diff --git "a/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt" "b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt" @@ -0,0 +1,751 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf,len=750 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11996v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='AP] 27 Jan 2023 L2 DIFFUSIVE EXPANSION FOR NEUTRON TRANSPORT EQUATION YAN GUO AND LEI WU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Grazing set singularity leads to a surprising counter-example and breakdown [24] of the classical mathematical theory for L∞ diffusive expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) of neutron transport equation with in-flow boundary condition in term of the Knudsen number ε, one of the most classical problems in the kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Even though a satisfactory new theory has been established by constructing new boundary layers with favorable ε-geometric correction for convex domains [24, 7, 8, 22, 23], the severe grazing singularity from non-convex domains has prevented any positive mathematical progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We develop a novel and optimal L2 expansion theory for general domain (including non-convex domain) by discovering a surprising ε 1 2 gain for the average of remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Analysis 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Equation 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Estimate 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proof of Main Theorem 13 References 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Problem Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We consider the steady neutron transport equation in a three-dimensional C3 bounded domain (convex or non-convex) with in-flow boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In the spatial domain Ω ∋ x = (x1, x2, x3) and the velocity domain S2 ∋ w = (w1, w2, w3), the neutron density uε(x, w) satisfies \uf8f1 \uf8f2 \uf8f3 w · ∇xuε + ε−1� uε − uε � = 0 in Ω × S2, uε(x0, w) = g(x0, w) for w · n < 0 and x0 ∈ ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) where g is a given function denoting the in-flow data, uε(x) := 1 4π � S2 uε(x, w)dw, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) n is the outward unit normal vector, with the Knudsen number 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We intend to study the asymptotic behavior of uε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on the flow direction, we can divide the boundary γ := � (x0, w) : x0 ∈ ∂Ω, w ∈ S2� into the incoming boundary γ−, the outgoing boundary γ+, and the grazing set γ0 based on the sign of w · n(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In particular, the boundary condition of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) is only given on γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Primary 35Q49, 82D75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Secondary 35Q62, 35Q20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' non-convex domains, transport equation, diffusive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo was supported by NSF Grant DMS-2106650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu was supported by NSF Grant DMS-2104775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Normal Chart near Boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We follow the approach in [8, 23] to define the geometric quantities, and the details can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For smooth manifold ∂Ω, there exists an orthogonal curvilinear coordinates system (ι1, ι2) such that the coordinate lines coincide with the principal directions at any x0 ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Assume ∂Ω is parameterized by r = r(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let the vector length be Li := |∂ιir| and unit vector ςi := L−1 i ∂ιir for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Consider the corresponding new coordinate system (µ, ι1, ι2), where µ denotes the normal distance to the boundary surface ∂Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' x = r − µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Define the orthogonal velocity substitution for w := (ϕ, ψ) as −w · n = sin ϕ, w · ς1 = cos ϕ sin ψ, w · ς2 = cos ϕ cos ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) Finally, we define the scaled normal variable η = µ ε , which implies ∂ ∂µ = 1 ε ∂ ∂η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Expansion and Remainder Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We seek a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) in the form uε =U + U B + R = � U0 + εU1 + ε2U2 � + U B 0 + R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) where the interior solution is U(x, w) := U0(x, w) + εU1(x, w) + ε2U2(x, w), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) and the boundary layer is U B(η, ι1, ι2, w) := U B 0 (η, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) Here U0, U1, U2 and U B 0 are constructed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, and R(x, v) is the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The study of the neutron transport equation in bounded domains, has attracted a lot of attention since the dawn of the atomic age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Besides its significance in nuclear sciences and medical imaging, neutron transport equation is usually regarded as a linear prototype of the more important yet more complicated nonlinear Boltzmann equation, and thus, is an ideal starting point to develop new theories and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We refer to [10, 11, 12, 13, 14, 15, 16, 17, 18] for the formal expansion with respect to ε and explicit solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The discussion on bounded domain and half-space cases can be found in [5, 4, 3, 1, 2, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The classical boundary layer of neutron transport equation dictates that U B 0 (η, ι1, ι2, w) satisfies the Milne problem sin ϕ∂U B 0 ∂η + U B 0 − U B 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) From the formal expansion in ε (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6)), it is natural to expect the remainder estimate [5] ∥R∥L∞ ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) Even though this is valid for domains with flat boundary, a counter-example is constructed [24] so that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) is invalid for a 2D disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This is due to the grazing set singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' To be more specific, in order to show the remainder estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9), the higher-order boundary layer expansion U B 1 ∈ L∞ is necessary, which further requires ∂ιiU B 0 ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Nevertheless, though U B 0 ∈ L∞, it is shown that the normal derivative ∂ηU B 0 is singular at the grazing set ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Furthermore, this singularity ∂ηU B 0 /∈ L∞ will be transferred to ∂ιiU B 0 /∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' A careful construction of boundary data [24] justifies this invalidity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' both the method and result of the boundary layer (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) are problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' A new construction of boundary layer [24] based on the ε-Milne problem with geometric correction for � U B 0 (η, ι1, ι2, w) sin ϕ∂� U B 0 ∂η − ε 1 − εη cos ϕ∂� U B 0 ∂ϕ + � U B 0 − � U B 0 = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) has been shown to provide the satisfactory characterization of the L∞ diffusive expansion in 2D disk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' With more detailed regularity analysis and boundary layer decomposition techniques for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), such result has been generalized to 2D/3D smooth convex domains [7, 8, 22, 23] and even 2D annulus domain [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 3 In non-convex domains, the boundary layer with geometric correction is essentially sin ϕ∂� U B 0 ∂η − ε 1 + εη cos ϕ∂� U B 0 ∂ϕ + � U B 0 − � U B 0 = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Compared to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), this sign flipping dramatically changes the characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 η φ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Characteristics in Convex Domains 0 1 2 3 4 5 6 7 8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 η φ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Characteristics in Non- Convex Domains In Figure 1 and Figure 2 [25], the horizontal direction represents the scaled normal variable η and the vertical direction represents the velocity ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' There exists a “hollow” region in Figure 2 that the characteristics may never track back to the left boundary η = 0 and ϕ > 0, making the W 1,∞ estimates impossible and thus preventing higher-order boundary layer expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In this paper, we will employ a fresh approach to design a cutoff boundary layer without the geometric correction and justify the L2 diffusive expansion in smooth non-convex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Notation and Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let ⟨ · , · ⟩w denote the inner product for w ∈ S2, ⟨ · , · ⟩x for x ∈ Ω, and ⟨ · , · ⟩ for (x, w) ∈ Ω × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, let ⟨ · , · ⟩γ± denote the inner product on γ± with measure dγ := |w · n| dwdSx = |sin ϕ| cos ϕdwdSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Denote the bulk and boundary norms ∥f∥L2 := ��� Ω×S2 |f(x, w)|2 dwdx � 1 2 , |f|L2 γ± := �� γ± |f(x, w)|2 dγ � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) Define the L∞ norms ∥f∥L∞ := ess sup (x,w)∈Ω×S2 ��f(x, w) ��, |f|L∞ γ± := ess sup (x,w)∈γ± ��f(x, w) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Let ∥·∥W k,p x denote the usual Sobolev norm for x ∈ Ω and |·|W k,p x for x ∈ ∂Ω, and ∥·∥W k,p x Lq w denote W k,p norm for x ∈ Ω and Lq norm for w ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The similar notation also applies when we replace Lq by Lq γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' When there is no possibility of confusion, we will ignore the (x, w) variables in the norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Throughout this paper, C > 0 denotes a constant that only depends on the domain Ω, but does not depend on the data or ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' It is referred as universal and can change from one inequality to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We write a ≲ b to denote a ≤ Cb and a ≳ b to denote a ≥ Cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, we write a ≃ b if a ≲ b and a ≳ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We will use o(1) to denote a sufficiently small constant independent of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption |g|W 3,∞L∞ γ− ≲ 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) there exists a unique solution uε(x, w) ∈ L∞(Ω × S2) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Moreover, the solution obeys the estimate ∥uε − U0∥L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Here U0(x) satisfies the Laplace equation with Dirichlet boundary condition � ∆xU0(x) = 0 in Ω, U0(x0) = Φ∞(x0) on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) in which Φ∞(ι1, ι2) = Φ∞(x0) for x0 ∈ ∂Ω is given by solving the Milne problem for Φ(η, ι1, ι2, w) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 sin ϕ∂Φ ∂η + Φ − Φ = 0, Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for sin ϕ > 0, lim η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In [24, 22, 23] for 2D/3D convex domains, as well as [25] for 2D annulus domain, it is justified that for any 0 < δ ≪ 1 ���uε − � U0 − � U B 0 ��� L2 ≲ ε 5 6 −δ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) where � U B 0 (η, ι1, ι2, w) is the boundary layer with geometric correction defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), and � U0 is the cor- responding interior solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1] reveals that the difference between two types of interior solutions ���� U0 − U0 ��� L2 ≲ ε 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Due to the rescaling η = ε−1µ, for general in-flow boundary data g, the boundary layer � U B 0 ̸= 0 satisfies ���� U B 0 ��� L2 ≃ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Hence, we conclude that ∥uε − U0∥L2 ≃ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) Therefore, this indicates that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 achieves the optimal L2 bound of the diffusive approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' It is well-known that the key of the remainder estimate is to control R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In a series of work [24, 25, 7, 8, 22, 23] based on a L2 → L∞ framework, it is shown that ��R �� L2 ≲ ε−1 ��R − R �� L2 ≲ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) combined from the expected energy (entropy production) bound for ε−1 ��R − R �� L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This bound requires the next-order ε expansion of boundary layer approximation, which is impossible for non-convex domains, and barely possible by the new boundary layer theory with the ε-geometric correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The key improvement in our work is ��R �� L2 ≲ ε 1 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) which is a consequence of the following conservation law for test function ξ(x) satisfying −∆xξ = R and ξ �� ∂Ω = 0: − � R, w · ∇xξ � = − � R − R, w · ∇xξ � = � S, ξ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) where � R, w ·∇xξ � = 0 thanks to the oddness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This conservation law exactly cancels the worst contribution of ε−1 ��R − R �� L2 in ��R �� L2 estimate, which comes from taking test function w · ∇xξ � γ R � w · ∇xξ � (w · n) − � R, w · ∇x � w · ∇xξ �� + ε−1� R − R, w · ∇xξ � = � S, w · ∇xξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Such a key cancellation produces an extra crucial gain of ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We then conclude the remainder estimate without any further expansion of the (singular) boundary layer approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 5 In addition, we construct a new cut-off boundary layer near ϕ = 0 to avoid the singularity, and are able to perform delicate and precise estimates to control the resulting complex forcing term S (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10)), in terms of the desired order ε for closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Analysis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Interior Solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Inserting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) and comparing the order of ε, following the analysis in [8, 23], we deduce that U0 = U 0, ∆xU 0 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) U1 = U 1 − w · ∇xU0, ∆xU 1 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) U2 = U 2 − w · ∇xU1, ∆xU 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) We need the boundary layer to determine the boundary conditions for U0, U 1 and U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Boundary Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Geometric Substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The construction of boundary layer requires a local description in a neigh- borhood of the physical boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We follow the procedure in [8, 23]: Substitution 1: Spacial Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Following the notation in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, under the coordinate system (µ, ι1, ι2), we have w · ∇x = −(w · n) ∂ ∂µ − w · ς1 L1(κ1µ − 1) ∂ ∂ι1 − w · ς2 L2(κ2µ − 1) ∂ ∂ι2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) where κi(ι1, ι2) for i = 1, 2 is the principal curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Substitution 2: Velocity Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the orthogonal velocity substitution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) for ϕ ∈ � −π 2 , π 2 � and ψ ∈ [−π, π], we have w · ∇x = sin ϕ ∂ ∂µ − � sin2 ψ R1 − µ + cos2 ψ R2 − µ � cos ϕ ∂ ∂ϕ + cos ϕ sin ψ L1(1 − κ1µ) ∂ ∂ι1 + cos ϕ cos ψ L2(1 − κ2µ) ∂ ∂ι2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) + sin ψ R1 − µ �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ � ∂ ∂ψ − cos ψ R2 − µ �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ � ∂ ∂ψ, where Ri = κ−1 i represents the radius of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that the Jacobian dw = cos ϕdϕdψ will be present when we perform integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Substitution 3: Scaling Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Considering the scaled normal variable η = ε−1µ, we have w · ∇x =ε−1 sin ϕ ∂ ∂η − � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ ∂ ∂ϕ + R1 cos ϕ sin ψ L1(R1 − εη) ∂ ∂ι1 + R2 cos ϕ cos ψ L2(R2 − εη) ∂ ∂ι2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) + sin ψ R1 − εη �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ � ∂ ∂ψ − cos ψ R2 − εη �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ � ∂ ∂ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Milne Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let Φ(η, ι1, ι2, w) be the solution to the Milne problem sin ϕ∂Φ ∂η + Φ − Φ =0, Φ(η, ι1, ι2) = 1 4π � π −π � π 2 − π 2 Φ(η, ι1, ι2, w) cos ϕdϕdψ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) with boundary condition Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for sin ϕ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU We are interested in the solution that satisfies lim η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) for some Φ∞(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on [8, Section 4], we have the well-posedness and regularity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), there exist Φ∞(ι1, ι2) and a unique solution Φ to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)such that Ψ := Φ − Φ∞ satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 sin ϕ∂Ψ ∂η + Ψ − Ψ = 0, Ψ(0, ι1, ι2, w) = g(ι1, ι2, w) − Φ∞(ι1, ι2), lim η→0 Ψ(η, ι1, ι2, w) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) and for some constant K > 0 and any 0 < r ≤ 3 |Φ∞|W 3,∞ ι1,ι2 + ��eKηΨ �� L∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) ����eKη sin ϕ∂Ψ ∂η ���� L∞ + ����eKη sin ϕ∂Ψ ∂ϕ ���� L∞ + ����eKη ∂Ψ ∂ψ ���� L∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) ����eKη ∂rΨ ∂ιr 1 ���� L∞ + ����eKη ∂rΨ ∂ιr 2 ���� L∞ ≲1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Let χ(y) ∈ C∞(R) and �χ(y) = 1 − χ(y) be smooth cut-off functions satisfying χ(y) = 1 if |y| ≤ 1 and χ(y) = 0 if |y| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We define the boundary layer U B 0 (η, ι1, ι2, w) := �χ � ε−1ϕ � χ(εη)Ψ(η, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Due to the cutoff in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have U B 0 (0, ι1, ι2, w) = �χ � ε−1ϕ �� g(ι1, ι2, w) − Φ∞(ι1, ι2) � = �χ � ε−1ϕ � Ψ(0, ι1, ι2, w), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) and sin ϕ∂U B 0 ∂η + U B 0 − U B 0 = −�χ � ε−1ϕ � χ(εη)Ψ + Ψ�χ(ε−1ϕ)χ(εη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Matching Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We plan to enforce the matching condition for x0 ∈ ∂Ω and w · n < 0 U0(x0) + U B 0 (x0, w) =g(x0, w) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Considering (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15), it suffices to require U0(x0) = Φ∞(x0) := Φ∞(ι1, ι2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) which yields U0(x0) + Ψ(x0, w) =g(x0, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Hence, we obtain U0(x0, w) + U B 0 (x0, w) = g(x0, w) + χ � ε−1ϕ � Ψ(0, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Construction of U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18), define U0(x) satisfying U0 = U0, ∆xU 0 = 0, U0(x0) = Φ∞(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) From standard elliptic estimates [9] and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have for any s ∈ [2, ∞) ∥U0∥W 3+ 1 s ,s + |U0|W 3,s ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) Construction of U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2), define U1(x, w) satisfying U1 = U 1 − w · ∇xU0, ∆xU 1 = 0, U 1(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22), we have for any s ∈ [2, ∞) ∥U1∥W 2+ 1 s ,sL∞ + |U1|W 2,sL∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 7 Construction of U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2), define U2(x, w) satisfying U2 = U 2 − w · ∇xU1, ∆xU 2 = 0, U 2(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24), we have for any s ∈ [2, ∞) ∥U2∥W 1+ 1 s ,sL∞ + |U2|W 1,sL∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) Summarizing the above analysis, we have the well-posedness and regularity estimates of the interior solution and boundary layer: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we can construct U0, U1, U2, U B 0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) satisfying for any s ∈ [2, ∞) ∥U0∥W 3+ 1 s ,s + |U0|W 3,s ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) ∥U1∥W 2+ 1 s ,sL∞ + |U1|W 2,sL∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) ∥U2∥W 1+ 1 s ,sL∞ + |U2|W 1,sL∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='29) and for some constant K > 0 and any 0 < r ≤ 3 ��eKηU B 0 �� L∞ + ����eKη ∂rU B 0 ∂ιr 1 ���� L∞ + ����eKη ∂rU B 0 ∂ιr 2 ���� L∞ ≲1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Equation Denote the approximate solution ua := � U0 + εU1 + ε2U2 � + U B 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) Inserting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1), we have w · ∇x � ua + R � + ε−1� ua + R � − ε−1� ua + R � = 0, � ua + R ���� γ− = g, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) which yields w · ∇xR + ε−1� R − R � = −w · ∇xua − ε−1� ua − ua � , R ��� γ− = � g − ua ���� γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Formulation of Remainder Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We consider the remainder equation \uf8f1 \uf8f2 \uf8f3 w · ∇xR + ε−1� R − R � = S in Ω × S2, R(x0, w) = h(x0, w) for w · n < 0 and x0 ∈ ∂Ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) where R(x) = 1 4π � S2 R(x, w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Here the boundary data h is given by h := −εw · ∇xU0 − ε2w · ∇xU1 − χ � ε−1ϕ � Ψ(0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) and the source term S is given by S := S0 + S1 + S2 + S3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU where S0 := − ε2w · ∇xU2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) S1 := � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂U B 0 ∂ϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) S2 :=ε−1 sin φ�χ � ε−1ϕ �∂χ(εη) ∂η Ψ + R1 cos ϕ sin ψ L1(R1 − εη) ∂U B 0 ∂ι1 + R2 cos ϕ cos ψ L2(R2 − εη) ∂U B 0 ∂ι2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) + sin ψ R1 − εη �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ �∂U B 0 ∂ψ − cos ψ R2 − εη �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ �∂U B 0 ∂ψ , S3 :=ε−1 � �χ � ε−1ϕ � χ(εη)Ψ − Ψ�χ � ε−1ϕ � χ(εη) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Weak Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 (Green’s Identity, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2 of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Assume f(x, w), g(x, w) ∈ L2(Ω × S2) and w · ∇xf, w · ∇xg ∈ L2(Ω × S2) with f, g ∈ L2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then �� Ω×S2 �� w · ∇xf � g + � w · ∇xg � f � dxdw = � γ fg(w · n) = � γ+ fgdγ − � γ− fgdγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we can derive the weak formulation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For any test function g(x, w) ∈ L2(Ω×S2) with w · ∇xg ∈ L2(Ω × S2) with g ∈ L2 γ, we have � γ Rg(w · n) − �� Ω×S2 R � w · ∇xg � + ε−1 �� Ω×S2 � R − R � g = �� Ω×S2 Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Estimates of Boundary and Source Terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for h defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5), we have |h|L2 γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have |εw · ∇xU0|L2 γ− + ��ε2w · ∇xU1 �� L2γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Noting the cutoff χ � ε−1ϕ � restricts the support to |ϕ| ≲ ε and dγ measure contributes an extra sin ϕ, we have ��χ � ε−1ϕ � Ψ(0) �� L2γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) Hence, our result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S0 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7), we have ∥S0∥L2 ≲ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8), we have ��� 1 + η � S1 �� L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Also, for the boundary layer U B 0 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ��� 1 + η � U B 0 �� L2 ≲ ε 1 2 , ��� 1 + η � U B 0 �� L2xL1w ≲ ε 1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) and ��� �� 1 + η � S1, g ���� ≲ ��� � 1 + η � ⟨v⟩2 U B 0 ��� L2 ∥∇wg∥L2 ≲ ε 1 2 ∥∇wg∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We split S1 = S11 + S12 := � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂Ψ ∂ϕ �χ � ε−1ϕ � χ(εη) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) + � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂�χ � ε−1ϕ � ∂ϕ χ(εη)Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that S11 is nonzero only when |ϕ| ≥ ε and thus based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we know ���� ∂Ψ ∂ϕ ���� ≤ |sin ϕ|−1 |Ψ| ≲ ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Hence, using dµ = εdη, we have ∥S11∥L2 ≲ ��� |ϕ|≥ε ���� ∂Ψ ∂ϕ ���� 2 dϕdµ � 1 2 ≲ ��� |ϕ|≥ε |sin ϕ|−2 |Ψ|2 dϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) ≲ ��� |ϕ|≥ε |sin ϕ|−2 e−2Kηdϕdµ � 1 2 ≲ � ε �� |ϕ|≥ε |sin ϕ|−2 e−2Kηdϕdη � 1 2 ≲ � εε−1� 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Noticing ∂�χ � ε−1ϕ � ∂ϕ = ε−1�χ′� ε−1ϕ � , and �χ′� ε−1ϕ � is nonzero only when ε < |ϕ| < 2ε, based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have ∥S12∥L2 ≲ε−1 ��� ε<|ϕ|<2ε |Ψ|2 dϕdµ � 1 2 ≲ ε−1 ��� ε<|ϕ|<2ε e−2Kηdϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) ≲ε−1 � ε �� ε<|ϕ|<2ε e−2Kηdϕdη � 1 2 ≲ ε−1 (εε) 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Collecting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that e−Kη will suppress the growth from the pre-factor 1 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) comes from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then we turn to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The most difficult term in �� ⟨S1, g⟩ �� is essentially ���� �∂U B 0 ∂ϕ , g �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Integration by parts with respect to ϕ implies ���� �∂U B 0 ∂ϕ , g ����� ≲ ���� � U B 0 , ∂g ∂ϕ ����� ≲ ��U B 0 �� L2 ���� ∂g ∂ϕ ���� L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) and ∂x ∂ϕ = 0, we know the substitution (µ, ι1, ι2, w) → (µ, ι1, ι2, w) implies −∂w ∂ϕ · n = cos ϕ, ∂w ∂ϕ · ς1 = − sin ϕ sin ψ, ∂w ∂ϕ · ς2 = − sin ϕ cos ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) Hence, we know ���� ∂w ∂ϕ ���� ≲ 1, and thus ���� ∂g ∂ϕ ���� ≲ |∇wg| ���� ∂w ∂ϕ ���� ≲ |∇wg| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Hence, we know that ���� �∂U B 0 ∂ϕ , g ����� ≲ ��U B 0 �� L2 ∥∇wg∥L2 ≲ ε 1 2 ∥∇wg∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9), we have ��� 1 + η � S2 �� L2 ≲ ε 1 2 , ��� 1 + η � S2 �� L2xL1w ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Notice that ����ε−1 sin φ�χ � ε−1ϕ �∂χ(εη) ∂η ���� ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we directly bound ∥S2∥L2 ≲ ��� � |Φ|2 + ���� ∂Φ ∂ι1 ���� 2 + ���� ∂Φ ∂ι2 ���� 2 + ���� ∂Φ ∂ψ ���� 2 � dϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) ≲ ��� e−2Kηdϕdµ � 1 2 ≲ � ε �� e−2Kηdϕdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then the L2 xL1 w estimate follows from a similar argument noting that there is no rescaling in w variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S3 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), we have ��� 1 + η � S3 �� L2 ≲ 1, ��� 1 + η � S3 �� L2xL1w ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Using χ = 1 − �χ, we split S3 = S31 + S32 :=ε−1Ψχ � ε−1ϕ � χ(εη) − ε−1χ � ε−1ϕ � χ(εη)Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='30) Noting that S31 is nonzero only when |ϕ| ≤ ε, based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have ∥S31∥L2 ≲ ��� |ϕ|≤ε ��ε−1Ψ ��2 dϕdµ � 1 2 ≲ � ε−2 �� |ϕ|≤ε e−2Kηdϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='31) ≲ � ε−1 �� |ϕ|≤ε e−2Kηdϕdη � 1 2 ≲ � ε−1ε � 1 2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Analogously, noting that S32 contains w integral, we have ∥S32∥L2 ≲ ��� ���ε−1Ψχ(ε−1ϕ) ��� 2 dϕdµ � 1 2 ≲ \uf8eb \uf8edε−2 �� ����� � |ϕ|≤ε Ψdϕ ����� 2 dϕdµ \uf8f6 \uf8f8 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='32) ≲ \uf8eb \uf8edε−2 �� ����� � |ϕ|≤ε e−Kηdϕ ����� 2 dϕdµ \uf8f6 \uf8f8 1 2 ≲ � ε−2 �� ε2e−2Kηdϕdµ � 1 2 ≲ ��� e−2Kηdϕdµ � 1 2 ≲ � ε �� e−2Kηdϕdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Collecting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='32), we have the L2 estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Similarly, we derive the L2 xL1 w bound: ∥S31∥L2xL1w ≲ �� � � |ϕ|≤ε ��ε−1Ψ �� dϕ �2 dµ � 1 2 ≲ �� e−2Kηdµ � 1 2 ≲ � ε � e−2Kηdη � 1 2 ≲ ε 1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='33) ∥S32∥L2xL1w ≲ �� � � ���ε−1Ψχ(ε−1ϕ) ��� dϕ �2 dµ � 1 2 ≲ � ε−2 � � � ����� � |ϕ|≤ε Ψdϕ ����� dϕ �2 dµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='34) ≲ � ε−2 � � � εe−Kηdϕ �2 dµ � 1 2 ≲ �� e−2Kηdµ � 1 2 ≲ � ε � e−2Kηdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Estimate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Basic Energy Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Taking g = ε−1R in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we obtain ε−1 2 � γ |R|2 (w · n) + ε−2� R, R − R � = ε−1� R, S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) Then using the orthogonality of R and R − R, we have ε−1 2 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 = ε−1� R, S � + ε−1 2 |h|2 L2γ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, we know ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ ε + ε−1� R, S0 + S1 + S2 + S3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have ���ε−1� R, S0 ���� ≲ ε−1 ∥R∥L2 ∥S0∥L2 ≲ ε ∥R∥L2 ≲ o(1) ∥R∥2 L2 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we have ���ε−1� R − R, S1 + S2 + S3 ���� ≲ε−1 ��R − R �� L2 ∥S1 + S2 + S3∥L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) ≲ε−1 ��R − R �� L2 ≲ o(1)ε−2 ��R − R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Finally, we turn to ε−1� R, S1 + S2 + S3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For S1, we integrate by parts with respect to ϕ and use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4 to obtain ���ε−1� R, S1 ���� =ε−1 ���� � R, � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂U B 0 ∂ϕ ����� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) =ε−1 ���� � R, � sin2 ψ R1 − εη + cos2 ψ R2 − εη � U B 0 sin ϕ ����� ≲ε−1 ��R �� L2 ��U B 1 �� L2xL1w ≲ ε− 1 2 ��R �� L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6 yield ���ε−1� R, S2 + S3 ���� ≲ε−1 ��R �� L2 � ∥S2∥L2xL1w + ∥S3∥L2xL1w � ≲ ε− 1 2 ��R �� L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) Collecting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8), we obtain ���ε−1� R, S0 + S1 + S2 + S3 ���� ≲ o(1)ε−2 ��R − R ��2 L2 + o(1)ε−1 ∥R∥2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Kernel Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ��R ��2 L2 ≲ ��R − R ��2 L2 + |R|2 L2γ+ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Denote ξ(x) satisfying � −∆xξ = R in Ω, ξ(x0) = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Based on standard elliptic estimates and trace estimates, we have ∥ξ∥H2 + |ξ|H 3 2 ≲ ��R �� L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) Taking g = ξ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we have � γ Rξ(w · n) − � R, w · ∇xξ � + ε−1� R − R, ξ � = � S, ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Using oddness, orthogonality and ξ �� ∂Ω = 0, we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then taking g = w · ∇xξ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Adding ε−1×(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) to eliminate ε−1� R − R, w · ∇xξ � , we obtain � γ R � w · ∇xξ � (w · n) − � R, w · ∇x � w · ∇xξ �� =ε−1� S, ξ � + � S, w · ∇xξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Notice that − � R, w · ∇x � w · ∇xξ �� = − � R, w · ∇x � w · ∇xξ �� − � R − R, w · ∇x � w · ∇xξ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) and Cauchy’s inequality yield − � R, w · ∇x � w · ∇xξ �� ≃ ��R ��2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) ��� � R − R, w · ∇x � w · ∇xξ ����� ≲ ��R − R ��2 L2 + o(1) ��R ��2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Also, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, we have ���� � γ R � w · ∇xξ � (w · n) ���� ≲ � |R|L2γ+ + |h|L2γ− � |∇xξ|L2 ≲ o(1) ��R ��2 L2 + |R|2 L2γ+ + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we obtain ��R ��2 L2 ≲ε2 + ��R − R ��2 L2 + |R|2 L2γ+ + ���ε−1� S, ξ ���� + ��� � S, w · ∇xξ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Then we turn to the estimate of source terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Cauchy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 yield ���ε−1� S0, ξ ���� + ��� � S0, w · ∇xξ ���� ≲ ε−1 ∥S0∥L2 ∥ξ∥H1 ≲ ε ��R �� L2 ≲ o(1) ��R ��2 L2 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7), we first integrate by parts with respect to ϕ in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Using ξ �� ∂Ω = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), Hardy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we have ���ε−1� S1 + S2 + S3, ξ ���� ≲ ����ε−1� U B 0 + S2 + S3, � µ 0 ∂ξ ∂µ ����� = ���� � ηU B 0 + ηS2 + ηS3, 1 µ � µ 0 ∂ξ ∂µ ����� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) ≲ ��ηU B 0 + ηS2 + ηS3 �� L2xL1w ���� 1 µ � µ 0 ∂ξ ∂µ ���� L2 ≲ ��ηU B 0 + ηS2 + ηS3 �� L2xL1w ���� ∂ξ ∂µ ���� L2 ≲ ε 1 2 ∥ξ∥H1 ≲ε 1 2 ��R �� L2 ≲ o(1) ��R ��2 L2 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Analogously, we integrate by parts with respect to ϕ in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), fundamental theorem of calculus, Hardy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we bound ��� � S1 + S2 + S3, w · ∇xξ ���� ≲ ����� � U B 0 + S2 + S3, ∇xξ ��� µ=0 + � µ 0 ∂ � ∇xξ � ∂µ ������ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) ≲ ���� � U B 0 + S2 + S3, ∇xξ ��� µ=0 ����� + �����ε � ηU B 0 + ηS2 + ηS3, 1 µ � µ 0 ∂ � ∇xξ � ∂µ ������ ≲ ��U B 0 + S2 + S3 �� L2xL1w |∇xξ|L2 + ε ��ηU B 0 + ηS2 + ηS3 �� L2 ����� ∂ � ∇xξ � ∂µ ����� L2 ≲ε 1 2 |∇xξ|L2 ∂Ω + ε ∥ξ∥H2 ≲ ε 1 2 ��R �� L2 ≲ o(1) ��R ��2 L2 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Hence, inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19), we have shown (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ε− 1 2 |R|L2γ+ + ε− 1 2 ��R �� L2 + ε−1 ��R − R �� L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), we have ��R ��2 L2 ≲ ��R − R ��2 L2 + |R|2 L2γ+ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25), we have ��R ��2 L2 ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) Hence, adding ε−1×(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26), we have ε−1 |R|2 L2γ+ + ε−1 ��R ��2 L2 + ε−2 ��R − R ��2 L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) Then our result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proof of Main Theorem The well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) is well-known [5, 4, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The construction of U0, Φ and Φ∞ follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, so we focus on the derivation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5), we have ��uε − U0 − εU1 − ε2U2 − U B 0 �� L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have ��εU1 + ε2U2 �� L2 ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 and the rescaling η = ε−1µ, we have ��U B 0 �� L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) follows from inserting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2)(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Golse, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Perthame, The Rosseland approximation for the radiative transfer equations, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 40 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 69–721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Golse, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Perthame, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sentis, The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 77 (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 434–460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phung, Observation estimate for kinetic transport equations by diffusion approximation, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Paris, 355 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 640–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Santos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sentis, Diffusion approximation and computation of the critical size, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 284 (1984), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 617–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bensoussan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lions, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Papanicolaou, Boundary layers and homogenization of transport processes, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 15 (1979), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 53–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Esposito, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Kim, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 323 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 177–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo and L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='edu (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu) Department of Mathematics, Lehigh University Email address: lew218@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'}