d\eta+2\beta^{\prime}\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\partial_{\eta}G_{B}^{\varepsilon}d\eta$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}\left(e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon})^{2}-2e^{2\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\sin\phi\varphi_{B}^{\varepsilon}\right)d\phi
d\eta$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta.$

The right-hand side of equation (4.3) can be estimated by

$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}S_{2}(\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B})d\phi
d\eta\leq\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$

Taking the above inequalities into (4.3), we obtain

$\displaystyle\frac{3}{4}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(F(\varepsilon;\eta)-2\beta^{\prime})\sin\phi
e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta$
$\displaystyle\quad+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon})^{2}(0,\cdot)d\phi\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$

For $\varepsilon$ sufficiently small, for example
$\varepsilon\leq(4-3\delta)/16$, inequality (3.11) implies for any
$\eta\in\mathbb{R}_{+}$,

$\displaystyle|F(\varepsilon;\eta)-2\beta^{\prime}|\leq
2\beta^{\prime}+\frac{4}{4-3\delta}\varepsilon\leq 1+\frac{1}{4}=\frac{5}{4}.$
(4.32)

With this, the previous inequality implies (16).

Moreover, we can get from equation (4.8) that

$\displaystyle\int_{0}^{B}e^{2\beta^{\prime}\eta}|\partial_{\eta}^{2}G_{B}^{\varepsilon}|^{2}d\eta$
$\displaystyle\quad\leq\int_{0}^{B}e^{2\beta^{\prime}\eta}|F(\varepsilon;\eta)\partial_{\eta}G_{B}^{\varepsilon}|^{2}d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|\langle\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}\rangle|^{2}d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq\int_{0}^{B}|F(\varepsilon;\eta)|^{2}e^{2\beta^{\prime}\eta}\langle\sin\phi(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})\rangle
d\eta+\int_{0}^{B}\left(\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi\cdot\int_{-\pi}^{\pi}d\phi\right)d\eta$
$\displaystyle\qquad+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq\|F\|_{L^{\infty}(\mathbb{R}_{+})}^{2}2\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+2\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta$
$\displaystyle\qquad+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq
2\pi\left(\frac{4}{4-3\delta}\varepsilon+1\right)\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta.$

For $\varepsilon$ sufficiently small, for example
$\varepsilon\leq(4-3\delta)/16$, the above inequality becomes

$\displaystyle\int_{0}^{B}e^{2\beta^{\prime}\eta}|\partial_{\eta}^{2}G_{B}^{\varepsilon}|^{2}d\eta$
$\displaystyle\quad\leq\frac{5}{2}\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq
10\pi\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+20\pi\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\|e^{\beta^{\prime}\eta}S_{1}\|_{L^{2}([0,B])}^{2},$

and hence is uniformly bounded in $B$. Let
$C_{1}:=10\pi\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+20\pi\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\|e^{\beta^{\prime}\eta}S_{1}\|_{L^{2}([0,B])}^{2}$.
Using the bounded condition $\partial_{\eta}G_{B}^{\varepsilon}(B)=0$, we have
for any $\eta\in[0,B]$,

$\displaystyle|\partial_{\eta}G_{B}^{\varepsilon}(\eta)|$
$\displaystyle=-\left|\int_{\eta}^{B}\partial_{s}^{2}G_{B}^{\varepsilon}(s)ds\right|=\left|\int_{\eta}^{B}e^{-\beta^{\prime}s}e^{\beta^{\prime}s}\partial_{s}^{2}G_{B}^{\varepsilon}(s)ds\right|$
$\displaystyle\leq\left(\int_{\eta}^{B}e^{-2\beta^{\prime}s}ds\right)^{\frac{1}{2}}\int_{0}^{B}e^{2\beta^{\prime}s}|\partial_{s}^{2}G_{B}^{\varepsilon}(s)|^{2}ds$
$\displaystyle\leq
C_{1}\frac{\sqrt{e^{-2\beta^{\prime}\eta}-e^{-2\eta^{\prime}B}}}{\sqrt{2\beta^{\prime}}}\leq
C_{1}(2\beta^{\prime})^{-\frac{1}{2}}e^{-\beta^{\prime}\eta}.$

Using the boundary condition $G_{B}^{\varepsilon}(0)=0$, we have

$\displaystyle|G_{B}^{\varepsilon}(\eta)|$
$\displaystyle=\left|\int_{0}^{\eta}\partial_{s}G_{B}^{\varepsilon}(s)ds\right|\leq\int_{0}^{\eta}|\partial_{s}G_{B}^{\varepsilon}(s)|ds\leq
C_{1}(2\beta^{\prime})^{-\frac{1}{2}}\int_{0}^{\eta}e^{-\beta^{\prime}s}ds$
$\displaystyle\leq\sqrt{2}C_{1}(\beta^{\prime})^{-\frac{3}{2}}(1-e^{-\beta^{\prime}\eta})\leq\sqrt{2}C_{1}(\beta^{\prime})^{-\frac{3}{2}}.$

###  4.4. Existence on the half line

Next we pass to the limit $B\to\infty$ in (4.8)-(4.9) to show the existence
for system (4.1)-(4.2) on the half line.

###### Proof of Theorem 13.

By the uniform estimate (4.29), $G_{B}^{\varepsilon}\in L^{\infty}_{\rm
loc}([0,\infty))$. Hence there exists a subsequence such that

$\displaystyle
G_{B}^{\varepsilon}\rightharpoonup^{*}G^{\varepsilon},\quad\text{weakly star
in }L_{\rm loc}^{\infty}([0,\infty)),$ (4.33) $\displaystyle
G_{B}^{\varepsilon}\rightharpoonup G^{\varepsilon},\quad\text{weakly in
}L_{\rm loc}^{2}([0,\infty)).$

Moreover, by the estimate (16),
$\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}$ is uniformly bounded in
$L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi)))$, hence

$\displaystyle\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}\rightharpoonup\varphi^{\varepsilon}-4h^{3}G^{\varepsilon},\quad\text{weakly
in }L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi))),$

and so

$\displaystyle\varphi_{B}^{\varepsilon}\rightharpoonup\varphi^{\varepsilon},\quad\text{weakly
in }L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi))).$ (4.34)

With (4.33) and (4.34), we can pass to the limit in (4.8)-(4.9) with boundary
conditions (4.10)-(4.11) and obtain that
$(G^{\varepsilon},\varphi^{\varepsilon})$ solve system (4.1)-(4.2) in the
sense of distributions and satisfy the boundary conditions (4.3)-(4.4).
Moreover, we can pass to the limit $B\to\infty$ in (4.30) and obtain

$\displaystyle\partial_{\eta}G^{\varepsilon}=\langle\sin\phi\varphi^{\varepsilon}(\eta,\cdot)\rangle$
(4.35)

for $\eta\in\mathbb{R}_{+}$ almost everywhere. By the semi-lower continuity of
the $L^{2}$ norm, we can take $\liminf$ in (16) and use

$\displaystyle\liminf_{B\to\infty}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta\geq\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi
d\eta$

and obtain (13). ∎

###  4.5. Decay properties of the solutions

We can use the above derived uniform estimates to show that solutions to
system (4.1)-(4.2) decay to constants as $\eta\to\infty$. We set

$\displaystyle
C_{2}:=4\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+8\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,\infty)\times[-\pi,\pi))}^{2}.$

Then

$\displaystyle\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi
d\eta\leq C_{2}.$ (4.36)

By the relation (4.35), we get

$\displaystyle\int_{0}^{\infty}e^{2\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta$
$\displaystyle=\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left|\int_{-\pi}^{\pi}\sin\phi(\varphi^{\varepsilon})d\phi\right|^{2}d\eta=\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left|\int_{-\pi}^{\pi}\sin\phi(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})d\phi\right|^{2}d\eta$
$\displaystyle\leq\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left(\int_{-\pi}^{\pi}\sin^{2}\phi
d\phi\right)\left(\int_{-\pi}^{\pi}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi\right)d\eta$
$\displaystyle=\pi\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi\leq\pi
C_{2}.$

Therefore, $\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$ exists and is finite.
To show this, we can take $M_{2}>M_{1}>0$ to be some large numbers and

$\displaystyle|G^{\varepsilon}(M_{1})-G^{\varepsilon}(M_{2})|$
$\displaystyle=\left|\int_{M_{1}}^{M_{2}}\partial_{\eta}G^{\varepsilon}d\eta\right|\leq\int_{M_{1}}^{M_{2}}e^{-\beta^{\prime}\eta}e^{\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|d\eta$
$\displaystyle\leq\left(\int_{M_{1}}^{M_{2}}e^{-2\beta^{\prime}\eta}d\eta\right)^{\frac{1}{2}}\left(\int_{M_{1}}^{M_{2}}e^{2\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta\right)^{\frac{1}{2}},$
$\displaystyle=\sqrt{(e^{-2\beta^{\prime}M_{1}}-e^{-2\beta^{\prime}M_{2}})}\pi\sqrt{C_{2}/(2\beta^{\prime})}.$

Since for $M_{1},M_{2}$ sufficiently large,
$e^{-2\beta^{\prime}M_{1}}-e^{-2\beta^{\prime}M_{2}}$ is sufficiently small.
$\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$ exists. Let
$G^{\varepsilon}_{\infty}=\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$. Then we
take $M_{1}=\eta$, $M_{2}=\infty$ in the above inequality and obtain

$\displaystyle|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|\leq\pi\sqrt{C_{2}/(2\beta^{\prime})}e^{-\beta^{\prime}\eta}.$

Moreover, taking $M_{1}=0$, $M_{2}=\infty$ gives

$\displaystyle|G^{\varepsilon}_{\infty}|\leq\pi\sqrt{C_{2}/(2\beta^{\prime})}.$

Due to the exponentially convergence of $h$, we have

$\displaystyle|4h^{3}(\eta)G^{\varepsilon}(\eta)-4h_{\infty}^{3}G^{\varepsilon}_{\infty}|$
$\displaystyle\leq
4h^{3}(\eta)|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|+4|h^{3}(\eta)-h_{\infty}^{3}||G^{\varepsilon}_{\infty}|\leq
Ce^{-\alpha\eta}+Ce^{-\beta^{\prime}\eta}$ $\displaystyle\leq
2Ce^{-\beta^{\prime}\eta},$ (4.37)

taking $1>\alpha>\beta^{\prime}$.

To show the exponential convergence of $\varphi^{\varepsilon}$, we use the
formulas (A.4), (A.5) and (20). In the case $\sin\phi>0$, we have

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-4h_{\infty}^{3}G_{\infty}^{\varepsilon}|$
$\displaystyle=\left|\int_{0}^{\eta}\frac{4h^{3}(\xi)G^{\varepsilon}(\xi)+S_{2}(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi-4h^{3}_{\infty}G_{\infty}^{\varepsilon}\right|$
$\displaystyle\leq\int_{0}^{\eta}\frac{|4h^{3}(\xi)G^{\varepsilon}(\xi)-4h_{\infty}^{3}G^{\varepsilon}_{\infty}|}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\int_{0}^{\eta}\frac{|S_{2}(\xi,\phi^{\prime}(\phi,\eta,\xi))|}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\left|1-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi\right||4h_{\infty}^{3}G_{\infty}^{\varepsilon}|$
$\displaystyle\quad\leq
2C\int_{0}^{\eta}\frac{e^{-\beta^{\prime}\xi}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq
2C\int_{0}^{\eta}\frac{1}{1-\beta^{\prime}\sin\phi^{\prime}(\phi,\eta,\xi)}de^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}-\frac{1}{2}\beta^{\prime}\xi}+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq\frac{2C}{1-\beta^{\prime}}(e^{-\beta^{\prime}\eta}-e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}})+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq\frac{3C}{1-\beta^{\prime}}e^{-\beta^{\prime}\eta},$

where we use $\beta^{\prime}<1/2$ and $\sin\phi^{\prime}\leq 1$ in the above
inequality. One can also check for $\sin\phi<0$, there exists positive
constants $C>0$,

$|\varphi^{\varepsilon}(\eta,\phi)-4h_{\infty}^{3}G_{\infty}^{\varepsilon}|\leq
Ce^{-\beta^{\prime}\eta},$

which follows a similar estimate as the case for $\sin\phi>0$ above and the
details are similar to the proof of (3.47).

###  4.6. Uniqueness

In order to show the uniqueness, let
$(G_{1}^{\varepsilon},\varphi_{1}^{\varepsilon})$ and
$(G_{2}^{\varepsilon},\varphi_{2}^{\varepsilon})$ be two solutions of the
problem (4.1)-(4.4). Let $g:=G_{1}^{\varepsilon}-G_{2}^{\varepsilon}$ and
$\psi:=\varphi_{1}^{\varepsilon}-\varphi_{2}^{\varepsilon}$, then they satisfy

$\displaystyle\partial_{\eta}^{2}g+F(\varepsilon;\eta)\partial_{\eta}g+\langle\psi-4h^{3}g\rangle=0,$
(4.38)
$\displaystyle\sin\phi\partial_{\eta}\psi+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi+\psi-4h^{3}g=0,$
(4.39)

with boundary conditions

$\displaystyle g(0)=0,\quad\psi(0,\phi)=0,\quad\text{for any }\sin\phi>0.$

Moreover, as $\eta\to\infty$, $G_{1}^{\varepsilon}\to G_{1,\infty}$,
$G_{2}^{\varepsilon}\to G_{2,\infty}$ and $\psi\to
4h_{\infty}^{3}(G_{1,\infty}-G_{2,\infty})$, where $G_{1,\infty}$,
$G_{2,\infty}$ are constants.

We multiply the first equation by $4h^{3}g$ and the second by $\psi$, and
integrate over $[0,\infty)\times[-\pi,\pi)$ to get

$\displaystyle\int_{0}^{\infty}4h^{3}|\partial_{\eta}g|^{2}d\eta+\int_{0}^{\infty}\partial_{\eta}(4h^{3})g\partial_{\eta}gd\eta-\int_{0}^{\infty}F(\varepsilon;\eta)4h^{3}g\partial_{\eta}gd\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi^{2}d\phi
d\eta$
$\displaystyle\quad+\int_{-\pi}^{0}|\sin\phi|\psi^{2}(0,\cdot)d\phi+\int_{0}^{\infty}\int_{-\pi}^{\pi}(\psi-4h^{3}g)^{2}d\phi
d\eta=0.$ (4.40)

From the relation (4.35), we have
$\partial_{\eta}g=\langle\sin\phi\varphi\rangle$. Thus

$\displaystyle-\int_{0}^{\infty}F(\varepsilon;\eta)4h^{3}g\partial_{\eta}gd\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi^{2}d\phi
d\eta$
$\displaystyle\quad=-\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)4h^{3}g\sin\phi\varphi
d\phi
d\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi\psi^{2}d\phi
d\eta$
$\displaystyle\quad=-\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\psi-4h^{3}g)^{2}d\phi
d\eta.$

The spectral assumption implies

$\displaystyle\int_{0}^{\infty}4h^{3}|\partial_{\eta}g|^{2}d\eta+\int_{0}^{\infty}\partial_{\eta}(4h^{3})g\partial_{\eta}gd\eta\geq
0.$

Therefore, we get from (4.6) that

$\displaystyle\int_{-\pi}^{0}|\sin\phi|\psi^{2}(0,\cdot)d\phi+\int_{0}^{\infty}\int_{-\pi}^{\pi}\left(1+\frac{1}{2}F(\varepsilon;\eta)\right)(\psi-4h^{3}g)^{2}d\phi
d\eta=0.$

For $\eta$ sufficiently small, $1+\tfrac{1}{2}F(\varepsilon;\eta)>0$ for any
$\eta\in\mathbb{R}_{+}$, hence the above equation implies $\psi=4h^{3}g$
almost everywhere on $\mathbb{R}_{+}\times[-\pi,\pi)$ and $\psi(0,\phi)=0$ for
almost every $\phi\in[-\pi,\pi)$. Hence
$\partial_{\eta}g=\langle\sin\phi\psi\rangle=\langle\sin\phi
4h^{3}g\rangle=0$, which with $g(0)=0$ implies $g=0$ on $\mathbb{R}_{+}$. We
also have $\psi=4h^{3}g=0$. Hence system (4.38)-(4.39) only has trival
solution and the solution to the problem (4.1)-(4.4).

∎

######  Remark 17.

Compared to the flat boundary case without geometric corrections [6], the
effect of the geometric corrections could be seen from (4.32). for
$\varepsilon$ not sufficiently small, one cannot get a $L^{2}$ estimate on
$\|e^{\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})\|_{L^{2}(\mathbb{R}_{+}\times[-\pi,\pi])}$.
However, for $\varepsilon$ sufficiently small, the geometric correction
contributes to a small error of this $L^{2}$ norm so does not affect the
boundness of this norm, which is crucial for proving Theorem 13.

##  5\. Convergence of the diffusive limit

###  5.1. Errors of the approximation

Next we calculate the approximation errors. Using systems (2.71)-(2.79) and
(2.56)-(2.67), we can calculate

$\displaystyle\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})\right)$
$\displaystyle\quad=\varepsilon^{2}\Delta\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})+\left\langle\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})+\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}\right\rangle$
$\displaystyle\quad=\varepsilon^{2}\Delta\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\langle\psi_{k}^{\varepsilon}\rangle+\partial_{\eta}^{2}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}-\frac{\chi_{0}\varepsilon^{2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}$
$\displaystyle\qquad+\left\langle\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}\right\rangle$
$\displaystyle\quad=\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)+\left\langle\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}\right)^{4}\right\rangle$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\partial_{\eta}^{2}\bar{T}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k}^{\varepsilon}+\langle\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)\rangle+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon})$
$\displaystyle\qquad+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)\rangle$
$\displaystyle\quad=\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)\rangle$
$\displaystyle\qquad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)\rangle$
$\displaystyle\qquad+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)-\mathcal{C}(P^{\varepsilon},k)\rangle$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\partial_{\eta}^{2}\bar{T}_{k}^{\varepsilon}-\frac{\varepsilon}{(1-\varepsilon\eta)}\partial_{\eta}\bar{T}_{k}^{\varepsilon}+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon}+\langle\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)\rangle),$
(5.1)

and

$\displaystyle\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})\right)$
$\displaystyle\quad=\varepsilon\beta\cdot\nabla\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})+\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}$
$\displaystyle\quad=\varepsilon\beta\cdot\nabla\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}+\sin\phi\partial_{\eta}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}$
$\displaystyle\qquad-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}$
$\displaystyle\quad=\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)+\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}\right)^{4}$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\sin\phi\partial_{\eta}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\frac{1}{(1-\varepsilon\eta)}\cos\phi\partial_{\theta}\bar{\psi}_{k-1})$
$\displaystyle\qquad-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}-\sum_{k=N+1}^{4N}\varepsilon^{k}\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)$
$\displaystyle\quad=\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k))-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k))$
$\displaystyle\qquad-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)-\mathcal{C}(P^{\varepsilon},k))$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\sin\phi\partial_{\eta}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}^{\varepsilon}-\frac{1}{1-\varepsilon\eta}\partial_{\theta}\bar{\psi}^{\varepsilon}_{k-1}+\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)).$
(5.2)

By the definition (2.56) of
$(\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon})$, we get (we drop
the superscripts $\varepsilon$ in the calculations below)

$\displaystyle E_{0}^{0}:=$
$\displaystyle\partial_{\eta}^{2}\bar{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{0}+\langle\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle\partial_{\eta}^{2}(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty}))-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty}))+\langle\chi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}+\chi\partial_{\eta}^{2}\tilde{T}_{0}-\frac{\varepsilon}{(1-\varepsilon\eta)}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})$
$\displaystyle-\frac{\varepsilon}{(1-\varepsilon\eta)}\chi\chi_{0}\partial_{\eta}\tilde{T}_{0}+\langle\chi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\chi(\partial_{\eta}^{2}\tilde{T}_{0}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty}\rangle)$
$\displaystyle-\langle(\bar{T}_{0}+T_{0}(0))^{4}-T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})$
$\displaystyle+\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle,$
(5.3)

where we use the fact that $\partial_{\eta}^{2}\chi=0$ and
$\chi\chi_{0}=\chi$. We can also calculate

$\displaystyle E_{0}^{1}:=$
$\displaystyle\sin\phi\partial_{\eta}\bar{\psi}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)$
$\displaystyle=$
$\displaystyle\sin\phi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})\partial_{\eta}\chi+\chi(\sin\phi\partial_{\eta}\tilde{\psi}_{0}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4})$
$\displaystyle+\chi(\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4})-((\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4})$
$\displaystyle=$
$\displaystyle\sin\phi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})\partial_{\eta}\chi+\chi(\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4})-((\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}).$
(5.4)

Similarly, by (2.67), we get

$\displaystyle E_{k}^{0}:=$
$\displaystyle\partial_{\eta}^{2}\bar{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k}+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}+\langle\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle$
(5.5) $\displaystyle=$
$\displaystyle\partial_{\eta}^{2}(\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty}))-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}(\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty}))+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
$\displaystyle+\langle\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}\rangle-\langle\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty})$
$\displaystyle+\chi\bigg{\langle}\partial_{\eta}^{2}\tilde{T}_{k}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}-4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+P,k-1)$
$\displaystyle+\mathcal{E}(P,k-1)+\frac{\chi_{0}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}\bigg{\rangle}+\chi\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}\rangle$
$\displaystyle-\langle
4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0)))\rangle$
$\displaystyle-(1-\chi)\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle+\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
(5.6) $\displaystyle=$
$\displaystyle(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty})+\chi\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}\rangle$
$\displaystyle-\langle
4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0)))\rangle$
$\displaystyle-(1-\chi)\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle+\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
(5.7)

and

$\displaystyle E_{k}^{1}:=$
$\displaystyle\sin\phi\partial_{\eta}\bar{\psi}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}-\frac{1}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}+\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)$
$\displaystyle=$
$\displaystyle\sin\phi\partial_{\eta}\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})+\chi\bigg{(}\sin\phi\partial_{\eta}\tilde{\psi}_{k}+F(\varepsilon;\eta)\cos\phi\partial_{\eta}\tilde{\psi}_{k}-\frac{\chi_{0}}{(1-\varepsilon\eta)}\partial_{\theta}\bar{\psi}_{k-1}$
$\displaystyle+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}-4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+P,k-1)+\mathcal{E}(P,k-1)\bigg{)}$
$\displaystyle+\chi(4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty})-(4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))))$
$\displaystyle-(1-\chi)(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1},$
$\displaystyle=$
$\displaystyle\sin\phi\partial_{\eta}\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})+\chi(4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty})$
$\displaystyle-(4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))))$
$\displaystyle-(1-\chi)(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}.$
(5.8)

Using the formulas (5.1) and (5.1), we get from the above equations

$\displaystyle\mathcal{R}_{1}(T^{a,\varepsilon},\psi^{a,\varepsilon})$
$\displaystyle\quad=\varepsilon^{N+1}\Delta T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}^{\varepsilon}-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)\rangle$
$\displaystyle\qquad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)\rangle+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{0},$
(5.9)

and

$\displaystyle\mathcal{R}_{2}(T^{a,\varepsilon},\psi^{a,\varepsilon})$
$\displaystyle=\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}^{\varepsilon}-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k))$
$\displaystyle\quad-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k))-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{1}.$
(5.10)

###  5.2. Estimation on the approximation errors

The approximation errors calculated are expected to be small for large $N$.
Here we estimate on the terms in (5.1)-(5.1) and prove the following lemma.

######  Lemma 18.

Assume the boundary data in (1.3)-(1.4) satisfy $T_{b}\in C^{2}(\Omega)$,
$\psi_{b}\in C^{1}(\Gamma_{-})$. Let
$(T^{a,\varepsilon},\psi^{a,\varepsilon})$ be the approximate solution
constructed in section 2. Assume the solution $\tilde{T}_{0}^{\varepsilon}$ to
(2.56) satisfy the spectral assumption (A) and
$\tilde{T}_{0}^{\varepsilon}\geq a$ for some constant $a>0$. Then for
sufficient large $N$,

$\displaystyle\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{2}(\Omega)},~{}\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}},$ (5.11)
$\displaystyle\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{2}(\Omega\times\mathbb{S}^{2})},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$ (5.12)

for some constant $0\leq\lambda\leq 1/2$, and $C>0$ a constant independent of
$\varepsilon$.

###### Proof.

The proof of the lemma follows from the decay property of solutions for the
nonlinear Milne and linear Milne problems (Theorem 4 and Theorem 13). Below we
give the estimates on the terms related to geometric corrections. Other terms
also appear in the flat case and the estimates can be found in [6]. Since the
solutions to Milne problems with geometric corrections decay to constant
solutions exponentially, same as the Milne problem without geometric
corrections in [6], these terms can be estimated in the same way.

First by Theorem 4 and Theorem 13, there exists a constant $0\leq\lambda<1/2$
such that

$\displaystyle|\tilde{T}_{k}^{\varepsilon}(\eta)-\tilde{T}_{k,\infty}^{\varepsilon}|\leq
Ce^{-\lambda\eta},\quad|\tilde{\psi}_{k}^{\varepsilon}(\eta)-\tilde{\psi}_{k,\infty}^{\varepsilon}|\leq
Ce^{-\lambda\eta}.$

By the definition of $\bar{T}_{k}^{\varepsilon},\bar{\psi}_{k}^{\varepsilon}$
The terms involving $\partial_{\theta}^{2}$ in (5.1)-(5.1) can be estimated by

$\displaystyle\left|\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}\right|+\left|\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}\right|+\left|\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}\right|\leq
C\varepsilon^{N+1},$

since $\bar{T}_{k}^{\varepsilon},\bar{\psi}_{k}^{\varepsilon}$ are bounded
functions. For the term $E_{0}^{0}$ given in (5.1), the term related to
geometric correction can be estimated by

$\displaystyle\left|\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}^{\varepsilon}-\tilde{T}_{0,\infty}^{\varepsilon})\right|\leq\frac{C\varepsilon}{1-\varepsilon\eta}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}.$

The term $E_{0}^{1}$ is the same as the case without geometric correction [6].
For $E_{k}^{0}$, the related terms are

$\displaystyle\left|-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}^{\varepsilon}-\tilde{T}_{k,\infty}^{\varepsilon})\right|+\left|\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon}\right|$
$\displaystyle\quad\leq\frac{C\varepsilon}{1-\varepsilon\eta}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}+\frac{1}{(1-\varepsilon\eta)^{2}}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}$
$\displaystyle\quad\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}+\frac{C}{(1-\tfrac{3}{8}\delta)^{2}}e^{-\frac{\lambda\delta}{4\varepsilon}}.$

The geometric correction related term in $E_{k}^{1}$ can be estimated
similarly by

$\displaystyle\left|-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}\right|\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}.$

Combing the above estimates, we can obtain

∎

###  5.3. Proof of Theorem 1

The proof of Theorem 1 is based on Banach fixed point theorem by showing that
system (1.1)-(1.2) has a unique solution around
$(T^{a,\varepsilon},\psi^{a,\varepsilon})$ and is given in [6, section 4]. A
crucial step in the proof is based on the following lemma.

######  Lemma 19.

Let $(T^{a,\varepsilon},\psi^{a,\varepsilon})$ be the approximate solution
constructed in the section 2. Assuming the spectral assumption (A) holds for
the solution $\tilde{T}_{0}^{\varepsilon}$ of the nonlinear Milne problem
(2.56) and $\tilde{T}_{0}^{\varepsilon}\geq a$ for some constant $a>0$, and
$T^{a,\varepsilon}$ and $\tilde{T}^{\varepsilon}_{0}$ are positive. The the
following inequality holds

$\displaystyle-\int_{\Omega}4(T^{a,\varepsilon})^{3}g\Delta
gdx=\int_{\Omega}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a,\varepsilon})^{3})\cdot g\nabla
gdx\geq\kappa\int_{\Omega}|\nabla g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2},$ (5.13)

for any function $g$ satisfying $g(0)=0$ and for some constants $\kappa>0$,
$C>0$ depending on $M$, where $M<1$ is the constant of the spectral
assumption.

###### Proof.

Note that
$T^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})$
where
$\bar{T}_{k}^{\varepsilon}=\chi(1-r)(\tilde{T}_{k}^{\varepsilon}-\tilde{T}_{k,\infty}^{\varepsilon})$.
In the domain $1-r>\tfrac{3}{8}\delta$, $\chi(1-r)=0$ and
$T^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}$, which
only contain the interior approximations. Since
$\|T_{k}^{\varepsilon}\|_{C^{s}(\Omega)}$ is bounded for any $s>0$ and
$k=1,\ldots,N$,

$\displaystyle\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}\nabla(4(T^{a,\varepsilon})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}2(T^{a,\varepsilon})^{3/2}\nabla
g\cdot 6(T^{a,\varepsilon})^{1/2}\nabla T^{a,\varepsilon}gdx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}36T^{a,\varepsilon}|\nabla
T^{a,\varepsilon}|^{2}g^{2}dx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}2(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$ (5.14)

In the domain $1-r\leq\frac{3}{8}\delta$, boundary layer effects play a role.
First we split the integral as

$\displaystyle\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\\}}\nabla(4(T^{a,\varepsilon})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\partial_{\eta}T^{a,\varepsilon}|^{2}dx+\int_{\Omega\cap}=:I_{1}+I_{2}.$

Since $\|\nabla_{x^{\prime}}(T^{a})\|_{L^{2}(\Omega)}$ is bounded, we can
estimate $I_{1}$ the same as (5.14):

$\displaystyle
I_{1}\geq\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$
(5.15)

To estimate $I_{2}$, we use the spectral assumption (A). The composite
approximate solution $(T^{a},\psi^{a})$ is close to the solution
$(\tilde{T}_{0},\tilde{\psi}_{0})$ of the nonlinear Milne problem (2.56).
Using the equation

$\displaystyle(T^{a})^{3}=\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{3}=(T_{0}+\bar{T}_{0})^{3}+\varepsilon
G,$

where
$G=3(T_{0}+\bar{T}_{0})^{2}\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k})+6(T_{0}+\bar{T}_{0})^{2}(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}+3(T_{0}+\bar{T}_{0})(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}$,
we can rewrite $I_{2}$ as

$\displaystyle I_{2}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)|\partial_{\eta}g|^{2}d\eta
dx^{\prime}-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)g\partial_{\eta}gd\eta dx^{\prime}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta$
$\displaystyle\quad-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta
dx^{\prime}+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta$
$\displaystyle=:I_{21}+I_{22}+I_{23}+I_{24}.$

The spectral assumption (A) implies

$\displaystyle I_{21}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta$
$\displaystyle\geq\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\frac{1}{2}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}+36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2}))d\eta$
$\displaystyle\geq\frac{1}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2})d\eta$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$

For $I_{22}$, since
$\bar{T}_{0}=T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))$, it holds
that

$\displaystyle(T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3}$
$\displaystyle=(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=((T_{0}-T_{0}(0))-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}).$

Since we are considering the integration over
$x_{1}=\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and
$(1-\chi(\varepsilon\eta))$ is supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$,

$\displaystyle
I_{22}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{\delta}{2\varepsilon}}|\partial_{\eta}g|^{2}d\eta.$

For $I_{23}$, due to

$\displaystyle\partial_{\eta}((T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3})$
$\displaystyle=\partial_{\eta}(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=\varepsilon\chi^{\prime}(\varepsilon\eta)(\tilde{T}_{0}-{T}_{0}(0))(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\chi(\varepsilon\eta)-1)\partial_{\eta}\tilde{T}_{0}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}),$

with consideration of $\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and and
$(1-\chi(\varepsilon\eta))$ being supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$, it holds that

$\displaystyle I_{23}$
$\displaystyle=-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta\leq\frac{C}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(\varepsilon|g||\partial_{\eta}g|+\frac{3\delta}{8}|g||\partial_{\eta}g|)d\eta$
$\displaystyle\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(g^{2}+|\partial_{\eta}g|^{2})d\eta.$

For $I_{24}$, we have

$\displaystyle
I_{24}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta\leq\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$

Combing the above estimates gives

$\displaystyle I_{2}$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta$
$\displaystyle\quad-\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$

By the assumption of the lemma, $\tilde{T}_{0}\geq a$, hence
$4\tilde{T}_{0}^{3}\geq 4a^{3}$ for some constant $a>0$. We can take
sufficiently small $\varepsilon$ and $\delta$ such that
$\varepsilon<(1-M)a^{3}/C$ and $3\delta C/8\leq(1-M)/8$, and we get from the
above inequality

$\displaystyle
I_{2}\geq\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$

Combing this with (5.14) and (5.15) implies

$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta gdx$
$\displaystyle=\int_{\Omega}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a})^{3})\cdot g\nabla gdx$
$\displaystyle\geq\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla
g|^{2}dx+\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx$
$\displaystyle\quad+\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-C\|g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\geq\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}-C\|g\|_{L^{2}(\Omega)}^{2},$

where $\kappa=\min\\{2a^{3},(1-M)a^{3}\\}$, which finishes the proof. ∎

With the above lemma, Theorem 1 can be proved in the same way as [6, Theorem
1].

##  Appendix A Transport equation with geometric correction in half-space

Consider the equation in $(\eta,\phi)\in\mathbb{R}_{+}\times[-\pi,\pi)$:

$\displaystyle\sin\phi\partial_{\eta}f+F(\eta)\cos\phi\partial_{\phi}f=H(\eta,\phi),$
(A.1) $\displaystyle f(0,\phi)=h(\phi),\quad\text{for }\sin\phi>0.$ (A.2)

The following lemma was proved in [10, Pages 1512-1514].

######  Lemma 20.

There exists a solution to the above problem given by

$\displaystyle f(\eta,\phi)=\mathcal{A}h(\phi)+\mathcal{T}H(\eta,\phi),$ (A.3)

with $\mathcal{A}$, $\mathcal{H}$ are defined in the following: for
$\sin\phi>0$,

$\displaystyle\mathcal{A}h(\phi)=h(\phi^{\prime}(\phi,\eta,0))e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi},\quad\mathcal{T}H(\eta,\phi)=\int_{0}^{\eta}\frac{h(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi,$
(A.4)

for $\sin\phi<0$ and $|E(\phi,\eta)|\leq e^{-V(\varepsilon;\infty)}$,

$\displaystyle\mathcal{A}h(\phi)$
$\displaystyle=0,\quad\mathcal{T}H(\eta,\phi)=\int_{\eta}^{\infty}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi,$
(A.5)

and for $\sin\phi<0$ and $|E(\phi,\eta)|\geq e^{-V(\varepsilon;\infty)}$,

$\displaystyle\mathcal{A}h(\phi)=h(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi},$
$\displaystyle\mathcal{T}H(\eta,\phi)=\int_{0}^{\eta_{+}}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\qquad\qquad\quad+\int_{\eta}^{\eta_{+}}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$
(A.6)

Moreover, $\mathcal{A}$ and $\mathcal{H}$ satisfies

$\displaystyle\|e^{\beta\eta}\mathcal{A}h\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))}\leq\|h\|_{L^{\infty}((0,\pi))},\quad\text{for
any }0\leq\beta\leq 1,$ (A.7)
$\displaystyle\|e^{\beta\eta}\mathcal{T}H\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))}\leq
C\|e^{\beta\eta}H\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))},\quad\text{for
any }0\leq\beta\leq\tfrac{1}{2}.$ (A.8)

##  Appendix B Existence for the linearized Milne problem without geometric
correction

Consider the following linear Milne problem on the half-line:

$\displaystyle\partial_{\eta}G+\langle\varphi-4h^{3}G\rangle=S_{1},$ (B.1)
$\displaystyle\sin\phi\partial_{\eta}\varphi+\varphi-4h^{3}G=S_{2},$ (B.2)

with boundary conditions

$\displaystyle
G(0)=G_{b},\quad\varphi(0,\phi)=\varphi_{b}(\phi),\quad\text{for }\sin\phi>0.$
(B.4)

The following theorem was proved in [6, Theorem 2].

######  Theorem 21.

Assume $h$ satisfies the spectral assumption (A). Assume $S_{1}=S_{1}(\eta)$,
$S_{2}S_{2}(\eta,\phi)$ satisfy $|S_{1}(\eta)|,|S_{2}(\eta,\phi)|\leq
C_{S}e^{-\beta\eta}$ for any $\eta\in\mathbb{R}_{+}$, $\phi\in[-\pi,\pi)$.
Then there exists a unique bounded solution $(G,\varphi)\in L^{2}_{\rm
loc}(\mathbb{R}_{+})\times L^{2}(\mathbb{R}_{+}\times[-\pi,\pi))$ to system
(B.1)-(B.2) with boundary conditions (B.4). Moreover there exists a constant
$G_{\infty}\in\mathbb{R}$ such that

$\displaystyle|G(\eta)-G_{\infty}|\leq
Ce^{-\beta\eta},\quad|\varphi(\eta,\phi)-4h^{3}_{\infty}G_{\infty}|\leq
Ce^{-\beta\eta},\quad\text{for any
}\eta\in\mathbb{R}_{+},~{}\phi\in[-\pi,\pi),$ (B.5)

where $C>0$ is a positive constant depending on $\beta$ and
$\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi$.

The theorem was proved in [6] by first showing the existence on bounded
interval and then using uniform estimate to extend the solution to the half-
line. Consider system (B.1)-(B.2) on the interval $\eta\in[0,B]$ with the
following boundary conditions at $\eta=B$,

$\displaystyle\partial_{\eta}G(B)=0,\quad\varphi(B,\phi)=\varphi(B,-\phi),\quad\text{for
}\sin\phi>0.$ (B.6)

The following theorem holds.

######  Theorem 22.

Let the assumptions of Theorem 22 hold. Then there exists a unique solution
$(G,\varphi)\in C^{2}([0,B])\times C^{1}([0,B]\times[-\pi,\pi))$ to system
(B.1)-(B.2) with boundary condtions (B.4)-(B.6).

## References

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#  Non-Standard Interactions of Supernova Neutrinos
and Mass Ordering Ambiguity at DUNE

Sudip Jana<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Yago Porto
<EMAIL_ADDRESS>Centro de Ciências Naturais e Humanas, Universidade
Federal do ABC, 09210-170, Santo André, SP, Brazil

###### Abstract

We show that non-standard neutrino interactions (NSI) can notably modify the
pattern of resonant flavor conversion of neutrinos within supernovae and
significantly impact the neutronization burst signal in forthcoming
experiments such as the Deep Underground Neutrino Experiment (DUNE). The
presence of NSI can invert the energy levels of neutrino matter eigenstates
and even induce a new resonance in the inner parts close to the proto-neutron
star. We demonstrate how DUNE can use these new configurations of energy
levels to have sensitivity to NSIs down to $\mathcal{O}(0.1)$. We also
elucidate how the effect may result in a puzzling confusion of normal and
inverted mass orderings by highlighting the emergence or vanishing of the
neutronization peak, which distinguishes between the two mass orderings.
Potential implications are analyzed thoroughly.

_Introduction_.– In recent decades, extensive efforts and data from solar,
atmospheric, reactor, and accelerator neutrino experiments have provided
robust evidence for neutrino oscillations, indicating the presence of neutrino
masses and mixing. However, the origin of these phenomena remains
unestablished, leaving room for potential new physics, particularly in the
form of non-standard neutrino interactions (NSIs). First introduced by
Wolfenstein in 1978 Wolfenstein:1977ue , NSIs have been the subject of intense
scrutiny since then, providing an avenue for exploring new aspects of neutrino
physics. These interactions involve higher-dimensional operators with
neutrinos and matter, as represented by the equations:

$\displaystyle\mathcal{L}_{\mathrm{NC}}=-2\sqrt{2}G_{F}\sum_{f,P,\alpha,\beta}\varepsilon_{\alpha\beta}^{f,P}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta}\right)\left(\bar{f}\gamma_{\mu}Pf\right)$
(1)
$\displaystyle\mathcal{L}_{\mathrm{CC}}=-2\sqrt{2}G_{F}\sum_{f,P,\alpha,\beta}\varepsilon_{\alpha\beta}^{f,P}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\ell_{\beta}\right)\left(\bar{f}\gamma_{\mu}Pf^{\prime}\right)$

where $\varepsilon$ represents the strength of NSI relative to the weak scale,
$P\in{P_{L},P_{R}}$ indicates chirality projection operators and the sum is
over matter fermions $f,f^{\prime}\in{e,u,d}$. Such NSIs modify the matter
potential experienced by neutrinos, adding substantial intricacy to the
determination of neutrino oscillation parameters. For instance, the existence
of NSI introduces ambiguity in the determination of $\theta_{12}$ from the
solar neutrino data Miranda:2004nb . Furthermore, NSI effects have been shown
to alleviate the tension between solar and KamLAND data, as it flattens the
solar neutrino spectrum at high energies ($>3$ MeV) and generates larger day-
night asymmetry Maltoni:2015kca . In this study, we advocate for the
utilization of supernova neutrinos to investigate the influence of these NSI
on the precise determination of neutrino oscillation parameters.

Figure 1: A simplified picture of flavor conversions of supernova neutrinos in
presence of NSI.

During a supernova (SN) explosion, as the progenitor’s core collapses to form
a neutron star, about $10^{53}$ erg of gravitational binding energy is
released in the form of neutrinos Colgate:1966ax ; Arnett:1966 ; Bethe:1985sox
; Wilson:1985 ; Janka:2017vlw . Originating in the SN core with energies in
the tens of MeV, these neutrinos travel through the stellar mantle and
envelope, undergoing modifications in mixing and oscillations through
neutrino-matter interactions111Matter effects can also influence the evolution
of high-energy astrophysical neutrinos with energies greater than 100 TeV
Dev:2023znd . Wolfenstein:1977ue ; Mikheyev:1985zog ; Mikheev:1986wj ;
Mikheev:1986if . The consequences of neutrino propagation in the SN-dense
environment offer significant potential for probing new physics, and these
effects are observable by flavor-sensitive Earth-based detectors Valle:1987gv
; Nunokawa:1997ct ; Nunokawa:1998vh ; Esteban-Pretel:2007zkv ;
deGouvea:2019goq ; Tang:2020pkp ; Jana:2022tsa ; Jana:2023ufy ;
dosSantos:2023skk ; Bendahman:2023hjj . A galactic SN, a rare event in the
Milky Way occurring once or twice per century, was last observed in 1987 from
the Large Magellanic Cloud, 51 Kpc away Tammann:1994ev ; Kamiokande-II:1987idp
; Bionta:1987qt ; Alekseev:1988gp . Neutrino detectors recorded about 25
$\bar{\nu}_{e}$ events during that event, significantly advancing our
understanding of core-collapse processes and neutrino emission Olsen:2021uvt ;
Li:2023ulf ; DedinNeto:2023hhp ; Fiorillo:2023frv . Future detectors,
especially the Deep Underground Neutrino Experiment (DUNE), are anticipated to
capture hundreds of thousands of events in various channels during a galactic
supernova IceCube:2011cwc ; DarkSide20k:2020ymr ; Hyper-Kamiokande:2021frf ;
KM3NeT:2021moe ; JUNO:2023dnp ; DUNE:2020zfm , with DUNE’s unique capability
to detect a clean $\nu_{e}$ signal in the critical first 40 ms after core
bounce, known as the neutronization burst phase DUNE:2020zfm ; Cuesta:2023nnt
; Zhu:2018rwc . In this article, we investigate the features of neutrino
emission in the neutronization burst phase, elucidate the anticipated resonant
flavor conversion within the stellar envelope triggered by NSI [cf.Fig 1] and
discuss the crucial role of DUNE in exploring new physics impacting supernova
neutrinos.

_Dynamics of Resonant Flavor Conversions_.–

Figure 2: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right). For each panel, neutrino lines are shown for
positive $V_{e}$, while antineutrinos are plotted with negative $V_{e}$. The
location of the MSW resonances is indicated by the letters $L$ and $H$. Solid
lines represent the effective matter eigenstates, while the dashed lines
follow the track of flavor states. Matter eigenstates mix with different
flavors at different locations; this is shown by the flavor tags in red, blue,
and green colors.

Understanding the flavor composition of SN neutrinos reaching the Earth during
the neutronization burst phase requires monitoring how the initial fluxes
generated in the core evolve while traveling outward within the star (see
Appendix for details). In the dense environment where neutrinos propagate
inside the star, vacuum oscillations are suppressed. However, the density
variation in the way out of the SN instigates a flavor evolution dictated by a
time (or space) dependent Hamiltonian that can trigger Mikheyev-Smirnov-
Wolfenstein (MSW) resonant flavor conversion in specific layers of the matter
profile Wolfenstein:1977ue ; Mikheyev:1985zog ; Mikheev:1986wj ;
Mikheev:1986if ; Mikheev:1987jp .

Assuming that neutrinos propagate radially outwards, the evolution equation
for the flavor state $\nu=(\nu_{e},\nu_{\mu},\nu_{\tau})^{T}$ is

$i\frac{d}{dr}\nu=\mathcal{H}\nu.$ (2)

$\mathcal{H}$ is the Hamiltonian in the flavor basis given by

$\mathcal{H}=\frac{1}{2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+V_{e}\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$ (3)

where $E$ and $U$ denote the neutrino energy and the Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) matrix. The matter potential due to charged current interactions
can be expressed as

$V_{e}=\sqrt{2}G_{F}\frac{\rho}{m_{N}}Y_{e},$ (4)

where $Y_{e}$ is the electron number fraction and $\rho$ is the matter density
(see Fig. 7 in the Appendix for the profile of $\rho$ and $Y_{e}$). The
antineutrino Hamiltonian is similar to $\mathcal{H}$, with the only difference
being that the matter potential inverts sign $\bar{V}_{e}=-V_{e}$.

In dense matter, $\rho>10^{5}$ $\text{g/cm}^{3}$, $V_{e}$ is much bigger than
all other matrix elements in Eq. 3, suppressing mixing and making $\nu_{e}$
the heaviest effective eigenstate in matter, $\nu_{e}\approx\nu_{3}^{m}$ for
normal ordering (NO) and $\nu_{e}\approx\nu_{2}^{m}$ for inverted ordering
(IO), the superscript $m$ denotes an effective eigenstate in matter. In a
vacuum, however, $\nu_{e}$ is mostly associated with $\nu_{1}$. Therefore, as
matter density decreases, so does $V_{e}$, and $\nu_{e}$ will transit from
$\nu_{3}^{m}$($\nu_{2}^{m})$ to $\nu_{1}^{m}$. In doing so, it has to cross
energy levels twice for NO
($\nu_{3}^{m}\rightarrow\nu_{2}^{m}\rightarrow\nu_{1}^{m}$) and once for IO
($\nu_{2}^{m}\rightarrow\nu_{1}^{m}$). The level crossing diagram is shown in
Fig. 2. The red dashed line follows the evolution of the electron flavor,
$\nu_{e}$ on the right of $V_{e}=0$ and $\bar{\nu}_{e}$ on the left. Note that
$\bar{\nu}_{e}\approx\bar{\nu}_{1}^{m}$ in the inner regions, and there is no
level crossing for antineutrinos in NO, while there is one level crossing in
IO: $\bar{\nu}_{3}^{m}\rightarrow\bar{\nu}_{1}^{m}$. While $\nu_{e}$ and
$\bar{\nu}_{e}$ are effective eigenstates in the SN environment, $\nu_{\mu}$,
$\nu_{\tau}$, and their antiparticles are not effective eigenstates due to
near maximal vacuum mixing. In this situation, it is convenient to diagonalize
the $\nu_{\mu}-\nu_{\tau}$ subspace and work with the effective eigenstates
$\nu^{\prime}_{\mu}$, $\nu^{\prime}_{\tau}$, and their antiparticles. The
primed states have constant energy levels as a function of $V_{e}$, as shown
in Fig. 2.

The crossing $\nu_{e}\leftrightarrow\nu^{\prime}_{\tau}$ is called
$H-$resonance and occurs when

$V_{e}(\rho_{H})\approx\frac{\Delta m_{31}^{2}}{2E}\cos\theta_{13},$ (5)

which corresponds to higher densities compared to the $L-$resonance,
$\nu_{e}\leftrightarrow\nu^{\prime}_{\mu}$, that happens when

$V_{e}(\rho_{L})\approx\frac{\Delta m_{21}^{2}}{2E}\cos\theta_{12}.$ (6)

Note that Eq. 5 is satisfied for neutrinos in NO and for antineutrinos in IO
($\Delta m^{2}_{31}=-|\Delta m^{2}_{31}|$ and $V_{e}$ is negative) while Eq. 6
is satisfied only for neutrinos in both orderings. Using the profile in Fig.
7, and oscillation parameters from Esteban:2018ppq , the $H-$resonance happens
at $r\sim 10^{4}$ Km ($\rho_{H}$ in the range $10^{3}-10^{4}$
$\text{g/cm}^{3}$) and the $L-$resonance at $r\sim 10^{5}$ Km ($\rho_{L}\sim
1-10$ $\text{g/cm}^{3}$). Efficient resonant conversion only happens if
neutrinos cross resonance layers adiabatically Dighe:1999bi . We will assume
perfect adiabaticity of the $H-$ and $L-$resonances so that transitions
between distinct energy eigenstates are negligible. Moreover, the produced
flux during the neutronization burst phase is mostly
$\nu_{e}\approx\nu_{3}^{m}(\nu_{2}^{m})$ for NO (IO). Due to adiabaticity, the
initial eigenstates are preserved and reach vacuum as $\nu_{3}(\nu_{2})$.
Consequently, for NO only $|U_{e3}|^{2}\approx 0.02$ of the initial $\nu_{e}$
flux, $F_{e}^{i}$, survives and reaches Earth, and some amount of $\nu_{e}$,
($1-|U_{e3}|^{2})F_{x}^{i}\approx 0.98F_{x}^{i}$, comes from the conversion of
non-electron flavors, $F_{\mu}^{i}=F_{\tau}^{i}=F_{x}^{i}$:

$F_{e}^{NO}=|U_{e3}|^{2}F_{e}^{i}+(1-|U_{e3}|^{2})F_{x}^{i}\approx
0.02F_{e}^{i}+0.98F_{x}^{i}.$ (7)

Similarly for IO,

$F_{e}^{IO}=|U_{e2}|^{2}F_{e}^{i}+(1-|U_{e2}|^{2})F_{x}^{i}\approx
0.3F_{e}^{i}+0.7F_{x}^{i}.$ (8)

Due to $F_{e}^{i}$ being approximately ten times $F_{x}^{i}$ at the
neutronization peak, the $\nu_{e}$ flux is greater in the Inverted Ordering
(IO) than in the Normal Ordering (NO) during the initial 20 milliseconds of
neutrino emission. This disparity enhances the visibility of the peak in
neutrino detectors for the IO scenario. In the following sections, we will
analyze signals derived from flux equations (7) and (8) in DUNE and discuss
the presence or absence of the peak as a discriminative criterion between the
two mass orderings.

Figure 3: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right) with the inclusion of
$\varepsilon_{\tau\tau}^{d}>0.33$. See text for details.

_Effect of NSI on the Energy Levels_.– The introduction of NSIs has a
pronounced impact on the configuration of energy levels and the flavor
composition of SN neutrinos detected on Earth. To demonstrate this effect, an
additional energy term involving NSI parameters must be incorporated into the
Hamiltonian mentioned in Eq. 3. Our focus here is on NSI specifically
influencing $\nu_{\tau}$, potentially originating from interactions with
electrons, $u$-quarks, and $d$-quarks. Recent systematic investigations
indicate the feasibility of obtaining substantial values for
$\varepsilon_{\tau\tau}$ in various radiative neutrino mass models
Babu:2019mfe . It is important to note that the approach to $\nu_{\mu}$ NSI is
analogous and yields numerically equivalent results.

The effective strength of the NSI parameter
$\varepsilon^{\text{eff}}_{\tau\tau}$ can be expressed in relation to
interactions involving electrons, $u$-quarks, and $d$-quarks as follows:

$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{e}+(2\varepsilon_{\tau\tau}^{u}+\varepsilon_{\tau\tau}^{d})\frac{n_{p}}{n_{e}}+(\varepsilon_{\tau\tau}^{u}+2\varepsilon_{\tau\tau}^{d})\frac{n_{n}}{n_{e}}.$
(9)

In various models of neutrino mass, new interactions emerge, giving rise to
diverse NSI scenarios where $\varepsilon_{\tau\tau}^{u}$,
$\varepsilon_{\tau\tau}^{d}$, or $\varepsilon_{\tau\tau}^{e}$ may be
exclusively generated, or in certain instances, combinations of two NSIs or
all three NSIs. For instance, a scalar leptoquark with Standard Model gauge
quantum numbers ($3,2,1/6$), denoted as $\tilde{R_{2}}$, can induce only
$\varepsilon_{\tau\tau}^{d}$, while the leptoquark ${R_{2}}$ with charges
($3,2,7/6$) exclusively leads to $\varepsilon_{\tau\tau}^{u}$. On the other
hand, the leptoquark ${S_{3}}$ with charges ($\bar{3},3,1/3$) can give rise to
both $\varepsilon_{\tau\tau}^{d}$ and $\varepsilon_{\tau\tau}^{u}$. Further
elaboration on these scenarios can be found in the detailed discussion
provided in reference Babu:2019mfe . To optimize the signal, we exclusively
consider quark NSIs here. Although leptonic NSIs may have some impact, they
are expected to be less significant compared to quark NSIs, and therefore,
they do not affect the phenomenology we are concentrating on. If only
$\varepsilon_{\tau\tau}^{d}$ exhibits a non-zero value, and under the
condition of charge neutrality ($n_{p}=n_{e}$), we obtain:

$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right).$
(10)

Considering non-zero values of $\varepsilon_{\tau\tau}^{u}$, we obtain:

$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{u}\left(\frac{1+Y_{e}}{Y_{e}}\right).$
(11)

Now, we focus on the region of the SN matter profile where $H-$ and $L-$
resonance occur, specifically for $r>10^{4}$ Km. In this particular zone,
$Y_{e}=0.5$ (see Fig. 7 in the Appendix). The existence of either
$\varepsilon_{\tau\tau}^{d}$ or $\varepsilon_{\tau\tau}^{u}>0.33$ indicates
that the effective matter potential for $\nu_{\tau}$ surpasses that of
$\nu_{e}$, thereby modifying the dynamics of flavor conversion. The
Hamiltonian in the presence of NSI is expressed as follows:

$\mathcal{H}^{NSI}=\frac{1}{2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+V_{e}\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&3\varepsilon^{u,d}_{\tau\tau}\end{array}\right).$ (12)

For $\varepsilon^{u,d}_{\tau\tau}>0.33$, $\nu_{\tau}$ is the heaviest
eigenstate in matter, $\nu_{\tau}\approx\nu_{3}^{m}(\nu_{2}^{m})$ for NO (IO).
Hence, the main difference from the standard case is that $\nu_{e}$ starts as
the second heaviest eigenstate, $\nu_{e}\approx\nu_{2}^{m}(\nu_{1}^{m})$ for
NO (IO), see Fig. 3. For NO, $\nu_{e}$ crosses level once through a
$L-$resonance, $\nu_{2}^{m}\rightarrow\nu_{1}^{m}$, at densities given by Eq.
6. While for IO, $\nu_{e}$ is already produced as $\nu_{1}^{m}$ and reaches
vacuum mostly as $\nu_{1}$ without crossing levels. For antineutrinos,
$\bar{\nu}_{e}\approx\bar{\nu}_{2}^{m}(\bar{\nu}_{1}^{m})$, and crosses levels
only for NO, via $H-$resonance. In this case, however, the potential of
$\nu_{\tau}$, $V_{\tau}=\varepsilon_{\tau\tau}V_{e}$, is higher than $V_{e}$,
so the difference $V_{e}-V_{\tau}$ is negative. The resonance condition is
found by modifying Eq. 5 with
$V_{e}\rightarrow(1-\varepsilon_{\tau\tau})V_{e}$. Neutrinos fail to satisfy
this new condition in NO. However, antineutrinos in NO can fulfill this
condition since their potentials exhibit an inverted sign, denoted as
$\bar{V_{e}}=-V_{e}$. Furthermore, the eigenstates are not represented in the
primed basis, as the potential for $\nu_{\tau}$ significantly surpasses the
mixing terms, distinctly separating $\nu_{\mu}$ and $\nu_{\tau}$.

Assuming perfect adiabaticity of $H-$ and $L-$ resonances, we find the final
flux for both orderings:

$F_{e}^{NO}(\varepsilon_{\tau\tau}^{u,d}>0.33)=|U_{e2}|^{2}F_{e}^{i}+(1-|U_{e2}|^{2})F_{x}^{i}\approx
0.3F_{e}^{i}+0.7F_{x}^{i},$ (13)

and,

$F_{e}^{IO}(\varepsilon_{\tau\tau}^{u,d}>0.33)=|U_{e1}|^{2}F_{e}^{i}+(1-|U_{e1}|^{2})F_{x}^{i}\approx
0.7F_{e}^{i}+0.3F_{x}^{i}.$ (14)

It is noteworthy that Eq.13 is equivalent to Eq.8 since, in both instances,
$\nu_{e}$ is generated as $\nu_{2}^{m}$. Consequently, the neutronization
burst phase in the NO scenario with NSI and the standard IO scenario without
NSI are indistinguishable. In the next sections, we analyze the signals from
Eq. 13 and Eq. 14 at DUNE, emphasizing potential confusion in discerning
between the two mass orderings in the presence of NSI.

_New Resonances triggered by NSI_.–

Figure 4: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right) with the inclusion of
$0.1<\varepsilon^{d}_{\tau\tau}<0.33$ and
$\varepsilon^{d}_{e\tau}>10^{-5}$.The MSW resonance locations are marked by
the letters $L$ and $H$, while the NSI-triggered resonance location is marked
by the letter $C$.

We will now explore scenarios in which NSI can induce an additional resonance,
designated as the $C$-resonance, alongside the existing $L-$ and $H-$
resonances. While examining NSI within the range
$0.1<\varepsilon_{\tau\tau}^{u,d}<0.33$, we find that it influences the flavor
evolution of neutrinos during the neutronization burst phase. Despite such
values of $\varepsilon_{\tau\tau}^{u,d}$ not modifying the energy level
pattern in the region $r>10^{3}$ Km where $L-$ and $H-$resonances occur
($Y_{e}\approx 0.5$), it does alter the energy levels inside the iron core
($r<10^{3}$ Km) where neutrinos are produced ($0.2<Y_{e}<0.5$). In this
configuration, the $C-$ resonance emerges inside the iron core, as illustrated
in Fig. 4.

Figure 5: Expected number of $\nu_{e}$ events per bin of $5$ ms at DUNE in the
time interval from $-5$ to $40$ ms corresponding to the neutronization burst
phase for NO (left) and IO (right). $0$ ms represents the time of core bounce.
Blue lines are computed from fluxes in Eqs. 7 and 8, assuming the distance
between the SN and Earth to be $10$ Kpc. Green lines are computed from fluxes
modified by the presence of $\varepsilon_{\tau\tau}^{d}>0.33$ in Eqs. 13 and
14. The similarity between the green line on the left panel and the blue line
on the right panel displays the degeneracy between the NO$+$NSI and standard
IO cases. Red lines are computed from fluxes modified by the presence of
$0.1<\varepsilon_{\tau\tau}<0.33$ and $10^{-6}<\varepsilon_{e\tau}<10^{-5}$.
For $\varepsilon_{e\tau}$ in such interval, counting rates are intermediate
between the NSI case with $\varepsilon_{\tau\tau}>0.33$ and the standard
scenario. For the totally adiabatic $C-$resonance
($\varepsilon_{e\tau}>10^{-5}$), results are degenerate with the ones for
$\varepsilon_{\tau\tau}>0.33$.

In order to examine the emergence of the $C-$resonance, we rewrite Eq. 12 by
incorporating the expression for $\varepsilon_{\tau\tau}$ from Eq. 10 for
$d$-quark NSI. Here, we concentrate exclusively on the matter potential term,
which dominates over the vacuum term in regions of high core densities:

$\mathcal{H}^{NSI}\approx V_{e}\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\end{array}\right).$
(15)

Note that our focus here is on $d$-quark NSI. Nevertheless, the qualitative
outcomes remain similar for $u$-quark NSI due to the resemblance between Eq.
10 and Eq. 11. In the subsequent section, we analyze the consequential effects
by concurrently considering NSIs involving both $d$-quark and $u$-quark. Using
the $Y_{e}$ values from Fig. 7 (See Appendix for details), a level crossing
occurs within the range of $50$ to $1000$ Km for
$0.1<\varepsilon^{d}_{\tau\tau}<0.33$ as $Y_{e}$ transitions from $0.2$ to
$0.5$. Prior to the crossing point,
$\varepsilon_{\tau\tau}^{d}(2-Y_{e})/Y_{e}>1$, rendering $\nu_{\tau}$ as the
heaviest state, and $\nu_{e}$ as the second heaviest:
$\nu_{e}\approx\nu_{2}^{m}(\nu_{1}^{m})$ for NO (IO). Subsequent to the
crossing point, the scenario inverts with
$\varepsilon_{\tau\tau}^{d}(2-Y_{e})/Y_{e}<1$, causing the energy levels of
$\nu_{e}$ and $\nu_{\tau}$ to intersect, as depicted in Fig. 4. This crossing
can result in resonant conversion if we introduce off-diagonal
NSI222Significant flavor off-diagonal NSI can resolve discrepancies in the
determination of the standard CP-phase $\delta_{CP}$ between the NO$\nu$A and
T2K long-baseline accelerator experiments Chatterjee:2020kkm ; Denton:2020uda
. $\varepsilon_{e\tau}$ and couple $\nu_{e}$ and $\nu_{\tau}$ states:

$\mathcal{H}^{NSI}\approx
V_{e}\left(\begin{array}[]{ccc}1&0&\varepsilon_{e\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\\\
0&0&0\\\
\varepsilon_{e\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)&0&\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\end{array}\right).$
(16)

In such a scenario, the resonance condition can be expressed as

$\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)=1,$ (17)

and can be satisfied by both neutrinos and antineutrinos simultaneously.
Furthermore, Eq. 17 is independent of matter density, energy, and mass
orderings. The adiabaticity condition is now formulated and can be expressed
as

$\gamma_{C}=\left|\frac{4\left(\mathcal{H}_{e\tau}^{NSI}\right)^{2}}{\dot{\mathcal{H}}_{\tau\tau}^{NSI}-\dot{\mathcal{H}}_{ee}^{NSI}}\right|\approx\left|\frac{16V_{e}(\varepsilon_{e\tau}^{d})^{2}}{Y_{e}\dot{Y_{e}}(1+\varepsilon_{\tau\tau}^{d})^{3}}\right|>1.$
(18)

Eq. 18 is satisfied for very small values of off-diagonal NSI:
$\varepsilon_{e\tau}>10^{-5}$. The resonance is partially adiabatic for
$\varepsilon_{e\tau}$ values within the range of $10^{-6}$ to $10^{-5}$. In
the limit of total adiabaticity, the produced $\nu_{e}$ reaches vacuum as
$\nu_{2}(\nu_{1})$ for NO (IO). The scenario is analogous to the one stated in
the last section by Eq. 13 and Eq. 14. When adiabaticity is totally violated,
the results are identical to the standard case in Eq. 7 and Eq. 8. Partial
adiabaticity provides fluxes, which are intermediary between the standard
scenario and the $\varepsilon^{d}_{\tau\tau}>0.33$ scenario. In the next
section, we analyze these signals in DUNE.

Figure 6: Projected sensitivities for NSI $\varepsilon_{\tau\tau}$ at DUNE.
For the totally adiabatic $C-$resonance, we consider
$\varepsilon_{e\tau}>10^{-5}$. Global-fit bound on NSI, as shown by the
hatched region, is adopted from Ref. Coloma:2023ixt .

We want to stress here that the search for $\varepsilon_{\tau\tau}^{u,d}$
during the neutronization burst phase is limited to values of
$\mathcal{O}(0.1)$. Conversely, in later phases of SN neutrino emission, when
$Y_{e}$ can attain values below $\mathcal{O}(0.1)$, smaller values of
$\varepsilon_{\tau\tau}^{u,d}$ could potentially alter observables. Note that
for $Y_{e}\ll 1$, Eq. 10 and Eq. 11 transforms as:

$\varepsilon_{\tau\tau}^{eff}\approx
2\frac{\varepsilon_{\tau\tau}^{d}}{Y_{e}}\hskip 14.22636pt\text{and}\hskip
14.22636pt\varepsilon_{\tau\tau}^{eff}\approx\frac{\varepsilon_{\tau\tau}^{u}}{Y_{e}}.$
(19)

Consequently, $\varepsilon_{\tau\tau}^{u,d}$ can be as small as $Y_{e}\sim
O(0.01)$, while maintaining the $\varepsilon^{eff}_{\tau\tau}$ value of order
one necessary to satisfy the resonance criterion in Eq. 17. Therefore,
sensitivity to $\varepsilon^{u,d}_{\tau\tau}$ can be improved to
$\mathcal{O}(0.01)$ by further investigating flavor conversion that occurs
during other phases of neutrino emission.

Signal analysis at DUNE.– Here, we investigate the expected SN neutrino signal
spectrum at DUNE under both the standard scenario and the consideration of NSI
effects. The graphical representation of the analysis, depicted in Fig. 5,
illustrates the projected number of $\nu_{e}$ events per 5 ms interval within
the time range of -5 to 40 ms (with 0 ms denoting the moment of core bounce).
The scenarios for Normal Ordering (NO) and Inverted Ordering (IO) are
presented on the left and right sides, respectively, assuming a distance of 10
Kpc between the supernova (SN) and Earth. For detailed information regarding
the computation of DUNE event spectra, see the Appendix. The initial neutrino
fluxes ($F_{e}^{i}$ and $F_{x}^{i}$) are taken from a simulation of a 15 solar
mass progenitor garching , as outlined in deGouvea:2019goq ; Jana:2022tsa .
The standard case, represented by the blue lines in Fig. 5 [cf. Eq. 7 and Eq.
8], highlights a notable feature—the visibility of the neutronization peak,
which distinguishes between the two mass orderings. The green lines are
computed from Eq. 13 and Eq. 14 which includes NSI
($\varepsilon_{\tau\tau}^{d}>0.33$). Crucially, as evident from Eq. 8 and Eq.
13, along with Fig. 5, the introduction of NSI leads to a scenario where
perfect confusion arises between the two mass orderings (green lines on the
left panel and blue lines on the right panel are identical).

DUNE can distinguish between the standard case and signals characterized by
$\varepsilon^{d}_{\tau\tau}>0.33$, as depicted in each panel of Fig. 5. We
analyze it using the $\chi^{2}$ estimator,

$\chi^{2}=\min_{\xi}\left(\sum_{i=1}^{n}2\left[(1+\xi)F_{i}-D_{i}+D_{i}\ln\left(\frac{D_{i}}{(1+\xi)F_{i}}\right)\right]\right).$
(20)

$F_{i}$ and $D_{i}$ are the number of $\nu_{e}$ events in the $i$-th time bin
for $\varepsilon_{\tau\tau}^{d}>0.33$ and $\varepsilon_{\tau\tau}^{d}=0$,
respectively. The parameter $\xi$ is allowed to vary in the range $-1<\xi\leq
1000$ with no penalty and we select the minimum to make our derived
sensitivity as much as possible independent of the overall normalization. We
find for NO,

$\chi^{2}(NO)\approx 27,$ (21)

and, for IO,

$\chi^{2}(IO)\approx 11.$ (22)

Notably, $\chi^{2}(NO)$ surpasses $\chi^{2}(IO)$ by a considerable margin.
This discrepancy is primarily attributed to the more pronounced
differentiation between the green and blue lines in Fig. 5 for NO, arising
from the presence (green) or absence (blue) of the peak around $\sim 5$ ms.
Conversely, for IO, both lines incorporate the peak, and the blue line could
mimic the green one even without NSI, contingent on different spectral
parameters of the supernova. We want to stress that the green lines in Fig. 5
also represent the case of NSI in the range
$0.1<\varepsilon_{\tau\tau}^{d}<0.33$ with adiabatic $C$-resonance
($\varepsilon_{e\tau}^{d}>10^{-5}$). Partial adiabaticity of the $C$-resonance
($10^{-6}<\varepsilon_{e\tau}^{d}<10^{-5}$) provides counting rates between
the standard scenario and the $\varepsilon^{d}_{\tau\tau}>0.33$ scenario; see
red lines in Fig. 5. We exclusively analyze scenarios with $d$-quark NSI.
However, for completeness, Fig. 6 displays projected sensitivities considering
the presence of both $d$-quark and $u$-quark NSIs.

NSI affecting the flavor evolution of SN neutrinos can also be studied in the
electron antineutrino ($\bar{\nu}_{e}$) channel using other future experiments
such as Hyper-Kanmiokande Hyper-Kamiokande:2021frf  and JUNO JUNO:2023dnp .
Although these experiments will certainly provide more statistics than DUNE,
their observations will lack some specific features of the time signal that
are present in the $\nu_{e}$ channel (such as the peak at $5$ ms) that can be
crucial in diagnosing the presence of NSI. The quark NSI we discussed, which
is capable of inducing resonant flavor conversion of supernova neutrinos, may
also be subject to complementary tests. For instance, it could produce a
Glashow-like resonance feature detectable by neutrino telescopes Babu:2022fje
; Babu:2019vff ; Huang:2021mki  or be examined in future collider experiments,
potentially resolving existing degeneracies Babu:2020nna .

Conclusions.– We have shown that the NSIs have a profound impact on supernova
neutrino flavor conversion, significantly impacting the neutronization burst
signals. The presence of NSI can invert the energy levels of neutrino matter
eigenstates and even induce a new resonance in the inner parts close to the
proto-neutron star. We showcase how the forthcoming experiments, such as DUNE,
can exploit these altered energy level configurations to achieve sensitivity
to NSIs at the order of $\mathcal{O}(0.1)$. Furthermore, we elaborate on how
this phenomenon may lead to a puzzling ambiguity between normal and inverted
mass orderings, and we thoroughly analyze its potential implications.

Acknowledgments.– We are grateful to André de Gouvêa for an illuminating
discussion and email correspondence. We also thank Pedro A. N. Machado and
Manibrata Sen for useful discussions. We wish to acknowledge the Center for
Theoretical Underground Physics and Related Areas (CETUP*) and the Institute
for Underground Science at SURF for hospitality and for providing a
stimulating environment. The work of YP was supported by the São Paulo
Research Foundation (FAPESP) Grant No. 2023/10734-3 and 2023/01467-1 and by
the National Council for Scientific and Technological Development (CNPq) Grant
No. 151168/2023-7. S.J. thanks Centro de Ciências Naturais e Humanas at
Universidade Federal do ABC and ICTP South American Institute for Fundamental
Research for their warm hospitality during the completion of this work.

## Appendices

###  .1 Neutronization burst phase

Figure 7: Status of the matter density $\rho$ and and electron number fraction
$Y_{e}$ at $4.37$ ms after core bounce. Serpico:2011ir ; Tang:2020pkp .

The SN neutronization burst phase encompasses the period of $\sim 40$ ms after
core bounce when the $\nu_{e}$ luminosity dominates over all other flavors,
$\bar{\nu}_{e}$, $\nu_{x}$ and $\bar{\nu}_{x}$, with $x=\mu,\tau$ ($\nu_{e}$
luminosity can be $10-100$ times higher than luminosity in other flavors, see
deGouvea:2019goq ; Jana:2022tsa ). During the core collapse, the inner core is
compressed and reaches nuclear densities, while the matter falling above it
bounces back and launches a shock wave, which dissociates nuclei into its
component nucleons as it travels outwards. The capture of electrons in the
environment by the dissociated protons, $e^{-}+p\rightarrow n+\nu_{e}$, is
responsible for generating the large $\nu_{e}$ burst during the neutronization
burst phase. Theoretical uncertainties in the calculation of the
neutronization fluxes are believed to be as small and $10\%$ Serpico:2011ir ;
Wallace:2015xma ; OConnor:2018sti ; Kachelriess:2004ds  and uncertainties
related to neutrino-neutrino refraction Mirizzi:2015eza ; Capanema:2024hdm
can also be avoided. Therefore, the burst phase presents a great opportunity
to search for new physics with SN neutrinos.

Neutrinos are produced in the region $r<100$ Km, which is opaque due to very
high densities ($\rho>10^{11}$ $\text{g/cm}^{3}$), and, at $r\sim 100$ Km,
they start free streaming. The efficient electron capture at the production
region reduces its electron number fraction, $Y_{e}=n_{e}/(n_{p}+n_{n})$,
where $n_{e}$, $n_{p}$ and $n_{n}$ are the electron, proton, and neutron
number densities, respectively, to levels below the one found in the envelope
($Y_{e}=0.5$). Precisely, this difference between $Y_{e}$ in inner and outer
layers might produce observable consequences for non-zero NSI. Further details
can be seen in Fig. 7, where we plot $\rho$ and $Y_{e}$ at $\sim 5$ ms
Fischer:2009af ; Tang:2020pkp . This instant is representative of the matter
profile during the neutronization peak, the most distinct feature of the burst
phase.

###  .2 DUNE: technical details

The Deep Underground Neutrino Experiment (DUNE) will be composed of four-time
projection chambers, each with $10$ Kton of liquid argon, placed underground
in the Long-Baseline Neutrino Facility (LNBF) in South Dakota, United States
DUNE:2020zfm ; Cuesta:2023nnt . DUNE will primarily detect neutrinos with
energies of GeV and higher coming from a beam produced at Fermilab.
Nevertheless, DUNE is also sensitive to SN neutrinos in the range from about
$5$ MeV to tens of MeV via the charged-current interaction

$\nu_{e}+{}^{40}\mathrm{Ar}\rightarrow{}^{40}\mathrm{~{}K}^{*}+e^{-},$ (23)

that separates the electron flavor and has a much higher cross-section than
the elastic scattering,
$\nu_{e,\mu,\tau}+e^{-}\rightarrow\nu_{e,\mu,\tau}+e^{-}$, which is the
current main detection channel for $\nu_{e}$ Scholberg:2012id ; Zhu:2018rwc .
We compute the event spectrum for $\nu_{e}$ at DUNE as Capozzi:2018rzl

$\frac{dN_{\nu_{e}}}{dE_{r}}=\frac{N_{\mathrm{Ar}}}{4\pi R^{2}}\int
dE_{\nu_{e}}F_{\nu_{e}}\left(E_{\nu_{e}}\right)\sigma_{\nu_{e}+\mathrm{Ar}}\left(E_{\nu_{e}}\right)W\left(E_{r},E_{\nu_{e}}\right)$
(24)

where $N_{\mathrm{Ar}}$ is the number of target ${}^{40}\mathrm{Ar}$ nuclei in
the chambers, $R$ is the distance between the SN and the Earth, $F_{\nu_{e}}$
is the $\nu_{e}$ flux leaving the SN with energy $E_{\nu_{e}}$,
$\sigma_{\nu_{e}+\mathrm{Ar}}$ is the cross section for the interaction in Eq.
23 generated by MARLEY Gardiner:2018zfg , $W$ is the gaussian energy
resolution with
$\sigma_{E}/\text{MeV}=0.11\sqrt{E_{r}/\text{MeV}}+0.02E_{r}/\text{MeV}$ and
$E_{r}$ is the reconstructed electron energy.

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