Announcement
Jan. 21 2019 Dear All,
Everyone is invited to attend the two Meetings. Please register on line for the Workshop which will help us to make a better preparation. The registration fee is NT$1000 for those who has grant of MOST. We also need a lot of chairs for the meetings. Please let us know if you are interested in hosting some sessions of the Meetings. We have a limited support to cover lodging fee and transportation fees for those who do not have any grant. Any suggestions and comments are welcome.
The 7th Trilateral Meeting (Australia‐Italy‐Taiwan) on Nonlinear PDEs and Applications Date: Jan. 23 (Wed) 2019 ~ Jan. 28 (Mon) 2019
Venue: Small lecture hall, Ger‐Jyh Hall, College of Science, Cheng Kung Campus, NCKU (國立成功大學成功校區，理化大樓，格致廳小講堂)
Scientific Committee:
Nicola Fusco <n.fusco@unina.it>
Tai‐Ping Liu <liu@math.stanford.edu>
Neil Trudinger <Neil.Trudinger@anu.edu.au>
Local Organizer:
Yung‐Fu Fang <yffang@mail.ncku.edu.tw>
Ching‐Lung Lin <cllin2@mail.ncku.edu.tw>
The 27th Annual Meeting on Differential Equations and Related Topics Date: Jan. 26 (Sat) 2019 ~ Jan. 27 (Sun) 2019
Date: Jan. 28 (Mon) 2019 , Informal Discussion Day
Venue: Department of Mathematics, National Cheng Kung University (NCKU), Tainan, Taiwan Local Organizer:
王辰樹 Chern‐Shuh Wang <chenshu@mail.ncku.edu.tw>
方永富 Yung‐Fu Fang <yffang@mail.ncku.edu.tw>
史習偉 Hsi‐Wei Shih <shihhw@mail.ncku.edu.tw>
林景隆 Ching‐Lung Lin <cllin2@mail.ncku.edu.tw>
陳旻宏 Min‐Hung Chen <chen0499@mail.ncku.edu.tw>
林敏雄 Matthew Lin <mhlin@mail.ncku.edu.tw>
Website: http://www.ncts.ntu.edu.tw/events_2_detail.php?nid=210 http://www.math.ncku.edu.tw/news/news.php
http://www.math.ncku.edu.tw/~fang/
Online Registration: https://goo.gl/forms/ToLtgwHU5KR1xwlv2
Hotel Informations:
Zenda Suites (成大會館)
(On Campus) (Single NT$2800 per night, Double NT$3600 per night) Address: No.2, Dasyue Rd., East District. Tainan City 701, Taiwan.
Tel \ +886‐6‐275‐8999 Fax \ +886‐6‐209‐3567 http://www.zendasuites.com.tw/en/about.php
GuangHaw Hotel (光華商務飯店)
(15 minutes walk) (Single NT$1200 per night, Double NT$1600 per night)
Address: No. 155 Sec. 1, Beimen Road, West Central Dist., Tainan city 700 Taiwan Tel: +886‐6‐226‐3171
Email: only4utw@gmail.com http://www.guanghaw.com.tw
Dynasty Hotel (新朝代大飯店)
(20 minutes walk) (Single NT$1800 per night, Double NT$2400 per night) (online reservation) Address: No. 46 Cheng Kung Road, North District, Tainan 704
Email: tndynasty@dynastyhotel.com.tw Tel: 06 225 8121
The 7th Trilateral Meeting (Australia‐Italy‐Taiwan) on Nonlinear PDEs and Applications
January 23‐28, 2019
Venue: Small lecture hall, Ger‐Jyh Hall, College of Science National Cheng Kung University (Cheng Kung Campus) 國立成功大學，成功校區，理化大樓，格致廳小講堂
Website: http://www.ncts.ntu.edu.tw/events_2_detail.php?nid=210 http://www.math.ncku.edu.tw/news/news.php
On‐Line Registration: https://goo.gl/forms/ToLtgwHU5KR1xwlv2
Speakers from Australia:
Julie Clutterbuck, Monash University Daniel Daners, University of Sydney Yihong Du, University of New England Xuan Duong, Macquarie University
Ramiro Lafuente, University of Queensland Zihua Guo, Monash University
Jiakun Liu, University of Wollongong
Neil Trudinger, Australian National University
Speakers from Italy:
Stefano Bianchini, SISSA, Trieste:
Lucio Boccardo, University of Roma 1:
Piermarco Cannarsa, University of Roma 2:
Nicola Fusco, University of Napoli
Giuseppe Mingione, University of Parma:
Massimiliano Morini, University of Parma:
Aldo Pratelli, University of Erlangen‐Nurnberg
Speakers from Taiwan:
Chiun‐Chuan Chen, National Taiwan Univ.
Volker Elling, Academia Sinica
Hsin‐Yuan Huang, National Chiao Tung Univ.
Jin‐Cheng Jiang, National Tsing Hua Univ.
Jenn‐Nan Wang, National Taiwan Univ.
Kung‐Chien Wu, National Cheng Kung Univ.
Shih‐Hsien Yu, National Univ. of Singapore
Scientific Committee:
Neil Trudinger, Australian National University Nicola Fusco, University of Napoli
Tai‐Ping Liu, Sinica & Stanford University Organizers:
Yung‐fu Fang, National Cheng Kung University Ching‐Lung Lin, National Cheng Kung University
Sponsors:
MOST
National Center for Theoretical Sciences TWSIAM
National Cheng Kung University NSCMRPC
The 27th Annual Meeting on Differential Equations and Related Topics
January 26‐27, 2019
Venue: Small lecture hall, Ger‐Jyh Hall, College of Science
National Cheng Kung University (NCKU) (Cheng Kung Campus) 國立成功大學，成功校區，理化大樓，格致廳小講堂
Website: http://www.ncts.ntu.edu.tw/events_2_detail.php?nid=210 http://www.math.ncku.edu.tw/news/news.php
On‐Line Registration: https://goo.gl/forms/ToLtgwHU5KR1xwlv2 Plenary Speakers:
Giuseppe Mingione, University of Parma:
Kenji Nakanishi (中西賢次), Kyoto University Neil Trudinger, Australian National University
Shih‐Hsien Yu (尤釋賢), National University of Singapore
Organizers:
Min‐Hung Chen (陳旻宏), NCKU Yung‐Fu Fang (方永富), NCKU Ching‐Lung Lin (林景隆), NCKU
Matthew Lin (林敏雄), NCKU Hsi‐Wei Shih (史習偉), NCKU Chern‐Shuh Wang (王辰樹), NCKU
Sponsors:
Speakers of DE:
Chi‐Hin Chan (陳子軒), NCTU Chueh‐Hsin Chang (張覺心), THU I‐Kun Chen (陳逸昆), NTU
John M. Hong (洪盟凱), NCU Chih‐Chiang Huang (黃志強), NTU Hung‐Wen Kuo (郭鴻文), NCKU Hsin‐Yi Lee (李信儀), NCU Yu‐Hao Liang (梁育豪), NUK Ying‐Chieh Lin (林英杰), NUK Bingying Lu (陸冰瀅), Sinica Cheng‐Fang Su (蘇承芳), NCTU Ryosuke Takahashi (楊劼之), NCKU Kuan‐Hsiang Wang (王冠祥), NUK
Speakers of Applied Math:
Po‐Yuan Chen (陳博源), MDI Center, NCKU Dean Chou (周鼎贏), NCU
Chia‐Yu Hsu (許佳璵), Feng Chia Univ.
Wei‐Qiang Huang (黃韋強), NCTU Ming‐Cheng Shiue (薛名成), NCTU Maxim Solovchuk, NHRI
Chun‐Hao Teng(鄧君豪), NCHU Yun‐Che Wang (王雲哲), NCKU Mei‐Heng Yueh (樂美亨), NTNU Zhengyang Zhang (張正陽), NTHU
MOST
National Center for Theoretical Sciences TWSIAM
National Cheng Kung University NSCMRPC
微方年會 24 年感言 撰文：許世壁 (清華大學數系教授 清華大學數系教授 )
微分方程年會 (Annual Workshopon Differential Equations)自 1993 年第一次在中正大學舉行 後迄今共 24 年 。當初是由中正大學林長壽教授召集台灣從事微分方程研究專家者提出每舉辨「偏微分方程研
討會」，後來經過一番論將常動態系統包括進去名稱 舉辨「偏微分方程研討會」，後來經過一番論將
常動態系統包括進去名稱 改為目前的「微分方程研討會」。自改為目前的「微分方程研討會」。自 1993
年迄今，舉辦三次有清大 (1996, 2004, 2014)，二次 的有中正大學 (1993, 2005)，交大 (1994 1 月, 2008)，
中研院 (1994 12 月, 2006 1 月)，中央大學 (1997, 2013)，成大 (2001, 2011)，中山大學 (2000, 2016)，
其餘的舉辦一次有師範大學 ，其餘的舉辦一次有師範大學 (1998)，中興大學 (1999)，彰師大 (2002)，
靜宜大學 ，靜宜大學 (2002 12 月)，南台科技大學 ，南台科技大學 (2006)，高雄 ，高雄大學 (2009)，
台大 (2010)，淡江大學 (2012)，台東大學 ，台東大學 (2015)。微方年會有時與國內分析研討會一起 辦， (2006 中研院 , 2004 清大)，有時會以國際研討的方式舉辦 (如 1994 年中研院， 暨中曰算子 理論聯合研討 2011 年成大 )。2004 年在清大微方會，我們藉此機向王懷權 教授退休致最大的敬意，
並且將論文收集發表在台灣數學期刊 (2005)。在 2014 年 微方年會，清大同仁藉此機順便祝賀我本 人 65 歲生日。目前在台灣數學界從事微分方程研究的人 口是最多 (見下列表格 )，其研究成果也有 目共睹，多人發表論文在頂尖期刊許同仁得過 國科會的傑出研究獎， 吳大猷數學術年輕家理論中心 年青學者 獎, 教育部 學術獎，國家講座中研院的年輕者士等榮譽。
回想我個人在 1979 年 8 月回交大任教，當時從事微分方程研究的人口不多後來由於丘 成桐應用偏 微分方程到幾何，才漸漸受到重視。在 1980 年暑假馬利蘭大學劉太平教授在中研院課介紹
Conservation law 及 gas dynamics。記得當時每星期一次，我、鄭國順、林松山從新竹坐巴士到台北 圓環，然後搭 305 號公車到南港。那時劉太平提出了一些 open problems 如 nonconvex conservation law，很快就被學物理出身的鄭國順 解決了。做事一向以動作快出名，吃飯、結婚生小孩學問都比般 人。而後在 1982 年劉太平 回中研院一年，每星期在交大 舉辦 gas dynamics 的 Seminar，那時參加 的人員有鄭國順、林紹雄、林松山及我本人，記得紹講述他在 gas dynamics 的研究，而劉太平也開 始做的研究，而劉太平也開始做 Boltzmann equation 的探討，這對他後來在 Boltzmann equation 的研 究有很大幫助。 劉院士 在 2006 年返台致力於培育青人研究雙曲型偏微分方程。
1987 年暑假 應清大王懷權 教授之邀請，明尼蘇達大學倪維教授 在清大講述 Semilinear Elliptic Equations，其內容涵蓋 ，其內容涵蓋 Semilinear Dirichlet Problems, Semilinear Neumann Problems, Semilinear Elliptic Equatons on R^n，討論的問題有 patternformation (Spiky pattern), Lane‐Emden equations, Conformal scalar curvature equations。倪維明有系統地介紹橢圓型偏微分方程與台灣 數學 界，而且留下了一本經典之作「 Some Aspects of Semilinear Elliptic Equation」。直到目前為止，許多人 從念這本講義開始學偏微分方程，為了感謝倪維明對台灣發展的貢獻，數學會於 2009 年頒發特殊貢 獻獎與倪教授。
中華民國數學會 電子報第 27 期
另一位台灣微分方程發展的重要人物是林長壽教授。早年研究幾何學局部 等距嵌入問題出名 。林教 授自 1987 年回台大任教 ，1990 年轉至中正大學 ，閉門研究 “肥皂 泡現象 ”的偏微分方程有重 大突破為國內外數學界肯定 。1997 林教授 擔任首任國家理論科學中心主任 ，推動國內數學的發展 有很大貢獻 。而後於 2006 年返回台大任教，創立數學科中心 (TIMS)，高等應用科學中心 ，高等應 用科學中心 (CAST)，林教授與其合作者陳俊全發表一系列文章在頂尖數學期刊研究 Conformal Scalar Curvature equation。近年來執行科技部攻頂計畫，研究非線性偏微分方程應用於解決數論、代幾何的 問題， 同時也 培育許多國內外偏微分方程的人才。
台灣微分方程界參與許多國際交流學術活動 。1985 年在交大舉辦中美 PDE 研討會 ，美方由劉太平
擔任團長，團員有丘成桐，Nirenberg, Glimm, Rabinowitz, Stroock 等國際知名學者。 這是一個非常不 對稱的研討會，我們很幸運地能跟當時 頂尖的數學家接觸，包括一起去中橫觀光，享受解決道路坍 方的樂趣 ；1990 年在中研院舉辦第一屆中日偏微分方程討會，
1994 年我代表台灣向日本交流協會申請舉辦中日偏微分方程研討會。研討會在京都舉行，
台灣代表有劉太平、倪維明、林長壽、鄭國順、林松山、郭忠勝、王慶安、王懷權、王信華、黃子偉，
日方代表以 Mimura 為首 。這是一個非常正式的研討會，台灣的亞東協會及日本交流協會皆有外交 代表在國宴致詞。從這以後台日交流頻繁，譬如每年春天台日輪流主辦年青數學家研討會，雙方的碩 士生、博士生、博士後研究人員能在一起切磋。為了感謝 Mimura 教授努力推動雙方交流 ,數學會 於 2008 年頒 給他特殊貢獻獎 ；法國是歐洲家中與台灣交流 最多的國家 。早在 1982 年大 Lions, Brezis 及 Temam 應法國在台協會之邀請訪問灣 應法國在台協會之邀請訪問灣 ，台灣與法國在 1988 年首 次由王懷權教授在清大主辦 台法微方研討會。1992 年我方組團到巴黎。直到目前台法交流持續進行，
由林長壽院士、郭忠勝教授主持；台灣、澳洲、義大利有三邊方程研討會 ，開始由郭紅珠 、Trudinger 辦台灣、澳洲研討會，目前三 邊研討會是由劉太平院士主導 。
微分方程是一門具有強大生命力的領域，許多問題源自於物理、工及幾何 學。從 1980 年至現在 2015 年， 35 年的期間台灣在微分方程研究有長足進步。國內兩位院士劉太平與林長壽在他們的研究領域 居國際導地位，而且也為內年輕人培育工作付出時間與心血。希望未來辛苦建立的國際名聲能繼續傳 承下去。
微分方程研討年會歷年舉辦地點
第 1 屆 1993 中正大學 第 2 屆 1994 交通大學
第 3 屆 1994‐12 月 中央研究院（與中日算子理論聯合研討會一併舉行）
第 4 屆 1996 清華大學 第 5 屆 1997 中央大學 第 6 屆 1998 師範大學 第 7 屆 1999 中興大學 第 8 屆 2000 中山大學 第 9 屆 2001 成功大學 第 10 屆 2002 彰化師範大學 第 11 屆 2002－12 月 靜宜大學
第 12 屆 2004 清華大學（與分析研討會一併舉行） 第 13 屆 2005 中正大學
第 14 屆 2006 中央研究院（與分析研討會一併舉行） 第 15 屆 2006‐12 月 南台科技大學 第 16 屆 2008 交通大學 第 17 屆 2009 高雄大學 第 18 屆 2010 台灣大學 第 19 屆 2011 成功大學 第 20 屆 2012 淡江大學 第 21 屆 2013 中央大學 第 22 屆 2014 清華大學 第 23 屆 2015 臺東大學 第 24 屆 2016 中山大學 第 25 屆 2017 交通大學 第 26 屆 2018 台灣大學
第 27 屆 2019 成功大學 第二十七屆微分方程及相關主題年會
第 28 屆 2020 中央研究院 (Academia Sinica) 第 29 屆 2021 中興大學
台灣、澳洲、義大利有三邊方程研討會
‧1st Trilateral Meeting in Taipei, Taiwan, 2000
‧2nd Trilateral Meeting in Murramarang, Australia, 2003
‧3rd Trilateral Meeting in Rome, Italy, 2006
‧4th Trilateral Meeting in Taipei, Taiwan, 2009
‧5th Trilateral Meeting in Wollongong, Australia, 2012
‧6th Trilateral Meeting in Parma, Italy, 2015
‧7th Trilateral Meeting in Tainan, Taiwan, 2019
The 7th Australia‐Italy‐Taiwan Trilateral Meeting on Partial Differential Equations and Applications
0810~0830 Registration
0830~0900 Opening Ceremony President Jenny Su
Tai-Ping Liu Nicola Fusco
Chair: Nicola Fusco Chair: Shin-Hwa Wang Chair: Tai-Ping Liu
0900~0950 Lucio Boccardo 0900~0950 Yihong Du 0900~0950 Nicola Fusco
0950~1010 Break 0950~1010 Break 0950~1010 Break
Chair: B Chair: Haitao Wang Chair: J
1010~1100 Xuan Duong 1010~1100 Stefano Bianchini 1010~1100 Zihua Guo
1110~1200 Aldo Pratelli 1110~1200 Kung-Chien Wu 1110~1200 Volker Elling
1200~1400 Lunch 1200~1400 Lunch 1200~1400 Lunch
Chair: Kuo-Chang Chen Chair: G Chair: K
1400~1450 Piermarco Cannarsa 1400~1450 Massimiliano Morini 1400~1450 Daniel Daners
1500~1550 Chiun-Chuan Chen 1500~1550 Jenn-Nan Wang 1500~1550 Jin-Cheng Jiang
1550~1610 Break 1550~1610 Break 1550~1610 Break & Group Photo
Chair: D Chair: H Chair: John M. Hong
1610~1700 Jiakun Liu 1610~1700 Julie Clutterbuck 1610~1700 Hsin-Yuan Huang
1710~1800 Ramiro A. Lafuente 1710~1800 1710~1800
Buffet 上海小籠湯包 1820~ 三采日式料理
會場：成大，成功校區，格致廳，小講堂
The 7th Trilateral Meeting (Australia-Italy-Taiwan) on Nonlinear PDEs and Applications
The 7th Trilateral Meeting (Australia - Italy - Taiwan) on Nonlinear PDEs and Applications &
The 27th Annual Meeting on Differential Equations and Related Topics,
Jan. 23 (Wed) 2019 ~ Jan. 28 (Mon) 2019
Venue: Small lecture hall, Ger-Jyh Hall, College of Science, NCKU, Tainan, Taiwan
Program
Wednesday (01/23) Thursday (01/24) Friday (01/25)
0830~0850 Registration
0850~0900 Chair:
0900~0950
0950~1010 Break 0950~1010 Break
Chair: Chang-Hong Wu Chair: Juan-Ming Yuan
1010~1050 John M. Hong 1010~1050 Ming-Cheng Shiue
1100~1130 Chih-Chiang Huang 1055~1135 Mei-Heng Yueh
1140~1220 Hung-Wen Kuo 1140~1220 Chia-Yu Hsu
1210~1400 Lunch 1220~1400 Lunch
Chair:
1400~1450
1450~1510 Break Break
Chair: Jen-Hsu Chang Chair:
1510~1550 Chi-Hin Chan 1510~1550 Maxim Solovchuk
1600~1630 Chueh-Hsin Chang 1550~1630 Wei-Qiang Huang
Group Photo Group Photo
1630~1650 Break 1630~1650 Break
Chair: Shih-wei Chou Chair: Chi-Tien Lin
1650~1720 I-Kun Chen 1650~1720 Zhengyang Zhang
1730~1800 Yu-Hao Liang 1720~1800 Po-Yuan Chen 1830~
Chair: Chair:
0900~0950 0900~0950
0950~1010 Break 0950~1010 Break 0950~1010
Chair: Chun-Kong Law Chair: Chin-Tien Wu Chair:
1010~1050 Ying-Chieh Lin 1010~1050 Yun-Che Wang 1010~1050
1100~1130 Cheng-Fang Su 1050~1130 Dean Chou 1050~1130
1140~1210 Bingying Lu 1140~1220 Chun-Hao Teng 1140~1220
1210~1400 Lunch 1220~1400 Lunch 1220~1400
Chair: Chair:
1400~1450 1400~1450
1450~1510 Break Break
Chair: Bo-Chih Huang Chair: Z
1510~1550 Ryosuke Takahashi 1510~1550 1510~1550
1600~1630 Hsin-Yi Lee 1550~1630 1550~1630
1630~1700 Kuan-Hsiang Wang 1650~1730 1650~1730
1700~1800 1730~1810
阿裕牛肉 Neil Trudinger, Tai-Ping Liu,
The 27th Annual Meeting on Differential Equations and Related Topics
Giuseppe Mingione Saturday (01/26)
Informal Discussion Shih-Hsien Yu
Banquet 雨荷舞水 Opening Ceremony: Sze-Bi Hsu
Informal Discussion Monday(01/28)
Sunday (01/27) Sze-Bi Hsu Kenji Nakanishi
Informal Discussion
X U P
The 7th Trilateral Meeting (Australia ‐ Italy ‐ Taiwan) on Nonlinear PDEs and Applications
and
The 27th Annual Meeting on Differential Equations and Related Topics,
Date: Jan. 23 (Wed) 2019 ~ Jan. 28 (Mon) 2019 Venue: Ger‐Jyh Hall, Small lecture hall, College of Science
National Cheng Kung University (Cheng Kung Campus) 國立成功大學成功校區，理化大樓，格致廳小講堂
Website: http://www.ncts.ntu.edu.tw/events_2_detail.php?nid=210 http://www.math.ncku.edu.tw/news/news.php
http://www.math.ncku.edu.tw/~fang/
Online Registration: https://goo.gl/forms/ToLtgwHU5KR1xwlv2
Title and Abstract:
01/23 (Wednesday)
"Lucio Boccardo" <boccardo@mat.uniroma1.it>, University of Roma 1:
Title : Some elliptic equations with W^{01,1} solutions
Abstract: We consider some nonlinear Dirichlet problems and we study how lower order terms can give a regularizing effect on the solutions: the existence of distributional solutions with minimal properties (solutions in , , functional space not so usual for finding solutions of elliptic problems) or finite energy solutions, even with nonregular data.
‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
"Xuan Duong" <xuan.duong@mq.edu.au>, Macquarie University
Title: Dispersive estimates for self‐adjoint operators
Abstract: Let X be a metric space with a doubling measure satisfying μ(B) ≳ for any ball B with radius . Let L be a non negative self‐adjoint operator on . We assume that the semigroup satisfies a
Gaussian upper bound and that the flow satisfies a typical dispersive estimate of the form ǁ ǁ → ≲ |t|
Then we prove a similar dispersive estimate for a general class of flows ϕ , with ϕ of power type near 0 and near ∞. In the case of fractional powers ϕ , ν ∈ (0,1), we deduce
dispersive estimates for with data in Sobolev, Besov, or Hardy spaces with p ∈ (0,1], associated to the operator L. This is a joint work with The Anh Bui, Piero D'Ancona and Detlef Müller.
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"Aldo Pratelli" <pratelli@math.fau.de>, University of Erlangen‐Nurnberg
Title: On the isoperimetric problem with double density in .
Abstract: We will discuss about the isoperimetric problem in with double density. This means, one aims to minimize the perimeter of sets of given volume, but volume and perimeter are given by the integral of two different functions, called densities, over the set and its boundary respectively. As usual, the main questions are existence and regularity of minimizers. This problem has been deeply studied in the last decades, because of some interesting applications. The problem has primarily been studied with a single density, but the case of two different ones is particularly important, specially when the density of the perimeter also depends on the direction of the boundary. In this talk, we will give an overview of the main classical results and open questions, and we will concentrate on some recent developments. Parts of the talk are based on several joint papers with De Philippis, Franzina, Jachan, Morgan, Saracco.
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"Piermarco Cannarsa" <cannarsa@mat.uniroma2.it>, uniroma2
Title: Mean field games with state constraints
Abstract: This talk will address deterministic mean field games for which agents are restricted in a closed domain of Euclidean space. In this case, the existence, uniqueness, and regularity of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of solutions to the minimization problem which is solved by each agent is no longer guaranteed.
We will therefore attack the problem by considering a relaxed version of it, for which the existence of equilibria can be proved by set‐valued fixed point arguments. We will then give a uniqueness result for such equilibria under a classical monotonicity assumption. Finally, we will analyze the regularity of the relaxed solution and show that it satisfies the typical first order PDE system of mean field games.
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"(陳俊全) Chiun‐Chuan Chen" <chchchen@math.ntu.edu.tw>, National Taiwan University
Title: A total mass estimate for the diffusive Lotka–Volterra system of competing species
Abstract: Using an elementary approach, we establish a new maximum principle for the diffusive Lotka–
Volterra system of competing species in 1‐dim case, which involves pointwise estimates of an elliptic equation consisting of the second derivative of one function, the first derivative of another function, and a quadratic nonlinear term. This maximum principle gives an a priori estimate for the total mass of the species in a traveling wave solution. Applying this estimate to the system of three competing species leads to a nonexistence theorem of traveling wave solutions.
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"Jiakun Liu" <jiakunl@uow.edu.au>, University of Wollongong
Title: A boundary value problem for Monge‐Ampere equations.
Abstract: In this talk, we will present a recent result on the global C , and W , regularity for the Monge‐Ampere equation subject to a natural boundary condition arising in optimal transportation.
This is a joint work with Shibing Chen and Xu‐Jia Wang.
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"Ramiro A. Lafuente" <r.lafuente@uq.edu.au>, University of Queensland,
Title: Homogeneous Einstein manifolds via a cohomogeneity‐one approach
Abstract: We establish non‐existence results on non‐compact homogeneous Einstein manifolds. The key idea in the proof is to consider non‐transitive group actions on these spaces (more precisely, actions with cohomogeneity one), and to find geometric monotone quantities for the ODE that results from writing the Einstein equation in such a setting. As an application, we show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. This is joint work with C. Bhm.
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01/24 (Thursday)
"Yihong Du" <ydu@une.edu.au>, University of New England
Title: The Dynamics of a Fisher‐Kpp Nonlocal Diffusion Model with Free Boundaries
Abstract: We introduce and discuss a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models considered by Du and Lin [SIAM J. Math. Anal., 2010] and elsewhere, where "local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution de fined for all time, and then examine its long‐time dynamical behavior when the growth function is of Fisher‐KPP type. We demonstrate that a spreading‐vanishing dichotomy holds, though for the
spreading‐vanishing criteria significant differences arise from the well‐known local diffusion model.
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"Stefano Bianchini" <bianchin@sissa.it>, SISSA, Trieste:
Title: A decomposition of vector fields in Abstract:
Given a vector field ρ 1, b ∈ X , ) such that , ρ 1, is a measure, we consider the problem of uniqueness of the representation η of ρ 1, as a superposition of characteristics γ: , → , γ′ t \b t, γ t . We give conditions in terms of a local structure of the representation η on suitable sets in order to prove that there is a partition of into disjoint trajectories P , a ∈ A, such that the PDE
, ρ 1, ∈ M , u ∈ X ,
can be disintegrated into a family of ODEs along P with measure r.h.s.. The decomposition P is
essentially unique. We finally show that b ∈ , particular, the renormalization property for nearly incompressible BV vector fields.
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"(吳恭儉)Kung‐Chien Wu" < kcwu@mail.ncku.edu.tw>, National Cheng Kung University
Title: Spatial behavior of the solution to the Boltzmann equation with hard potentials
Abstract: The main goal of this talk is to understand the quantitative spatial decay of the solution to the Boltzmann equation with hard potentials for both linear and nonlinear problems.
For the nonlinear study, we get the spatial behavior by using the nonlinear weighted energy estimate. For the linear study, we get the quantitative space‐time behavior under some slow velocity decay assumption, but without regularity assumption on the initial data. Both results reveal that hard sphere and hard
potential models differ in their spatial behaviors. This is a joint work with Yu‐Chu Lin and Haitao Wang.
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"Massimiliano Morini" <morini73@gmail.com>, <massimiliano.morini@unipr.it>, University of Parma
Title: Existence and uniqueness for anisotropic and crystalline mean curvature flows
Abstract: An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor and Wang (1993), to a unique (up to fattening) “flat flow”.
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"(王振男)Jenn‐Nan Wang" <jnwang@math.ntu.edu.tw>, National Taiwan University
Title: Quantitative unique continuation for the fractional Schrödinger operator
Abstract: In this talk, I would like to discuss some quantitative uniqueness estimates related to the strong unique continuation property and the unique continuation at infinity for the fractional Schrödinger
operator. These kinds of estimates are useful in understanding the local properties of the solution. For the classical Schrödinger operator, these estimates have been extensively studied and successfully applied to other problems. Recently, the study of the local properties of solutions to the fractional equation became possible thanks to the Caffarelli‐Silvestre extension theorem. For the fractional Schrödinger operator, we are especially interested in the dependence of the estimates on the size of the potential. Besides of mathematical interests, fractional equations arise naturally from super‐diffusion and can be used in modeling a lot of physical phenomena involving long jumps.
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"Julie Clutterbuck" <julie.clutterbuck@monash.edu>, Monash University
Title: "The shape of the ground state for the Robin eigenvalue problem".
Abstract:
We consider the first eigenfunction of the Laplace operator with Robin boundary values. In the case of Neumann boundary values, the first eigenfunction is constant. In the case of Dirichlet boundary values, the first eigenfunction is log‐concave. The Robin case is often considered to interpolate between these two, and so it is reasonable to ask whether the first Robin eigenfunction is also log‐concave. We show that in general it is not, and classify the limited situations in which it is. This is joint work with Ben Andrews and Daniel Hauer.
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01/25 (Friday)
"Nicola Fusco" <n.fusco@unina.it>, University of Napoli:
Title: Asymptotic stability of the gradient flow of nonlocal energies
Abstract: I will discuss short time existence and long‐time stability of a class of equations modeling the evolution of the interface between an elastic material and a material void, controlled by mass diffusion within the surface. These equations appear as the H ‐gradient flow of an energy given by the sum of the area of the interface plus a nonlocal volume term. Our stability results are new even in the simplest case of the surface diffusion equation.
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"Zihua Guo" <zihua.guo@monash.edu>, Monash University
Title: Scattering for the 3D Gross‐Pitaevskii equation
Abstract: We study the Cauchy problem for the 3D Gross‐Pitaevskii equation. Global well‐posedness in the natural energy space was proved by Gerard.
We prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering. The crucial ingredients are new generalized Strichartz estimates and some new observed "NULL" structures of the Gross‐Pitaevskii equation after some normal form type transform. This is a joint work with Zaher Hani and Kenji Nakanishi.
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"Volker Elling" <velling@math.sinica.edu.tw>, Sinica
Title: Vortex cusps
Abstract: Vortex cusps are pairs of vortex sheets with opposite circulation that merge in a cusp. Such solutions are observed in engineering flows, for example vortex sheets in Mach reflection at a solid wall.
We present modelling and numerics of vortex cusps, calculate the cusp exponent and discuss possible rigorous existence proofs.
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"Daniel Daners" <daniel.daners@sydney.edu.au>, University of Sydney,
Title: Degenerate periodic‐parabolic evolution equations of logistic type
Abstract: We consider periodic‐parabolic evolution equations with a logistic nonlinearity allowing spacial and temporal degeneration as a parameter becomes large. We characterize the existence and stability of positive periodic‐parabolic solutions with the help of a parabolic maximum principle on non‐cylindrical domains. This is joint work with Julian Lopez‐Gomez.
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"(江金城) Jin‐Cheng Jiang" <jcjiang@math.nthu.edu.tw>, National Tsing Hua University
Title: On the Cauchy problem for the Boltzmann equation
Abstract: In this talk, we will present some recent progress on the Cauchy problem for the Boltzmann equation. We will begin with the introduction of the Boltzmann equation, its connection with fluid
dynamics. Then the result of the local well‐posedness for the Cauchy problem of the non‐cutoff Boltzmann equation in the weighted Sobolev space will be presented. The quasi‐linear method instead of linearization method is used to prove the existence and the non‐negativity of the solution.
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"(黃信元)Hsin‐Yuan Huang" <hyhuang@math.nctu.edu.tw>, National Chiao Tung University
Title: On the Bubbling Solutions to the Liouville System
Abstract: In this talk, I will briefly introduce the recent developments on the Liouville system. The system is related to several models of Chemistry, Ecology and Physics. My recent result on the bubbling solutions will be present.
=================================
The 27th Annual Meeting on Differential Equations and Related Topics
01/26 (Saturday)
Plenary Talks:
"Kenji Nakanishi" <kenjinakanishi@gmail.com>, Kyoto University
Title: Randomized final data problem for the nonlinear Schrӧdinger and the Gross‐Pitaevskii equations Abstract: This is based on joint work with Takuto Yamamoto. We study large time behavior of solutions to the nonlinear Schrodinger equations with power‐type interactions. For powers between the mass critical and the Fujita exponents, there exists a global solution asymptotic (at time infinity) to any free solution of finite mass in three or higher space dimensions. A scaling argument suggests that the uniqueness is a super‐critical problem beyond the reach of standard perturbation arguments. Randomizing the final state, however, Murphy proved that one can almost surely find a unique asymptotic solution in a certain function space, if the power is above the Strauss exponent.
In this talk, it is shown that we can go slightly below the Strauss exponent by using another function space.
In particular, it allows us to treat quadratic interactions in three space dimensions, which often appears in physical models. The same argument applies to the asymptotic form consisting of a plane wave and a linearized dispersive wave with finite energy for the defocusing cubic equation.
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Differential Equations:
"P‐洪盟凱 John M. Hong" <jhong@math.ncu.edu.tw>, National Central University
Title: The Global Escape Phenomenon of Transonic Gas-like Fluids with Self-gravitation in Spherically Symmetric Space-times.
Abstract: In this talk, the global escape phenomenon of gas-like transonic fluids with self-gravitation in Spherically symmetric space-times is studied. The escape phenomenon is governed by an initial-boundary problem of one-dimensional compressible Euler-Poisson equations which form a mixed-type nonlinear partial differential system of balance laws. The compressible Euler-Poisson system is reformulated as a 3 _ 3 hyperbolic system of balance laws by the equations of fluid's density and the gravitational potential. The
global existence to the shock wave solutions of fluid's density-momentum and the Lipschitz continuous solution to the gradient of potential, is established by a new version of generalized Glimm scheme (GGS for short). The new approximate solutions of generalized Riemann and boundary-Riemann problems, which are the building block of GGS, are constructed by the de-coupling process of fluid's quantities and potential's gradient. For the global boundedness of approximate solution by GGS, the key conditions to the momentum and potential's gradient on the boundary are provided. Finally, the modified wave interaction estimates are shown for the decay of Glimm functionals, which leads to the global existence of solutions.
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"PD-黃志強 Chih-Chiang Huang" <loveworldsteven@hotmail.com>, NCTS
Title: Traveling waves for the FitzHugh-Naumo system with monostable or bistable nonlinearity Abstract: In this talk, we will study the FitzHugh-Naumo system (FHN) with monostable and bistable nonlinearity, respectively. We also consider steady states of (FHN) in a bounded domain and traveling waves of (FHN) in a cylinder. By a variational method, we would like to construct traveling waves for a scalar equation and generalize this approach to an equation with a nonlocal term arising from the
FitzHugh-Nagumo system (FHN). In addition, Turing patterns for (FHN) are discussed in the talk. This is a joint with Chiun-Chuan Chen and Chao-Nien Chen.
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"AP-郭鴻文 Hung-Wen Kuo" <hwkuo@mail.ncku.edu.tw>, National Cheng Kung University
Title: Singularity of Free Molecular Flow in Bounded Domains
Abstract: We study the singularity of free molecualr flow in the spherical symmetric domains. First, we show the singularities caused by the effects of the specular reflection boundary condition and the diffuse reflection boundary condition. Then we try to study whether the solution is smooth upon imposing some suitable conditions on initial data.
--- Applied Math:
"AP-薛名成 Ming‐Cheng Shiue" <mshiue@math.nctu.edu.tw>, National Chiao Tung University Title: Data assimilation algorithms based on Synchronization of truth and models
Abstract: In this talk, we first recall continuous and discrete data assimilation algorithms that were proposed for designing finite‐dimensional feedback controls for 2D Navier‐Stokes equations. Then, two new nudging methods, hybrid nonlinear and delay‐coordinate nudging are considered and studied.
In the first part, hybrid nonlinear continuous data assimilation algorithms for Lorenz systems will be studied and presented. It is shown that the approximate solutions converge to the unknown reference solutions over time provided that the first or second variable of Lorenz systems is synchronized. This is a joint work
with Yi Juna Du.
In the second part, two new continuous and discrete data assimilation algorithms for two‐dimensional Navier‐Stokes equations are presented and studied. The explicit use of present and past observations at each time step provides a way that new methods might outperform the old one, which was successfully tested for Lorenz 96 model.
In this talk, we will give preliminary results that provide sufficient conditions on the finite‐dimensional spatial resolution of the collected data and observational measurements to make sure that the
approximate solutions obtained from the new algorithms converge to the unknown reference solutions over time.
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"aP‐樂美亨 Mei‐Heng Yueh" <yue@ntnu.edu.tw>, National Taiwan Normal University
Title: Computational Conformal Geometry with Applications
Abstract: Computational conformal geometry is an interdisciplinary field based on the theories of
conformal geometry as well as computational algorithms. It has been widely applied to carry out 3D image processing tasks, such as surface resampling, remeshing, registration, rendering, and alignment. Especially when the geometry is complicated, a suitable parameterization of the surface can be used to simplify the shape of the domain. In this talk, I will introduce my recent works on the computation of surface
parameterizations, and demonstrate some applications on computer graphics and visualization of medical images.
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"aP-許佳璵 Chia-Yu Hsu" <cyuhsu@fcu.edu.tw>, <chiahsutw@gmail.com>, Feng Chia University Title: The Strategy for Schooling Pattern of Lampreys
Abstract: The numerical computational solutions for schooling of lampreys' swimming under some specific conditions, such as spacing in between fishes and initial body activation waves pattern next to or in front each one, are presented in this talk. The schooloing pattern [1] in marine ecology is a common migration pattern for fishes of different swimming styles, such as carangiform of makrells, subcarangiform of salmonids or anguiliform of eels [2].
In particular, to school is one strategy to reduce energy consumption during migration [3], not to mention, to survival from predators [4]. In this talk, a model of multiple anguiliform swimmers, such as lamprey, is created to simulate the schooling pattern. The adaptive mesh refinement immersed boundary method is used to solve the numerical solution for the simulations. Moreover, is there possibility of synchronized schooling for paralleled multi-swimmers or what is the strategy to have the schooling pattern stabilized? Those are questions will be discussed in this talk.
Keywords: lamprey, schooling pattern, adaptive mesh refinement immersed boundary method [1]A.D. Becker, H. Masoud, J. W. Newbolt1, M. Shelley, L. Ristroph1, Hydrodynamic schooling of flapping swimmers, Natural Communication, (2015), 1-8
[2]Eric D. Tytell, The hydrodynamics of eel swimming, II. Effect of swimming speed, J. of
Exp. Biol., 207 (2004), 3265-3279.
[3]E. Burgerhout , C. Tudorache, S. A. Brittijn , A. P. Palstra , R. P. Dirks, G. E.E.J.M.
van den Thillart , Schooling reduces energy consumption in swimming male European eels, Anguilla anguilla L. J. Exp. Mar. Bio. and Eco. 448 (2013) 66–71
[4] T. Oboshi, S. Kato, A. Mutoh, H. Itoh, A simulation study on the form of fish schooling for escape from predator, CiNii,(2003), 18, 119-131
--- Plenary Talks:
"P‐(尤釋賢) Shih‐Hsien Yu", <matysh@nus.edu.sg>, National University of Singapore
Title: Heat equation with Bounded Variation heat conductivity
Abstract: In this talk, a new constructive procedure to establish the Green's function for heat equation with a BV function heat conductivity; and the pointwise structure of the Green's function will be established.
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Differential Equations:
"AP-陳子軒 Chi-Hin Chan" <cchan@math.nctu.edu.tw>, National Chiao Tung University
Title: Anti-Thesis to the Stokes paradox on the hyperbolic plane.
Abstract: In this talk, we will discuss a recent result which is due to Chi Hin Chan and Magdalena Czubak in which we proved the existence of a nontrivial Stationary Navier-Stokes flow on an exterior domain of a hyperbolic plane which satisfies both the no-slip boundary condition and the finite Dirichlet norm property.
This shows that there is no Stokes paradox in the hyperbolic plane setting.
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"aP-張覺心 Chueh-Hsin Chang" <changjuexin@thu.edu.tw>, Tung Hai University
Title: Attractive interaction of 2-species traveling waves for the 3 components competition-diffusion systems Abstract: In this talk we consider the weak interaction between two traveling wave solutions of the
threes-species competition-diffusion systems. Each of the two traveling wave solutions has one trivial component (called trivial waves). By the invariant manifold theory and asymptotic behavior of kernels of linearized operators, we can prove the existence and instability of non-monotonic traveling wave solutions for three-species.
This is a joint work with Prof. Chiun-Chuan Chen and Prof. Shin-Ichiro Ei.
---
"aP-陳逸昆 I-Kun Chen" <ikun.chen@gmail.com>, National Taiwan University
Title: Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography
Abstract: We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct attenuation coefficient from jumps in boundary measurements.
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"aP-梁育豪 Yu-Hao Liang" <yhliang@nuk.edu.tw>, National University of Kaohsiung
Title: The effects of awareness on the epidemic models
Abstract: The rapid advance of technology has brought the communication between individuals more and more accessible and diverse. This also makes people have more chance to be aware of an infectious disease outbreak and hence reduce the risk of infection. In this talk, we will propose an epidemic model by taking into account the influence of awareness. In our model, a multiplex network for which the spreading of the disease and information occurs, respectively, in two different layers of networks, i.e., the physical network and the virtual network. In addition, these two diffusive processes are assumed to interact and affect each other. Some theoretical results on this model will be introduced. This is a joint work with Prof. Jonq Juang.
--- Applied Math；
"aP‐Maxim Solovchuk" <solovchuk@gmail.com>, National Health Research Institutes (NHRI)
Title: A Nonlinear Conservative System for Describing Highly Nonlinear Acoustic Waves in Heterogeneous Media
Abstract: A new system of hyperbolic PDEs capable of describing the nonlinear nature of acoustic
fluctuations that propagate over inhomogeneous and heterogeneous fluid media is formulated. This novel system model is initially derived by using the traditional principles of nonlinear acoustics [1], i.e. the finite‐amplitude methodology, to yield a general system for describing acoustic fluctuations from the Navier‐Stokes‐Fourier equations. Here, by incorporating the special substitution technique of [2], it is found that the classical result can be closed into a conservative system of nonlinear PDEs.
However, the resulting system is then found to be in a general form of the conservation laws, namely the capacitive‐conservative differential form [3]. A closer look at the Rankine‐Hugoniot relations that result from the system’s associated flux function indicates that the system model is consistent with the physical expectations inside the acoustic regime. As a result, we extend the high‐order shock‐capturing numerical
approach used in [4,5] so that the nonlinear nature of the acoustic propagation in heterogeneous fluid media (including shocks) can be captured without numerical artifacts while keeping any numerical
dissipation to a minimum. To verify and illustrate the capabilities of the proposed nonlinear system model, one‐ and two‐dimensional benchmark problems of the literature are studied [3,6]. Applications of the proposed system for the simulation of high intensity focused ultrasound treatment of liver cancer will be presented [7].
References
[1] Hamilton, Mark F., and David T. Blackstock, eds. “Nonlinear acoustics”. Vol. 1. San Diego: Academic press, 1998.
[2] Christov, Ivan, C. I. Christov, and P. M. Jordan. “Modeling weakly nonlinear acoustic wave propagation.” The Quarterly Journal of Mechanics & Applied Mathematics 60.4 (2007): 473‐495.
[3] LeVeque, Randall J. “Finite volume methods for hyperbolic problems.” Vol. 31. Cambridge university press,2002.
[4] Manuel A. Diaz, Maxim A. Solovchuk, Tony W.H. Sheu. “A Conservative Numerical Scheme for Modeling Nonlinear Acoustic Propagations in Thermoviscous Homogeneous Media.” Journal of Computational Physics, 2018,
https://doi.org/10.1016/j.jcp.2018.02.005.
[5] Manuel A. Diaz, Maxim A. Solovchuk, Tony W.H. Sheu, “High‐Performance MultiGPU Solver for Describing Nonlinear Acoustic Waves in Homogeneous Thermoviscous Media.” Computers & Fluids, 2018, https://doi.org/10.1016/j.compfluid.2018.03.008.
[6] LeVeque, Randall J. “Wave propagation algorithms for multidimensional hyperbolic systems.” Journal of Computational Physics 131.2 (1997): 327‐353.
[7] Solovchuk M, Sheu TW, Thiriet M. Multiphysics modeling of liver tumor ablation by high intensity focused ultrasound.
Communications in Computational Physics. 2015 Oct;18(4):1050‐71.
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“PD‐黃韋強 Wei‐Qiang Huang” <wqhuang@math.nctu.edu.tw>, National Chiao Tung University
Title: An Integrated Eigensolver for Graph Laplacian Eigenvalue Problem
Abstract: The eigenvalue problem of a graph Laplacian matrix arising from a simple, connected and undirected graph has been given more attention due to its extensive applications in the field of machine learning. The associated graph Laplacian matrix is symmetric, positive semi‐definite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest for either clustering or dimensionality reduction.
However, its singularity makes the classical eigensolvers inefficient since we need to solve related linear systems. Moreover, for large‐scaled networks from the real world, such as social media, transactional databases, and sensor systems, there are in general not only local connections. Therefore, it is usually time‐consuming, or even unable, to directly find the matrix factorization for solving involved linear systems exactly. In this talk, we propose an inner‐outer iterative eigensolver, iSIRA, based on the residual Arnoldi method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments demonstrate that the integrated eigensolver outperforms the classical methods, especially in the case when the matrix factorization is not available.
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"PD‐張正陽 Zhengyang Zhang" <zhengyang.zhang@math.nthu.edu.tw>, National Tsing Hua University
Title: A class of state‐dependent delay differential equations and applications to forest growth
Abstract: We consider a state‐dependent delay differential equation that describes the dynamics of a population of trees in a forest. This model comes from a size‐structured population dynamical model. This class of state‐dependent delay differential equation is compared with a computer model called SORTIE (which is an individual‐based model). The main ingredient taken into account in both models is the competition for light between trees. The comparison suggests that state‐dependent delay differential equations can help to understand the dynamics of forest, since we get pretty good fit to the SORTIE model.
Therefore it makes sense to analyze the state‐dependent delay differential equation. The second and third parts are devoted to the properties of the semi‐flow generated by such a state‐dependent delay differential equation and the boundedness and dissipativity of the solutions. In the last part, motivated by the
nematode destruction in a pine forest, we construct a predator‐prey system including the above
state‐dependent delay differential equation and we present numerical simulations of this system in several cases and scenarios.
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"Dr‐陳博源 Po‐Yuan Chen" <pyrobertchen@gmail.com>, Medical Device Innovation Center, NCKU (成大前瞻醫療器材科技中心)
Title: Quadratic Adaptive Algorithm for Solving Cardiac Action Potential Models
Abstract: In this talk, I will give a short introduction to the numerical simulation of cardiac cell models and present a new adaptive integration method for computing cardiac action potential models. Time steps are adaptively chosen by solving a quadratic formula involving the first and second derivatives of the
membrane action potential. To improve the numerical accuracy, we devise an extremum‐locator (el) function to predict the local extremum when approaching the peak amplitude of the action potential. In addition, the time step restriction (tsr) technique is designed to limit the increase in time steps, and thus prevent the membrane potential from changing abruptly. The performance of the proposed method is tested using the Luo‐Rudy phase 1 (LR1), dynamic (LR2), and human O’Hara‐Rudy dynamic (ORd)
ventricular action potential models, and the Courtemanche atrial model incorporating a Markov sodium channel model. Numerical experiments demonstrate that the action potential generated using the
proposed method is more accurate than that using the traditional Hybrid method, especially near the peak region. The traditional Hybrid method may choose large time steps near to the peak region, and sometimes causes the action potential to become distorted. In contrast, the proposed new method chooses very fine time steps in the peak region, but large time steps in the smooth region, and the profiles are smoother and closer to the reference solution.
In the test on the stiff Markov ionic channel model, the Hybrid blows up if the allowable time step is set to be greater than 0.1 ms. In contrast, our method can adjust the time step size automatically, and is stable.
Overall, the proposed method is more accurate than and as efficient as the traditional Hybrid method, especially for the human ORd model. The proposed method shows improvement for action potentials with a non‐smooth morphology, and it needs further investigation to determine whether the method is helpful during propagation of the action potential. ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
01/27 (Sunday)
Plenary Talks:
"Giuseppe Mingione" <rosariomingione@gmail.com>, University of Parma:
Title: Lipschitz estimates for every taste
Abstract: I will focus on gradient estimates for solutions to non‐homogeneous, possibly degenerate equations and systems. I will give a survey of results on Lipschitz estimates starting from the uniformly elliptic case, where linear and nonlinear potentials come into the play. I will then switch to the case of non‐uniformly elliptic equations, where a new and optimal theory can be developed.
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Differential Equations:
"aP-林英杰 Ying-Chieh Lin" <linyj@nuk.edu.tw>, National University of Kaohsiung
Title : Concentration of source terms in generalized Glimm scheme for initial-boundary problem of nonlinear hyperbolic balance laws
Abstract: In this talk, we investigate the initial-boundary value problem for a nonlinear hyperbolic system of balance laws with sources and . To get the approximate solutions of our problem, we consider a version of generalized Riemann problem that concentrates the variation of a on a thin T-shaped region of each grid. A new version of Glimm scheme is introduced to construct the approximate solutions and its stability is proved by considering two types of conditions on a. Finally, we verify the consistency of the scheme and the entropy inequality to establish the global existence of entropy solutions.
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"PD-蘇承芳 Cheng-Fang Su" < scf1204@nctu.edu.tw>, National Chiao Tung University, Taiwan
Title: Incompressible inviscid limit of the viscous two-fluid model on expanding domains with general initial data
Abstract: This talk is about that the incompressible inviscid limit of the viscous two-fluid model on the expanding domains with general initial data in the framework of weak solutions. We prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the
incompressible Euler equations in the time interval provided that the latter exists and the tool is based on the refined relative entropy method. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. My talk will be based on a joint work with Professor Young-Sam Kwon.
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"PD‐陸冰瀅 Bing‐Ying Lu" <bylu@gate.sinica.edu.tw>, Sinica
Title: The universality of the semi‐classical sine‐Gordon equation at the gradient catastrophe
Abstract: We study the semi‐classical sine‐Gordon equation with pure impulse initial data below the threshold of rotation:
ε ‐ ε + sin(u) = 0, u(x, 0) ≡ 0, ε (x, 0) = G(x) ≦ 0, and |G(0)| < 2.
A dispersive‐regularized shock forms in finite time. Using Riemann–Hilbert analysis, we rigorously studied the asymptotics near a certain gradient catastrophe. In accordance with a conjecture made by Dubrovin et.
al., the asymptotics in this region is universally (insensitive to initial condition) described by the tritronquée solution to the PainlevéI equation. Furthermore, we are able to universally characterize the shapes of the spike‐like local structures (rogue wave on periodic background) on top of the poles of the tritronquée solution. (Joint with Peter Miller)
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Applied Math:
"P‐王雲哲 Yun‐Che Wang" <yunche@mail.ncku.edu.tw>, Civil Engineering, National Cheng Kung University
Title: On the extreme viscoelastic properties in composite materials due to fields governed by Allen‐Cahn type PDEs
Abstract: In the framework of the Ginzburg‐Landau phase transition theory, ferroelastic solid‐solid phase transformations are phenomenologically modeled by the Allen‐Cahn‐type parabolic partial differential equations that govern the order‐parameter fields. In the vicinity of the phase transition, the energy
landscape of the system changes from a convex to non‐convex profile, hence the interactions between the transforming domains and their surroundings give rise to extreme effective physical properties, such as unbounded viscoelastic modulus and damping. Effective negative stiffness arises in the domains with non‐convex energy landscape. In this work, it is shown that our finite‐element‐based phase‐field modeling numerical results are consistent with experimental findings. Effects of microstructure on the extreme properties are to be discussed. In addition, a machine‐learning method to numerically solve the Allen‐Cahn PDEs, along with viscoelasticity equations, will also be discussed. (Joint work with H.W. Lai and P.C. Cheng) References
1. M.E. Gurtin, Generalized Ginzburg‐Landau and Cahn‐Hilliard equations based on a microforce balance, Physica D 92, 178‐192 (1996)
2. Y.C. Wang, H.W. Lai, M.W. Shen, Effects of cracks on anomalous mechanical behavior and energy dissipation of negative‐stiffness plates, Physica Status Solidi B, 1800489 (2018)
3. J. Han, A. Jentzen, W. E, Solving high‐dimensional partial differential equations using deep learning, Proceedings of the National Academy of Sciences of the United States of America 115, 8505‐8510 (2018)
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