$L^2$ diffusive expansion for neutron transport equation

1.   Introduction

1.1.   Problem formulation

We consider the steady neutron transport equation in a three-dimensional $C^3$ bounded domain (convex or non-convex) with in-flow boundary condition. In the spatial domain $\Omega\ni{x} = (x_1, x_2, x_3)$ and the velocity domain ${{\mathbb{S}}}^2\ni{w} = (w_1, w_2, w_3)$, the neutron density $u^{{\varepsilon}}({x}, {w})$ satisfies (0.1) with the Knudsen number $0 < {\varepsilon}\ll1$. We intend to study the asymptotic behavior of $u^{{\varepsilon}}$ as ${\varepsilon}{\rightarrow}0$.

Based on the flow direction, we can divide the boundary $\gamma: = \big\{({x}_0, {w}):\ {x}_0\in{\partial}\Omega, {w}\in{{\mathbb{S}}}^2\big\}$ into the incoming boundary $\gamma_-$, the outgoing boundary $\gamma_+$, and the grazing set $\gamma_0$ according to the sign of ${w}\cdot{n}({x}_0)$. In particular, the boundary condition of (0.1) is only given on $\gamma_{-}$.

1.2.   Normal chart near boundary

We follow the approach in ^[4,6] to define the geometric quantities, and the details can be found in Section 2.2. For smooth manifold ${\partial}\Omega$, there exists an orthogonal curvilinear coordinates system $(\iota_1, \iota_2)$ such that the coordinate lines coincide with the principal directions at any ${x}_0\in{\partial}\Omega$. Assume that ${\partial}\Omega$ is parameterized by ${\bf{r}} = {\bf{r}}(\iota_1, \iota_2)$. Let the vector length be ${L}_i: = \left\vert{{\partial}_{\iota_i}{\bf{r}}}\right\vert$ and let the unit vector be ${\varsigma}_i: = {L}_i^{-1}{\partial}_{\iota_i}{\bf{r}}$ for $i = 1, 2$.

Consider the corresponding new coordinate system $({\mu}, \iota_1, \iota_2)$, where ${\mu}$ denotes the normal distance to the boundary surface ${\partial}\Omega$, i.e.

Define the orthogonal velocity substitution for ${\mathfrak{w}}: = (\varphi, \psi)$ as

Finally, we define the scaled normal variable $\eta = \dfrac{{\mu}}{{\varepsilon}}$, which implies $\dfrac{{\partial}}{{\partial}{\mu}} = \dfrac{1}{{\varepsilon}}\dfrac{{\partial}}{{\partial}\eta}$. We then write ${\mathfrak{x}}: = (\eta, \iota_1, \iota_2)$.

1.3.   Asymptotic expansion and remainder equation

We seek a solution to (0.1) in the form

where the interior solution is

and the boundary layer is

Here ${U}_0$, ${U}_1$, ${U}_2$, and $U^B_0$ are constructed in Section 2.1 and Section 2.2, and ${R}(x, w)$ is the remainder satisfying

where ${h}$ and ${S}$ are as defined in (3.4) and (3.6)–(3.9).

1.4.   Literature

The study of the neutron transport equation in bounded domains has received a lot of attention since the dawn of the atomic age. This equation is not only significant in nuclear sciences and medical imaging but is also regarded as a linear prototype of the more important and complicated nonlinear Boltzmann equation. As a result, it serves as an ideal starting point to develop new theories and techniques.

For the formal expansion with respect to ${\varepsilon}$ and its explicit solution, ^{[7,8,9,10,11,12,13,14,15]} provide relevant literature. The discussion on the bounded domain and half-space cases can be found in ^{[16,17,18,19,20,21,22,23]}. In the more general context, we refer to ^{[24,25,26,27,28,29,30,31,32,33,34]} for the hydrodynamic limits of Boltzmann equations in bounded domains, and the recent progress on the diffusive limit of the transfer equation (which is a coupled system of the transport equation and the heat equation) ^[35,36].

The classical boundary layer analysis of the neutron transport equation leads to the Milne problem, which dictates that $U^B_0(\eta, \iota_1, \iota_2, {\mathfrak{w}})$ satisfies the equation given by

Based on the formal expansion in ${\varepsilon}$ (see (2.6)), it is natural to expect the following remainder estimate ^[16]:

While this estimate holds for domains with flat boundaries, a surprising counter-example was constructed ^[1] that shows (1.8) to be invalid for a two-dimensional (2D) disk due to the grazing set singularity.

To provide more specific details, demonstrating the validity of the remainder estimates (1.8), necessitates the use of the higher-order boundary layer expansion $U^B_1\in L^{\infty}$. In this case, the bound ${\partial}_{\iota_i}U^B_0\in L^{\infty}$ is required, and even though $U^B_0\in L^{\infty}$, it has been proven that the normal derivative ${\partial}_{\eta}U^B_0$ is singular at the grazing set $\varphi = 0$. This singularity is then transferred to ${\partial}_{\iota_i}U^B_0\notin L^{\infty}$. A meticulous construction of the boundary data ^[1] reveals that both the method and the result of the boundary layer (1.7) are problematic, which justifies this invalidity.

A new approach to constructing the boundary layer has been proposed in recent works ^[1,3,4,5,6]. It is based on the ${\varepsilon}$-Milne problem with geometric corrections for $\widetilde{U^B_0}({\mathfrak{x}}, {\mathfrak{w}})$, given by

where $\mathcal{R}_{\kappa}(\iota_1, \iota_2) > 0$ denotes the radius of curvature on ${\partial}\Omega$. This new construction has been shown to provide a satisfactory characterization of the $L^{\infty}$ diffusive expansion in two-dimension (2D) or three-dimensional (3D) convex domains. The proof relies on a detailed analysis of $W^{1, \infty}$ regularity and boundary layer decomposition techniques for (1.9).

In non-convex domains, where $\mathcal{R}_{\kappa}(\iota_1, \iota_2) < 0$, the boundary layer with geometric correction is described in ^[2] as follows:

This sign flipping of the geometric correction term in contrast to (1.9) dramatically changes the characteristics of the boundary layer.

In Figure 1 and Figure 2 ^[2], the horizontal axis represents the scaled normal variable $\eta$, while the vertical axis represents the velocity $\varphi$. The inflow boundary is located on the left boundary where $\eta = 0$ and $\varphi > 0$. It is apparent from Figure 2 that there exists a "hollow" region where the characteristics may never track back to the inflow boundary. This discrepancy in the information source results in a strong discontinuity across the boundary of the "hollow" region, making it impossible to obtain $W^{1, \infty}$ estimates, which, in turn, prevents higher-order boundary layer expansion.

In this paper, we employ a fresh approach to design a cutoff boundary layer without the geometric correction and justify the $L^2$ diffusive expansion in smooth non-convex domains.

1.5.   Notation and convention

Let $\langle {\ \cdot\, \ \cdot\ } \rangle_w$ denote the inner product for $w\in{{\mathbb{S}}}^2$, $\langle {\ \cdot\, \ \cdot\ } \rangle_x$ for $x\in\Omega$, and $\langle {\ \cdot\, \ \cdot\ } \rangle$ for $(x, w)\in\Omega\times{{\mathbb{S}}}^2$. Moreover, let $\langle {\ \cdot\, \ \cdot\ } \rangle_{\gamma_{\pm}}$ denote the inner product on $\gamma_{\pm}$ with the measure ${\mathrm{d}}\gamma: = \left\vert{w\cdot n}\right\vert{\mathrm{d}} w{\mathrm{d}} S_x = \left\vert{\sin\varphi}\right\vert\cos\varphi{\mathrm{d}}{\mathfrak{w}}{\mathrm{d}} S_x$. Denote the bulk and boundary norms as follows:

Define the $L^{\infty}$ norms as follows:

Let $\left\Vert{\cdot}\right\Vert_{W^{k, p}_x}$ denote the usual Sobolev norm for $x\in\Omega$ and $\left\vert{\cdot}\right\vert_{W^{k, p}_x}$ for $x\in{\partial}\Omega$, and $\left\Vert{\cdot}\right\Vert_{W^{k, p}_xL^q_w}$ denote the $W^{k, p}$ norm for $x\in\Omega$ and the $L^q$ norm for ${w}\in{{\mathbb{S}}}^2$. Similar notation also applies when we replace $L^q$ by $L^q_{\gamma}$. When there is no possibility of confusion, we will ignore the $(x, w)$ variables in the norms.

Throughout this paper, $C > 0$ denotes a constant that only depends on the domain $\Omega$, but does not depend on the data or ${\varepsilon}$. It is referred to as universal and can change from one inequality to another. We write $a{\lesssim} b$ to denote $a\leq Cb$ and $a{\gtrsim} b$ to denote $a\geq Cb$. We also write $a\simeq b$ if $a{\lesssim} b$ and $a{\gtrsim} b$. We use ${o(1)}$ to denote a sufficiently small constant that is independent of the data.

1.6.   Main results

Theorem 1.1. Under the assumption

there exists a unique solution $u^{{\varepsilon}}({x}, {w})\in L^{\infty}(\Omega\times{{\mathbb{S}}}^2)$ to (0.1). Moreover, the solution obeys the estimate

Here, ${U}_0({x})$ satisfies the Laplace equation with the Dirichlet boundary condition

in which ${\Phi}_{\infty}(\iota_1, \iota_2) = {\Phi}_{\infty}({x}_0)$ for ${x}_0\in{\partial}\Omega$ is given by solving the Milne problem for ${\Phi}({\mathfrak{x}}, {\mathfrak{w}})$

Remark 1.2. In ^[1,5,6] for 2D/3D convex domains, as well as ^[2] for a 2D annulus domain, it is justified that for any $0 < {\delta}\ll1$

where $\widetilde{U^B_0}({\mathfrak{x}}, {\mathfrak{w}})$ is the boundary layer with geometric correction defined in (1.9), and $\widetilde{{U}_0}$ is the corresponding interior solution. Previous work [23, Theorem 2.1] has revealed that the difference between two types of interior solutions satisfies

Due to the rescaling $\eta = {\varepsilon}^{-1}{\mu}$, for the general in-flow boundary data $g$, the boundary layer $\widetilde{U^B_0}\neq0$ satisfies

Hence, we conclude that

Therefore, this indicates that (1.14) in Theorem 1.1 achieves the optimal $L^2$ bound of the diffusive approximation.

1.7.   Methodology

It is well-known that the key of the remainder estimate is to control ${\overline{{R}}}$. In a series of works ^{[1,2,3,4,5,6]}, it has been shown that the kernel estimate

and the basic energy (entropy production) bound

provide a full control of the remainder ${R}$:

Although (1.23) alone is not enough to justify

expanding the boundary layer approximation beyond the leading order $U^B_0$ further improves the righ-hand side (RHS) of (1.23) to ${\varepsilon}^{\alpha}$ with $\alpha > 0$, and leads to (1.24). While this technique works well for convex domains, as our previous analysis revealed, it is impossible to expand to $U^B_1$ for non-convex domains due to the lack of the $W^{1, \infty}$ estimate in (1.10).

The bottleneck of the $L^2$ bound (1.23) lies in the kernel estimate (1.21), which stems from the weak formulation of (1.6) with the test function $w\cdot{\nabla_x}{\xi}$

Here, the auxiliary function ${\xi}({x})$ satisfies

In (1.25), $\Big\langle {{R}, w\cdot{\nabla_x}\big(w\cdot{\nabla_x}{\xi}\big)} \Big\rangle\simeq \left\Vert{{\overline{{R}}}}\right\Vert_{L^2}^2$, and ${\varepsilon}^{-1}\Big\langle {{{R}-{\overline{{R}}}}, w\cdot{\nabla_x}{\xi}} \Big\rangle$ leads to the worst and critical contribution ${\varepsilon}^{-1}\left\Vert{{{R}-{\overline{{R}}}}}\right\Vert_{L^2}$.

The key improvement in our work is an upgraded (compared with (1.21)) kernel estimate

which, combined with (1.22), leads to

The extra ${\varepsilon}^{\frac{1}{2}}$ gain in (1.27) follows from a crucial observation. We consider the conservation law of (1.6) with the test function ${\xi}$ defined in (1.26):

We discover that the combination ${\varepsilon}^{-1}\times$(1.29) and (1.25) yields

which exactly cancels out the worst contribution ${\varepsilon}^{-1}\Big\langle {{{R}-{\overline{{R}}}}, w\cdot{\nabla_x}{\xi}} \Big\rangle$. This key cancellation leads to an additional crucial gain of ${\varepsilon}^{\frac{1}{2}}$ in (1.27). Consequently, we can deduce the remainder estimate (1.28) without any further expansion of the (singular) boundary layer approximation.

Technically, to estimate the source terms ${\varepsilon}^{-1}\Big\langle {{S}, {\xi}} \Big\rangle$ and $\Big\langle {{S}, w\cdot{\nabla_x}{\xi}} \Big\rangle$ in (1.30), and ${\varepsilon}^{-1}\Big\langle {{S}, {R}} \Big\rangle$ in deriving (1.22), particularly the derivatives of the boundary layers, we construct a new cut-off boundary layer

where ${\Psi}$ solves the Milne problem. With the grazing set cutoff ${\widetilde{\chi}}\big({\varepsilon}^{-1}\varphi\big)$, we are able to perform delicate and precise estimates to control the resulting complex forcing term ${S}$ (see (3.6)–(3.9)). In addition, with the help of integration by parts in $\varphi$ and Hardy's inequality ^[37,38] in the ${\mu}$ direction, we have

we can bound all source contributions in terms of the desired order of ${\varepsilon}$ for closure.

2.   Asymptotic analysis

2.1.   Interior solution

Inserting (1.4) into (0.1) and comparing the order of ${\varepsilon}$, and following the analysis in ^[4,6], we deduce that

We need the boundary layer to determine the boundary conditions for ${\overline{{U}}}_0$, ${\overline{{U}}}_1$, and ${\overline{{U}}}_2$.

2.2.   Boundary layer

2.2.1.   Geometric substitutions

The construction of the boundary layer requires a local description in a neighborhood of the physical boundary ${\partial}\Omega$. We follow the procedure in ^[4,6].

Substitution 1: Spatial Substitution Following the notation in Section 1.2, under the coordinate system $(\mu, \iota_1, \iota_2)$, we have

where ${\kappa}_i(\iota_1, \iota_2)$ for $i = 1, 2$ is the principal curvature.

Substitution 2: Velocity Substitution Under the orthogonal velocity substitution (1.2) for $\varphi\in\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ and $\psi\in[-\pi, \pi]$, we have

where $R_i = {\kappa}_i^{-1}$ represents the radius of curvature. Note that the Jacobian ${\mathrm{d}}{w} = \cos\varphi{\mathrm{d}}\varphi{\mathrm{d}}\psi$ will be present when we perform integration.

Substitution 3: Scaling Substitution Considering the scaled normal variable $\eta = {\varepsilon}^{-1}\mu$, we have

2.2.2.   Milne problem

Let ${\Phi}({\mathfrak{x}}, {\mathfrak{w}})$ be the solution to the Milne problem

with the boundary condition

We are interested in the solution that satisfies

for some ${\Phi}_{\infty}(\iota_1, \iota_2)$. Based on [4, Section 4], we have the well-posedness and regularity of (2.7).

Proposition 2.1. Under the assumption (1.13), there exist ${\Phi}_{\infty}(\iota_1, \iota_2)$ and a unique solution ${\Phi}$ to (2.7)such that ${\Psi}: = {\Phi}-{\Phi}_{\infty}$ satisfies

and for some constant $K > 0$ and any $0 < {r}\leq 3$, we have

Let $\chi(y)\in C^{\infty}({\mathbb{R}})$ and ${\widetilde{\chi}}(y) = 1-\chi(y)$ be smooth cut-off functions satisfying $\chi(y) = 1$ if $\left\vert{y}\right\vert\leq1$ and $\chi(y) = 0$ if $\left\vert{y}\right\vert\geq2$. We define the boundary layer as follows:

Remark 2.2. Due to the cutoff in (2.14), we have

and

2.3.   Matching procedure

We plan to enforce the matching condition for ${x}_0\in{\partial}\Omega$ and ${w}\cdot{n} < 0$

Considering (2.15), it suffices to require

which yields

Hence, we obtain

${\underline{{\bf{Construction\ of }}\ {U}_0 }}$ Based on (2.1) and (2.18), we define ${U}_0({x})$ satisfying

From standard elliptic estimates [39, Chapter 9: Section 2], trace theorem, and Proposition 2.1, we have for any ${s}\in[2, \infty)$

${\underline{{\bf{Construction\ of}}\ {U}_1 }}$ Based on (2.2), we define ${U}_1({x}, {w})$ satisfying

From (2.22), we have that for any ${s}\in[2, \infty)$

${\underline{{\bf{Construction\ of }}\ {U}_2 }}$ Based on (2.2), define ${U}_2({x}, {w})$ satisfying

From (2.24), we have for any ${s}\in[2, \infty)$

Summarizing the analysis above, we have the well-posedness and regularity estimates of the interior solution and boundary layer.

Proposition 2.3. Under the assumption (1.13), we can construct ${U}_0, {U}_1, {U}_2$, and $U^B_0$ as in (2.21), (2.23), (2.25), and (2.14) satisfying for any ${s}\in[2, \infty)$

and, for some constant $K > 0$ and any $0 < {r}\leq 3$, we have

3.   Remainder equation

Denote the approximate solution

Inserting (1.3) into (0.1), we have

which yields

3.1.   Formulation of remainder equation

Now we consider the remainder equation (1.6), where the boundary data ${h}$ is given by

and the source term ${S}$ is given by

where

3.2.   Weak formulation

Lemma 3.1 (Green's identity, Lemma 2.2 of ^[40]). Assume $f({x}, {w}), \ g({x}, {w})\in L^2(\Omega\times{{\mathbb{S}}}^2)$, and ${w}\cdot{\nabla_x} f, \ {w}\cdot{\nabla_x} g\in L^2(\Omega\times{{\mathbb{S}}}^2)$ with $f, \ g\in L^2_{\gamma}$. Then

Using Lemma 3.1, we can derive the weak formulation of (1.6). For any test function ${\mathfrak{g}}(x, w)\in L^2(\Omega\times{{\mathbb{S}}}^2)$ with ${w}\cdot{\nabla_x}{\mathfrak{g}}\in L^2(\Omega\times{{\mathbb{S}}}^2)$ with ${\mathfrak{g}}\in L^2_{\gamma}$, we have

3.3.   Estimates of boundary and source terms

Lemma 3.2. Under the assumption (1.13), for ${h}$ defined in (3.4), we have

Proof. Based on Proposition 2.3, we have

Noting that the cutoff $\chi\big({\varepsilon}^{-1}\varphi\big)$ restricts the support to $\left\vert{\varphi}\right\vert{\lesssim}{\varepsilon}$ and that the ${\mathrm{d}}\gamma$ measure contributes an extra $\sin\varphi$, we have

Hence, our result follows. □

Lemma 3.3. Under the assumption (1.13), for ${S}_0$ defined in (3.6), we have

Proof. This follows from Proposition 2.3. □

Lemma 3.4. Under the assumption (1.13), for ${S}_1$ defined in (3.7), we have

Also, for the boundary layer $U^B_0$ defined in (2.14), we have

and

Proof. We split

Note that ${S}_{11}$ is nonzero only when $\left\vert{\varphi}\right\vert\geq{\varepsilon}$ and thus based on Proposition 2.1, we know $\left\vert{\dfrac{{\partial}{\Psi}}{{\partial}\varphi}}\right\vert\leq\left\vert{\sin\varphi}\right\vert^{-1}\left\vert{{\Psi}}\right\vert{\lesssim}{\varepsilon}^{-1}$. Hence, using ${\mathrm{d}}\mu = {\varepsilon}{\mathrm{d}}\eta$, we have

Noticing that $\dfrac{{\partial}{\widetilde{\chi}}\big({\varepsilon}^{-1}\varphi\big)}{{\partial}\varphi} = {\varepsilon}^{-1}{\widetilde{\chi}}'\big({\varepsilon}^{-1}\varphi\big)$ and ${\widetilde{\chi}}'\big({\varepsilon}^{-1}\varphi\big)$ is nonzero only when ${\varepsilon} < \left\vert{\varphi}\right\vert < 2{\varepsilon}$, based on Proposition 2.1, we have

Combining (3.20) and (3.21), we have (3.16). Note that ${\mathrm{e}}^{-K\eta}$ will suppress the growth from the pre-factor $1+\eta$.

Naturally, (3.17) comes from Proposition 2.1. We now turn to (3.18). The most difficult term in $\big\vert{\langle {{S}_1, g} \rangle}\big\vert$ is essentially $\left\vert{\langle {\dfrac{{\partial}U^B_0}{{\partial}\varphi}, g} \rangle}\right\vert$. Integration by parts with respect to $\varphi$ implies that

From (1.2) and $\dfrac{{\partial}{x}}{{\partial}\varphi} = 0$, we know that the substitution $({\mu}, \iota_1, \iota_2, {w}){\rightarrow} ({\mu}, \iota_1, \iota_2, {\mathfrak{w}})$ implies

Hence, we know $\left\vert{\dfrac{{\partial}{w}}{{\partial}\varphi}}\right\vert{\lesssim} 1$, and thus

Hence, we know that

□

Lemma 3.5. Under the assumption (1.13), for ${S}_2$ defined in (3.8), we have

Proof. Notice that $\left\vert{{\varepsilon}^{-1}\sin\phi{\widetilde{\chi}}\big({\varepsilon}^{-1}\varphi\big)\dfrac{{\partial}\chi({\varepsilon}\eta)}{{\partial}\eta}}\right\vert{\lesssim} 1$. Based on Proposition 2.1 and Proposition 2.3, we directly bound

Then the $L^2_xL^1_w$ estimate follows from a similar argument, noting that there is no rescaling in ${w}$ variables. □

Lemma 3.6. Under the assumption (1.13), for ${S}_3$ defined in (3.9), we have

Proof. Using $\chi = 1-{\widetilde{\chi}}$, we split

Noting that ${S}_{31}$ is nonzero only when $\left\vert{\varphi}\right\vert\leq{\varepsilon}$, based on Proposition 2.1, we have

Analogously, noting that ${S}_{32}$ contains a ${w}$ integral, we have

Combining (3.30) and (3.31), we have the $L^2$ estimate. Similarly, we derive the $L^2_xL^1_w$ bound:

□

4.   Remainder estimate

4.1.   Basic energy estimate

Lemma 4.1. Under the assumption (1.13), we have

Proof. Taking ${\mathfrak{g}} = {\varepsilon}^{-1}{R}$ in (3.11), we obtain

By using the orthogonality of ${\overline{{R}}}$ and ${{R}-{\overline{{R}}}}$, we have

Using Lemma 3.2, we know that

Using Lemma 3.3, we have

Using Lemma 3.4, Lemma 3.5, and Lemma 3.6, we have

Finally, we turn to ${\varepsilon}^{-1}\Big\langle {{\overline{{R}}}, {S}_1+{S}_2+{S}_3} \Big\rangle$. For ${S}_1$, we integrate by parts with respect to $\varphi$ and use Lemma 3.4 to obtain

In addition, Lemma 3.5 and Lemma 3.6 yield

Combining (4.5)–(4.8), we obtain

Combining (4.9) and (4.4), we have (4.1). □

4.2.   Kernel estimate

Lemma 4.2. Under the assumption (1.13), we have

Proof. Denote ${\xi}(x)$ satisfying

Based on standard elliptic estimates and trace estimates [41, Chapter 6: Section 6.3], we have

Taking ${\mathfrak{g}} = {\xi}$ in (3.11), we have

Using oddness, orthogonality, and ${\xi}\big|_{{\partial}\Omega} = 0$, we obtain (1.29)

Then taking ${\mathfrak{g}} = w\cdot{\nabla_x}{\xi}$ in (3.11), we obtain (1.25)

Adding ${\varepsilon}^{-1}\times$(4.14) and (4.15) to eliminate ${\varepsilon}^{-1}\Big\langle {{{R}-{\overline{{R}}}}, w\cdot{\nabla_x}{\xi}} \Big\rangle$, we obtain

Notice that

where (4.12) and Cauchy's inequality yield

Also, using (4.12) and Lemma 3.2, we have

Inserting (4.17)–(4.20) into (4.16), we obtain

Then we turn to the estimate of source terms in (4.21). Cauchy's inequality and Lemma 3.3 yield

Similar to (4.7), we first integrate by parts with respect to $\varphi$ in ${S}_1$. Using ${\xi}\big|_{{\partial}\Omega} = 0$, (4.12), Hardy's inequality, Lemma 3.4, Lemma 3.5, and Lemma 3.6, we have

Analogously, we integrate by parts with respect to $\varphi$ in ${S}_1$. Then using (4.12), fundamental theorem of calculus, Hardy's inequality, Lemma 3.4, Lemma 3.5, and Lemma 3.6, we bound

Hence, inserting (4.22), (4.23), and (4.24) into (4.21), we have shown (4.10). □

4.3.   Synthesis

Proposition 4.3. Under the assumption (1.13), we have

Proof. From (4.1), we have

From (4.10), we have

Inserting (4.27) into (4.26), we have

Inserting (4.28) into (4.27), we have

Hence, adding ${\varepsilon}^{-1}\times$(4.29) and (4.28), we have

Then our result follows. □

5.   Proof of the main theorem

The well-posedness of (0.1) is well-known ^[1,16,17]. The construction of ${U}_0$, ${\Phi}$, and ${\Phi}_{\infty}$ follows from Proposition 2.1 and Proposition 2.3, so we focus on the derivation of (1.14).

Based on Proposition 4.3 and (1.3), we have

Using Proposition 2.3, we have

Using Proposition 2.3 and the rescaling $\eta = {\varepsilon}^{-1}{\mu}$, we have

Then (1.14) follows from inserting (5.2) and (5.3) into (5.1).

Author contributions

The authors have equal contribution to this paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

We sincerely thank the editor and referees for their careful checking and proofreading. Yan Guo's research is supported by NSF grant DMS-2405051. Lei Wu's research is supported by NSF grants DMS-2104775 and DMS-2405161.

Conflict of interest

The authors claim that there is no conflict of interest.