that $\widehat { \mu }=\widehat { q }_t-\widehat { q }_r$ . In the last estimate, we apply Lemma 2.1 to get the factor $\langle r+t \rangle ^{-1}$ . Meanwhile, we recall that $\widehat { \mu }+2\in S^{0,-1}$ . In fact, the results in (3.23) hold because it is already known that $\widehat { q }\in S^{0,1}$ from [Reference Yu75, Lemma 7.7]. Thus, we have $\sum _{|I|\leq N}|Z^I(\widehat { \mu }+2)|\lesssim \langle r-t \rangle ^{-1}t^{C\varepsilon }$ in $\Omega $ .

We can now use Gronwall’s inequality to finish the proof. Fix an integer $N\geq 0$ and a point $(t_0,x_0)\in \Omega \cap \{|x_0|-t_0<2R\}$ . For each $\tau \in [\frac {t_0+|x_0|}{2}-R,t_0]$ , we set

$$ \begin{align*} z_N(\tau)&=\sum_{|I|\leq N}|Z^I(\widehat{ q }-r+t)(\tau,(t_0+|x_0|-\tau)\frac{x_0}{|x_0|};\varepsilon)|. \end{align*} $$

Note that $(\tau ,(t_0+|x_0|-2\tau )\frac {x_0}{|x_0|})\in \Omega \cap \{r-t<2R\}$ for all $\tau \in [\frac {t_0+|x_0|}{2}-R,t_0]$ . Moreover, given a function $h(t,x)$ , for $\widetilde { z }(\tau )=h(\tau ,(|x_0|+t_0-\tau )\cdot \frac {x_0}{|x_0|})$ , we have $\frac {d}{d\tau }\widetilde { z }=(h_t-h_r)(\tau ,(|x_0|+t_0-\tau )\cdot \frac {x_0}{|x_0|})$ . Thus, the inequality (3.34) implies that for each $\tau _0\in [\frac {t_0+|x_0|}{2}-R,t_0]$ ,

$$ \begin{align*} z_N(\tau_0)&\lesssim z_N(\frac{t_0+|x_0|}{2}-R)+\int_{\frac{t_0+|x_0|}{2}-R}^{\tau_0}\langle |x_0|+t_0-2\tau \rangle^{-1}\tau^{C\varepsilon}+\langle |x_0|+t_0 \rangle^{-1}z_N(\tau)\ d\tau\\ &\lesssim\langle |x_0|+t_0 \rangle^{-1}\int_{\frac{t_0+|x_0|}{2}-R}^{\tau_0}z_N(\tau)\ d\tau+ t_0^{C\varepsilon}(1+\ln\langle |x_0|-t_0 \rangle). \end{align*} $$

To obtain the second estimate, we notice that $(\frac {t_0+|x_0|}{2}-R,(\frac {t_0+|x_0|}{2}-R)\frac {x_0}{|x_0|})\in \{(t,x):\ |x|-t>R\}$ . Thus, we have $\widehat { q }=r-t$ near this point and thus $z_N(\frac {t_0+|x_0|}{2}-R)=0$ . Moreover, for $\frac {t_0+|x_0|}{2}-R\leq \tau \leq t_0$ , we have $\tau ^{C\varepsilon }\leq t_0^{C\varepsilon }$ and $t+x_0-2\tau \leq 2R$ . It follows that

$$ \begin{align*} \int_{\frac{t_0+|x_0|}{2}-R}^{\tau_0}\langle |x_0|+t_0-2\tau \rangle^{-1}\tau^{C\varepsilon}\ d\tau&\lesssim t_0^{C\varepsilon}\int_{\frac{t_0+|x_0|}{2}-R}^{\tau_0}(2\tau-|x_0|-t_0+2R+1)^{-1}\ d\tau\\ &\lesssim t_0^{C\varepsilon}\ln(t_0-|x_0|+2R+1)\lesssim t_0^{C\varepsilon}(1+\ln\langle t_0-|x_0| \rangle). \end{align*} $$

Thus, by Gronwall’s inequality, we have

$$ \begin{align*} z_N(t_0)&\lesssim t_0^{C\varepsilon}(1+\ln\langle t_0-|x_0| \rangle)\exp(\int_{\frac{t_0+|x_0|}{2}-R}^{t_0}\langle |x_0|+t_0 \rangle^{-1}\ d\tau)\\&\lesssim t_0^{C\varepsilon}(1+\ln\langle t_0-|x_0| \rangle)\exp(C\cdot \frac{\frac{t_0-|x_0|}{2}+R}{|x_0|+t_0+1})\lesssim t_0^{C\varepsilon}(1+\ln\langle t_0-|x_0| \rangle).\\[-41pt] \end{align*} $$

Using Lemma 3.4 and the bounds for $\widehat { U },\widehat { A }$ , we now show (3.33) which implies (3.31).

Lemma 3.5. Fix an integer $N\geq 0$ . Then, for $\varepsilon \ll _N1$ and all $(t,x)\in \Omega $ , we have

$$ \begin{align*} \sum_{|I|\leq N}|Z^I(\widehat{ U }(\varepsilon\ln t-\delta,\widehat{ q },\omega;\varepsilon)-\widehat{ U }(\varepsilon\ln t-\delta,r-t,\omega;\varepsilon))|\lesssim (1+\ln\langle r-t \rangle)\langle r-t \rangle^{-1}t^{C\varepsilon},\\ \sum_{|I|\leq N}|Z^I(\widehat{ A }(\widehat{ q },\omega)-\widehat{ A }(r-t,\omega))|\lesssim (1+\ln\langle r-t \rangle)\langle r-t \rangle^{-2}t^{C\varepsilon}. \end{align*} $$

Proof. Since $\widehat { q }=r-t$ for $r-t\geq R$ , it remains to prove this lemma in $\Omega \cap \{r-t<2R\}$ . We will prove by induction on $|I|$ .

When $|I|=0$ , by the mean value theorem, we have (here we set $s=\varepsilon \ln t-\delta $ and omit the parameter $\varepsilon $ )

$$ \begin{align*} &|\widehat{ U }(s,\widehat{ q },\omega)-\widehat{ U }(s,r-t,\omega)|\lesssim |\widehat{ q }-r+t|\cdot \sup_{\rho\ \text{lies between }\widehat{ q },r-t}|\widehat{ U }_q(s,\rho,\omega)|\\&\quad \lesssim (1+\ln\langle r-t \rangle)t^{C\varepsilon}\cdot \langle r-t \rangle^{-1+C\varepsilon}t^{C\varepsilon}\lesssim (1+\ln\langle r-t \rangle)\langle r-t \rangle^{-1}t^{C\varepsilon},\\&|\widehat{ A }(\widehat{ q },\omega)-\widehat{ A }(r-t,\omega)|\lesssim |\widehat{ q }-r+t|\cdot \sup_{\rho\ \text{lies between }\widehat{ q },r-t}|\widehat{ A }_q(\rho,\omega)|\\&\quad \lesssim (1+\ln\langle r-t \rangle)t^{C\varepsilon}\cdot \langle r-t \rangle^{-2+C\varepsilon}t^{C\varepsilon}\lesssim (1+\ln\langle r-t \rangle)\langle r-t \rangle^{-2}t^{C\varepsilon}. \end{align*} $$

Here, we use (3.17), (3.18), and Lemma 3.4. Besides, we also use the fact that if $\rho $ lies between $\widehat { q }$ and $r-t$ , then $\langle r-t \rangle \lesssim \langle \rho \rangle t^{C\varepsilon }$ . In fact, since $\widehat { q }<2R$ and $r-t<2R$ in $\Omega \cap \{r-t<2R\}$ , we also have $\rho <2R$ . And since $t^{-C\varepsilon }\lesssim \langle \widehat { q } \rangle /\langle r-t \rangle \lesssim t^{C\varepsilon }$ , we have

$$ \begin{align*} \langle \rho \rangle^{-1}&\sim (1+2R-\rho)^{-1}\leq (1+2R-\widehat{ q })^{-1}+(1+2R-r+t)^{-1}\\&\lesssim \langle \widehat{ q } \rangle^{-1}+\langle r-t \rangle^{-1}\lesssim \langle r-t \rangle^{-1}t^{C\varepsilon}. \end{align*} $$

We thus finish the proof when $|I|=0$ .

In general, we fix a multiindex I with $|I|>0$ . By the chain rule and Leibniz’s rule, we can write $Z^I(\widehat { U }(s,\widehat { q },\omega )-\widehat { U }(s,r-t,\omega ))$ as a linear combination (with real constant coefficients depending on I) of terms of the form

$$ \begin{align*} &(\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,\widehat{ q },\omega)\cdot \prod_{j=1}^b Z^{I_j}(\varepsilon\ln t-\delta)\cdot\prod_{j=1}^a Z^{J_j}\widehat{ q }\cdot\prod_{j=1}^cZ^{K_j}\omega \\&\quad-(\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,r-t,\omega)\cdot \prod_{j=1}^b Z^{I_j}(\varepsilon\ln t-\delta)\cdot\prod_{j=1}^a Z^{J_j}(r-t)\cdot\prod_{j=1}^cZ^{K_j}\omega. \end{align*} $$

Here, $a+b+c>0$ , the sum of all the multiindices is $|I|$ , and each of these multiindices is nonzero. This term equals

$$ \begin{align*} &[(\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,\widehat{ q },\omega)-(\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,r-t,\omega)]\cdot\prod_{j=1}^bZ^{I_j}(\varepsilon\ln t)\cdot\prod_{j=1}^a Z^{J_j}\widehat{ q }\cdot\prod_{j=1}^cZ^{K_j}\omega\\&\quad +(\partial_s^b\partial_q^a\partial_\omega^c \widehat{ U })(s,r-t,\omega)\cdot \left( \sum_{k=1}^a\prod_{j=1}^k Z^{J_j}\widehat{ q }\cdot Z^{J_k}(\widehat{ q }-r+t)\cdot \prod_{j=k+1}^a Z^{J_j}(r-t) \right)\\&\hspace{25em}\cdot\prod_{j=1}^bZ^{I_j}(\varepsilon\ln t)\cdot\prod_{j=1}^cZ^{K_j}\omega. \end{align*} $$

If $a=0$ , then the second row vanishes. We claim that this term is $O((1+\ln \langle r-t \rangle )\langle r-t \rangle ^{-1}t^{C\varepsilon })$ . In fact, by (3.18), we follow a similar proof in the case $|I|=0$ to show that

$$ \begin{align*} (\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,\widehat{ q },\omega)-(\partial_q^a\partial_s^b\partial_\omega^c \widehat{ U })(s,r-t,\omega)=O((1+\ln\langle r-t \rangle)\langle r-t \rangle^{-1-a}t^{C\varepsilon}). \end{align*} $$

Moreover, by (3.18) and (3.19), we have $(\partial _q^a\partial _s^b\partial _\omega ^c \widehat { U })(s,r-t,\omega )=O(\langle r-t \rangle ^{-a}t^{C\varepsilon })$ in $\Omega \cap \{r-t<2R\}$ . We also have $Z^J\omega =O(1)$ , $Z^J(\widehat { q },r-t)=O(\langle r-t \rangle t^{C\varepsilon })$ , and $Z^J(\widehat { q }-r+t)=O((1+\ln \langle r-t \rangle )t^{C\varepsilon })$ . Combine all these estimates and we obtain the estimates for $\widehat { U }$ in the lemma. The estimates for $\widehat { A }$ can be proved similarly.

3.5.2 Proof of (3.32)

We induct on $|I|$ . The case when $|I|=0$ has been proved; see (3.31). Now, fix a multiindex I with $|I|>0$ and write $Z^I=ZZ^{I'}$ . We have

(3.35) $$ \begin{align}\begin{aligned} rZ^I\widehat{ u }&= Z(rZ^{I'}\widehat{ u })+[r,Z]Z^{I'}\widehat{ u }\\ &=\varepsilon (\partial_sU_{I'})\cdot Z(\varepsilon\ln t)+\varepsilon (\partial_qU_{I'})\cdot Z(r-t)\\ &\quad +\sum_{k=1}^3\varepsilon (\partial_{\omega_k}U_{I'})\cdot Z\omega_k-Zr\cdot Z^{I'}\widehat{ u }+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1}, \end{aligned}\end{align} $$

(3.36) $$ \begin{align}\begin{aligned} (\partial_t-\partial_r)(rZ^I\widehat{ u })&=Z[(\partial_t-\partial_r)(rZ^{I'}\widehat{ u })]+[\partial_t-\partial_r,Z](rZ^{I'}\widehat{ u })+(\partial_t-\partial_r)([r,Z]Z^{I'}\widehat{ u })\\ &=\varepsilon (\partial_qA_{I'})\cdot Z(r-t)+\sum_{k=1}^3\varepsilon (\partial_{\omega_k}A_{I'})\cdot Z\omega_k\\ &\quad+[\partial_t-\partial_r,Z](rZ^{I'}\widehat{ u })-(\partial_t-\partial_r)(Zr\cdot Z^{I'}\widehat{ u })+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{aligned}\end{align} $$

On the right sides, every function of $(s,q,\omega )$ is evaluated at $(s,q,\omega )=(\varepsilon \ln t-\delta ,r-t,\omega )$ . For example, $\partial _sU_{I'}=(\partial _sU_{I'})(\varepsilon \ln t-\delta ,r-t,\omega ;\varepsilon )$ ; similarly for $U_{I'},A_{I'},\partial _q(U_{I'},A_{I'}),\partial _\omega (U_{I'}, A_{I'})$ . We apply the induction hypotheses and the chain rule to compute $Z(rZ^{I'}\widehat { u })$ and $Z((\partial _t-\partial _r)(rZ^{I'}\widehat { u }))$ . Now we can prove (3.32).

Lemma 3.6. Fix a multiindex I with $|I|>0$ and $Z^I=ZZ^{I'}$ . Assuming that (3.32) holds with I replaced by $I'$ , we have (3.32) for I. Here, the functions $U_I$ and $A_I$ are defined in the statement of Proposition 3.3.

Proof. In this proof, every function of $(s,q,\omega )$ (such as $U_{I'}$ and $A_{I'}$ ) is evaluated at $(s,q,\omega )=(\varepsilon \ln t-\delta ,r-t,\omega )$ .

Since the $U_I$ ’s and $A_I$ ’s are defined inductively by (3.25) and (3.26), we can prove (3.27) and (3.28) without proving (3.24). Thus, for each multiindex I, we have

(3.37) $$ \begin{align}\begin{aligned} U_I,\varepsilon \partial_sU_I,\partial_\omega U_I\in S^{0,0};\quad \partial_qU_I,\partial_\omega A_I\in S^{0,-1};\quad \partial_qA_I\in S^{0,-2}. \end{aligned}\end{align} $$

Here, we apply Lemma 2.7 (with $\rho =r-t$ ), (3.27), and (3.28).

(a) Suppose that $Z^I=\partial Z^{I'}$ . We have $\partial (\varepsilon \ln t),\partial \omega \in \varepsilon S^{-1,0}$ and $\partial (r-t)\in S^{0,0}$ . By (3.37) and Lemma 2.6, we have

$$ \begin{align*} & \varepsilon (\partial_sU_{I'})\cdot \partial(\varepsilon\ln t)+\varepsilon (\partial_qU_{I'}) \cdot \partial(r-t)+\sum_{k=1}^3\varepsilon (\partial_{\omega_k}U_{I'}) \cdot \partial\omega_k\\&\quad \in S^{0,0}\cdot\varepsilon S^{-1,0}+\varepsilon S^{0,-1}\cdot S^{0,0}+\varepsilon S^{0,0}\cdot S^{-1,0}\subset \varepsilon S^{0,-1},\\& \varepsilon (\partial_qA_{I'}) \cdot \partial(r-t)+\sum_{k=1}^3\varepsilon (\partial_{\omega_k}A_{I'}) \cdot \partial\omega_k\in \varepsilon S^{0,-2}\cdot S^{0,0}+\varepsilon S^{0,-1}\cdot S^{-1,0}\subset \varepsilon S^{0,-2}. \end{align*} $$

Moreover, we have $[\partial _t-\partial _r,\partial _i]=r^{-1}(\partial _i-\omega _i\partial _r)=S^{-1,0}\cdot \partial $ (by $S^{-1,0}\cdot \partial $ we mean $f^\alpha (t,x) \partial _\alpha $ where $f^\alpha \in S^{0,0}$ ), $\partial _ir=\omega _i\in S^{0,0}$ and $[\partial _t-\partial _r,\partial _t]=\partial _tr=0$ . Since $\widehat { u }\in \varepsilon S^{-1,0}$ , by Lemma 2.6, we have

$$ \begin{align*} \partial r\cdot Z^{I'}\widehat{ u }&\in S^{0,0}\cdot \varepsilon S^{-1,0}\subset \varepsilon S^{-1,0},\\ [\partial_t-\partial_r,\partial](rZ^{I'}\widehat{ u })&\in S^{-1,0}\cdot \partial(\varepsilon S^{0,0})\subset \varepsilon S^{-1,-1},\\ (\partial_t-\partial_r)(\partial r\cdot Z^{I'}\widehat{ u })&\in \partial (S^{0,0}\cdot \varepsilon S^{-1,0})\subset \varepsilon S^{-1,-1}. \end{align*} $$

In summary, by (3.35) and (3.36), we have $ r\partial Z^{I'}\widehat { u }\in \varepsilon S^{0,-1}_{\ln }+\varepsilon S^{-1,1}$ and $(\partial _t-\partial _r)(r\partial Z^{I'}\widehat { u })\in \varepsilon S^{0,-2}_{\ln }+\varepsilon S^{-1,0}$ , i.e. $U_I\equiv A_I\equiv 0$ .

(b) Suppose that $Z^I=SZ^{I'}$ . In this case, we have $S(\varepsilon \ln t)=\varepsilon $ , $S(r-t)=r-t$ , $S\omega =0$ , $[\partial _t-\partial _r,S]=\partial _t-\partial _r$ , and $Sr=r$ . By (3.35) and (3.36), we have

$$ \begin{align*} rSZ^{I'}\widehat{ u }&=\varepsilon^2 (\partial_sU_{I'}) +\varepsilon (\partial_qU_{I'}) \cdot (r-t) -r\cdot Z^{I'}\widehat{ u }+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1}\\ &=\varepsilon (\varepsilon \partial_sU_{I'}+q\partial_qU_{I'}-U_{I'}) +\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\ (\partial_t-\partial_r)(rSZ^{I'}\widehat{ u }) &=\varepsilon (\partial_qA_{I'}) \cdot (r-t)+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}=\varepsilon(q\partial_qA_{I'})+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{align*} $$

That is, $U_I=\varepsilon \partial _sU_{I'}+q\partial _qU_{I'}-U_{I'}$ and $A_I=q\partial _qA_{I'}$ . In these estimates, we apply the induction hypotheses to compute $rZ^{I'}\widehat { u }$ . We also notice that $[S,r(\partial _t-\partial _r)]=0$ .

(c) Suppose that $Z^I=\Omega _{ij}Z^{I'}$ . In this case, we have $\Omega _{ij}(\varepsilon \ln t)=\Omega _{ij}(r-t)=\Omega _{ij}r=0$ , $\Omega _{ij}\omega _k=\omega _i\delta _{jk}-\omega _j\delta _{ik}$ , and $[\partial _t-\partial _r,\Omega _{ij}]=0$ . By (3.35) and (3.36), we have

$$ \begin{align*} r\Omega_{ij}Z^{I'}\widehat{ u }&=\sum_{k=1}^3\varepsilon (\partial_{\omega_k}U_{I'}) \cdot (\omega_{i}\delta_{jk}-\omega_j\delta_{ik})+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1}\\ &=\varepsilon ((\omega_i\partial_{\omega_j}-\omega_j\partial_{\omega_i})U_{I'}) +\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\ (\partial_t-\partial_r)(r\Omega_{ij}Z^{I'}\widehat{ u })&=\sum_{k=1}^3\varepsilon (\partial_{\omega_k}A_{I'}) \cdot (\omega_i\delta_{jk}-\omega_j\delta_{ik})+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}\\ &=\varepsilon ((\omega_i\partial_{\omega_j}-\omega_j\partial_{\omega_i})A_{I'}) +\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{align*} $$

That is, $U_I=(\omega _i\partial _{\omega _j}-\omega _j\partial _{\omega _i})U_{I'}$ and $A_I=(\omega _i\partial _{\omega _j}-\omega _j\partial _{\omega _i})A_{I'}$ .

(d) Suppose that $Z^I=\Omega _{0i}Z^{I'}$ . In this case, we have $\Omega _{0i}(\varepsilon \ln t)=\varepsilon \omega _i(1+\frac {r-t}{t}) $ , $\Omega _{0i}(r-t)=-(r-t)\omega _i$ , $\Omega _{0i}\omega _k=(1-\frac {r-t}{r})(\delta _{ik}-\omega _i\omega _k)$ , $[\partial _t-\partial _r,\Omega _{0i}]=-\omega _i(\partial _t-\partial _r)-\frac {r+t}{r^2}\sum _{j=1}^3\omega _j\Omega _{ij}$ , and $\Omega _{0i}r=t\omega _i=(1-\frac {r-t}{r})r\omega _i$ . By (3.35) and (3.36), we have

$$ \begin{align*} &r\Omega_{0i}Z^{I'}\widehat{ u }\\&\quad =\varepsilon (\partial_sU_{I'}) \cdot \varepsilon\omega_i(1+\frac{r-t}{t})-\varepsilon (\partial_qU_{I'}) \cdot (r-t)\omega_i\\&\qquad +\sum_{k=1}^3\varepsilon (\partial_{\omega_k}U_{I'}) \cdot (1-\frac{r-t}{r})(\delta_{ik}-\omega_i\omega_k)-(1-\frac{r-t}{r})r\omega_i\cdot Z^{I'}\widehat{ u }+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1}\\&\quad =\varepsilon(\varepsilon\omega_i\partial_sU_{I'}-q\omega_i\partial_qU_{I'}+\partial_{\omega_i}U_{I'}-\omega_iU_{I'})\\&\qquad+\varepsilon^2 (\partial_sU_{I'})\omega_i \cdot \frac{r-t}{t}-\varepsilon (\partial_{\omega_i}U_{I'}) \cdot \frac{r-t}{r}+(r-t)\omega_iZ^{I'}\widehat{ u }+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\&(\partial_t-\partial_r)(r\Omega_{0i}Z^{I'}\widehat{ u })\\&\quad =-\varepsilon (\partial_qA_{I'}) \cdot (r-t)\omega_i+\sum_{k=1}^3\varepsilon (\partial_{\omega_k}A_{I'}) \cdot (1-\frac{r-t}{r})(\delta_{ik}-\omega_i\omega_k)\\&\qquad-\omega_i(\partial_t-\partial_r)((r+t)Z^{I'}\widehat{ u })-\frac{r+t}{r^2}\sum_{j=1}^3\omega_j\Omega_{ij}(rZ^{I'}\widehat{ u })+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}\\&\quad =\varepsilon (-q\omega_i\partial_qA_{I'}+\partial_{\omega_i}A_{I'}-2\omega_iA_{I'}) -\varepsilon (\partial_{\omega_i}A_{I'}) \cdot \frac{r-t}{r}\\&\qquad+\omega_i(\partial_t-\partial_r)((r-t)Z^{I'}\widehat{ u })-\frac{r+t}{r^2}\sum_{j=1}^3x_j\Omega_{ij}Z^{I'}\widehat{ u }+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{align*} $$

By (3.37), we have

$$ \begin{align*} \varepsilon^2 (\partial_sU_{I'})\omega_i \cdot \frac{r-t}{t}-\varepsilon (\partial_{\omega_i}U_{I'}) \cdot \frac{r-t}{r}+(r-t)\omega_iZ^{I'}\widehat{ u }&\in \varepsilon S^{-1,1},\\-\varepsilon (\partial_{\omega_i}A_{I'}) \cdot \frac{r-t}{r}+\omega_i(\partial_t-\partial_r)((r-t)Z^{I'}\widehat{ u })-\frac{r+t}{r^2}\sum_{j=1}^3x_j\Omega_{ij}Z^{I'}\widehat{ u }&\in \varepsilon S^{-1,0}. \end{align*} $$

In summary, we have

$$ \begin{align*} r\Omega_{0i}Z^{I'}\widehat{ u }&=\varepsilon(\varepsilon\omega_i\partial_sU_{I'}-q\omega_i\partial_qU_{I'}+\partial_{\omega_i}U_{I'}-\omega_iU_{I'})+\varepsilon S^{0,-1}_{\ln}+\varepsilon S^{-1,1},\\ (\partial_t-\partial_r)(r\Omega_{0i}Z^{I'}\widehat{ u }) &=\varepsilon (-q\omega_i\partial_qA_{I'}+\partial_{\omega_i}A_{I'}-2\omega_iA_{I'})+\varepsilon S^{0,-2}_{\ln}+\varepsilon S^{-1,0}. \end{align*} $$

That is, $U_I=\varepsilon \omega _i\partial _sU_{I'}-q\omega _i\partial _qU_{I'}+\partial _{\omega _i}U_{I'}-\omega _iU_{I'}$ and $A_I=-q\omega _i\partial _qA_{I'}+\partial _{\omega _i}A_{I'}-2\omega _iA_{I'}$ .

3.6 Gauge independence

In Section 3.2, we construct the optical function $q=q(t,x;\varepsilon )$ by solving the eikonal equation (3.3). There, we consider the region $\Omega =\Omega _{\delta ,R,\varepsilon }$ defined by (3.2) and assign the boundary condition $q|_{\partial \Omega }=r-t$ . Meanwhile, we will obtain different optical functions by choosing different regions $\Omega $ and by assigning different boundary conditions. For example, for $\overline {\delta },\kappa \in (0,1)$ , we can set

$$ \begin{align*} \overline{\Omega}_{\overline{\delta},R,\varepsilon,\kappa}=\{(t,x)\in\mathbb{R}^{1+3}:\ t>e^{\overline{\delta}/\varepsilon},\ |x|-e^{\overline{\delta}/\varepsilon}-2R>\kappa(t-e^{\overline{\delta}/\varepsilon})\} \end{align*} $$

and solve the eikonal equation (3.3) with $\Omega $ replaced by $\overline {\Omega }$ . Note that $\Omega _{\delta ,R,\varepsilon }=\overline {\Omega }_{\delta ,R,\varepsilon ,1/2}$ . All the computations and proofs in Sections 3.2–3.5 will work, so we will again obtain functions such as $\widehat { \overline {A} }$ (from (3.16)) and $\overline {A}_I$ (from Proposition 3.3), etc. It is natural to ask how these new functions are related to the old ones. We seek to obtain a gauge independence result; see [Reference Yu75, Section 6].

For convenience, we make the following definition.

Definition 3.7. Fix $\delta ,\kappa \in (0,1)$ . Let $q=q(t,x;\varepsilon )$ be a $C^2$ optical function defined for all $\varepsilon \ll _{\kappa ,\delta }1$ and for all $(t,x)\in \mathbb {R}^{1+3}$ with $t>e^{\delta /\varepsilon }$ and $|x|>\kappa t$ . That is, we have

$$ \begin{align*} g^{\alpha\beta}(u)\partial_\alpha q\partial_\beta q=0,\qquad \forall t>e^{\delta/\varepsilon}, |x|>\kappa t. \end{align*} $$

Also suppose that $q|_{r-t\geq R}=r-t$ . We say that the optical function q is $(\delta ,\kappa )$ -admissible if it satisfies the following assumptions:

1. a) For all $t>e^{\delta /\varepsilon }$ and $|x|>\kappa t$ , we have

   $$ \begin{align*} \sum_{|I|\leq 1}|Z^I(q_t-q_r,q_r^{-1},(q_t-q_r)^{-1})|&\lesssim t^{C\varepsilon};\\ \sum_{|I|\leq 1}\sum_{i=1,2,3}|Z^I(q_i+\omega_iq_t)|&\lesssim t^{-1+C\varepsilon}; \end{align*} $$

2. b) The map $(s,q,\omega )=(\varepsilon \ln t-\delta ,q(t,x;\varepsilon ),x/|x|)$ induces a $C^1$ diffeomorphism from $\{t>e^{\delta /\varepsilon },\ |x|>\kappa t\}$ to a subset of $[0,\infty )\times \mathbb {R}\times \mathbb {S}^2$ , so every $C^1$ function of $(t,x)$ induces a $C^1$ function of $(s,q,\omega )$ ;

3. c) Define $(\mu ,U)(t,x)=(q_t-q_r,\varepsilon ^{-1}ru)(t,x)$ and consider the induced functions $(\mu ,U)(s,q,\omega ;\varepsilon )$ . The limits in (3.5) exist; that is,

   $$ \begin{align*} \left\{\begin{array}{l} \displaystyle A(q,\omega;\varepsilon):=-\frac{1}{2}\lim_{s\to\infty}(\mu U_q)(s,q,\omega;\varepsilon),\\ \displaystyle A_1(q,\omega;\varepsilon):=\lim_{s\to\infty}\exp(\frac{1}{2}G(\omega)A(q,\omega;\varepsilon)s)\mu(s,q,\omega;\varepsilon),\\ \displaystyle A_2(q,\omega;\varepsilon):=\lim_{s\to\infty}\exp(-\frac{1}{2}G(\omega)A(q,\omega;\varepsilon)s)U_q(s,q,\omega;\varepsilon). \end{array}\right. \end{align*} $$