NTU-KU Joint Funding
Final Report

Section 1

NTU principle investigator
Name (last name, first name)
Position
Faculty/Department

Chen, I-Kun
Associate Professor
Department of Mathematics

KU principle investigator
Name (last name, first name)
Position
Faculty/Department
Visiting ECR*
Name (last name, first name)
Position
Faculty/Department

Kawagoe, Daisuke
Assistant Professor
Graduate School of Informatics
Su, Jhe-Kuan
PhD student
Department of Mathematics*Please complete this section if the KU principal investigators hosted ECRs from NTU.

Host researcher*
Name (last name, first name)
PositionFaculty/Department*Please complete this section if the host researcher is different from the KU principal
investigator.
Section 2Project title
Regularity of the solution to the stationary
Boltzmann equation in a bounded domain

Section 3
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NTU-KU Joint Funding Final Report

Period of project
From dd/mm/yy
to dd/mm/yy

From 19/10/2022 to 01/11/2022

Section 4

Summary of the project (approx. 100 words)*KU PIs are required to submit a summary of the project in Japanese in addition to theEnglish summary (approx. 200–300 characters).
(Please enter the summary of the project)
During the ECR’s stay, we discussed singularity of the first derivatives of solutions to theinflow boundary value problem of the stationary linearized Boltzmann equation. So far, heshowed that the solution belongs to the Sobolev space H^1, which means that its first
derivatives are square-integrable, if the domain is bounded convex of positive Gaussian
curvature and if it is small enough. In this project, we consider a thin slab domain andconstructed an example of a solution whose first derivative is not square-integrable. This
example supports that the positive Gaussian curvature condition is crucial in the previousresult.
入射境界条件を伴う定常線型 Boltzmann 方程式の解の偏導関数の特異性を議論した. ECR の Su 氏はこれまでに, 方程式を考える領域が有界凸でその境界の Gauss曲率が一様に正である場合に, 領域の直径が十分小さければ, 滑らかな入射に対して解は Sobolev 空間 H1 に属することを示していた. これは, 解の偏導関数が2乗可積分であることを意味する. 本プロジェクトでは, この Gauss 曲率に関する条件に焦点を当て, 曲率が0である平行平板の問題を考え, 解の偏導関数が2乗可積分でないような例を構成した. この例は, 境界の Gauss 曲率が解の偏導関数に影響を与えることを示唆している.
Section 5 (Please complete this section if ECRs from NTU participated in collaborative research at KU)

Achievements and Outcomes of ECRs’ Stay (approx. 100–250 words)*This section should be filled by each of the ECR(s) (one paragraph per ECR) based on
his/her experience of staying in Japan.
(Please enter the achievements and outcomes for each of the ECR(s).)
In the stay in Kyoto, I discussed with Professor Kawagoe about the H1 regularity
problem for stationary linearized Boltzmann equation with incoming boundary
condition in slab domain. Initially we construct an L2 solution and decomposed it intosome parts and examine the regularity of each term. In the process, I learned some
new ways to estimate the integral operator, which can be used for simpler
mathematical argument. As a result, we finished some crucial step for constructingsome counterexample for H1 regularity of solution, which implies that the assumptionthat the boundary of domain is of positive Gaussian curvature is crucial for the H1
regularity of the stationary solution.
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NTU-KU Joint Funding Final Report

Section 6

Photographs with captions*Please submit digital files (such as JPEG or GIF files) of the photographs used in yourreport as attachments. The size of each image should be at least 4MB, so that it can be used
for printed materials. Please ensure that none of the photographs submitted will cause any
issues relating to portrait rights.

URL at which project outcomes can be viewed (Optional)*E.g. workshop notifications/programs/reports, evidence of academic papers published orotherwise made available, etc.
URL:Page 3 of 3