Analysis on nonlinear diffusion and dynamic singular structure

• Project/Area Number: 19H00639
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 儀我 美一 東京大学, 大学院数理科学研究科, 特任教授 (70144110)
• Co-Investigator(Kenkyū-buntansha): 利根川 吉廣 東京工業大学, 理学院, 教授 (80296748) ; 山本 昌宏 東京大学, 大学院数理科学研究科, 教授 (50182647) ; 三竹 大寿 東京大学, 大学院数理科学研究科, 准教授 (90631979)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Project Status: Granted (Fiscal Year 2023)
• Budget Amount: ¥43,810,000 (Direct Cost: ¥33,700,000、Indirect Cost: ¥10,110,000); Fiscal Year 2023: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2022: ¥10,920,000 (Direct Cost: ¥8,400,000、Indirect Cost: ¥2,520,000); Fiscal Year 2021: ¥11,050,000 (Direct Cost: ¥8,500,000、Indirect Cost: ¥2,550,000); Fiscal Year 2020: ¥7,020,000 (Direct Cost: ¥5,400,000、Indirect Cost: ¥1,620,000); Fiscal Year 2019: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
• Keywords: 界面運動方程式 / 弱解 / 粘性解析 / 変分解析 / 実解析 / クリスタライン平均曲率流方程式 / 小林・ワレン・カーターエネルギー4階方程式 / 有界平均振動関数 / クリスタライン曲率流方程式 / 鋭敏界面極限 / 分数冪時間微分方程式

Outline of Research at the Start

発展型偏微分方程式で記述される現象においては、何らかの平滑化効果があっても、ちぎれる液滴のように動的特異構造が生じうる。このような動的特異構造が発生した後も解を追跡するためには、微分できない関数を解とみなせるような解概念の拡張が必要となる。
本研究では、非線形性や非局所性が強い拡散型偏微分方程式に対して、動的特異構造を許容する新たな解（弱解）の概念を導入し、方程式の時間大域可解性および解の挙動を、粘性解析、変分解析、関数解析、漸近解析、実解析の新手法開拓により解明する。また結晶成長分野、画像処理分野、流体力学分野等の諸科学に展開する。

Outline of Annual Research Achievements

拡散・平滑化効果が非局所的な非線形拡散方程式を中心に、動的特異構造を許す弱解の概念を構築し、その解の性質を調べることを目的とする。典型的な動的特異構造として結晶成長分野の「ファセットを伴う成長」を扱った。次の5つのテーマについて研究成果をあげた。
1. 特異拡散方程式とその応用：クリスタライン平均曲率流方程式についてのこれまでの研究成果をまとめたサーベイ論文を執筆した。
2. 弱解の安定性と長時間挙動：4階のクリスタライン方程式について特殊な場合ではあるが、区分的線形の形状変化について記述することに成功した。また、任意の多角形から出発した平面クリスタライン流方程式の弱解（等高面解）は、瞬時にある種の許容多角形になることを示した。
3. 多粒界モデル：典型的なエネルギーである小林・ワレン・カーターエネルギーの特異極限を特徴づける。
4. 異常拡散現象：粘性解理論と超関数理論の比較を行った。
5. ナヴィエ・ストークス方程式：プリミティブ方程式を異方的ナヴィエ・ストークス方程式より厳密に導出した。方程式の数学解析の基礎であるヘルムホルツ分解を、有界平均振動関数の空間に対して行うことを可能とした。

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

研究課題ごとに着実に成果をあげている。コロナ禍で対面での国際会議出席などは一部自粛してきたが、オンラインでの参加等によりその分を補ってきた。

Strategy for Future Research Activity

課題ごとに成果を取りまとめつつ、国際会議等で積極的に成果を発表していく予定である。

Research Products

• [Int'l Joint Research] ダルムシュタット工科大学(ドイツ)
• [Int'l Joint Research] ダルムシュタット工科大学(ドイツ)
• [Int'l Joint Research] ワルシャワ大学/ヴロツワフ大学(ポーランド)
• [Int'l Joint Research] ウィスコンシン大学(米国)
• [Journal Article] Crystalline surface diffusion flow for graph-like curves (2023)
  • Author(s): Giga Mi-Ho、Giga Yoshikazu
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 43 Issue: 3&4 Pages: 1436-1468
  • DOI: 10.3934/dcds.2022160
  • Peer Reviewed / Open Access
• [Journal Article] The Helmholtz decomposition of a space of vector fields with bounded mean oscillation in a bounded domain (2022)
  • Author(s): Giga Yoshikazu、Gu Zhongyang
  • Journal Title: Mathematische Annalen Volume: － Issue: 1-2 Pages: 673-712
  • DOI: 10.1007/s00208-022-02410-y
  • Peer Reviewed / Open Access
• [Journal Article] Exhaustive existence and non-existence results for Hardy-Henon equations in R^n (2022)
  • Author(s): Giga Yoshikazu、Ngo Quoc Anh
  • Journal Title: Partial Differential Equations and Applications Volume: 3 Issue: 6
  • DOI: 10.1007/s42985-022-00190-3
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Motion by crystalline-like mean curvature: A survey (2022)
  • Author(s): Giga Yoshikazu、Pozar Norbert
  • Journal Title: Bulletin of Mathematical Sciences Volume: 12 Issue: 02 Pages: 1-68
  • DOI: 10.1142/s1664360722300043
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] 相の構造を考慮した拡散界面エネルギーとその特異極限 (2022)
  • Author(s): 儀我 美一、上坂 正晃
  • Journal Title: 応用数理 Volume: 32 Pages: 186-197
  • Peer Reviewed
• [Journal Article] Crystalline flow starting from a general polygon (2022)
  • Author(s): Giga Mi-Ho、Giga Yoshikazu、Kuroda Ryo、Ochiai Yusuke
  • Journal Title: Discrete & Continuous Dynamical Systems Volume: 42 Issue: 4 Pages: 2027-2027
  • DOI: 10.3934/dcds.2021182
  • Peer Reviewed / Open Access
• [Journal Article] Normal Trace for a Vector Field of Bounded Mean Oscillation (2022)
  • Author(s): Giga Yoshikazu、Gu Zhongyang
  • Journal Title: Potential Analysis Volume: － Issue: 1 Pages: 409-434
  • DOI: 10.1007/s11118-021-09973-6
  • Peer Reviewed
• [Journal Article] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2022)
  • Author(s): Giga Yoshikazu、Mitake Hiroyoshi、Sato Shoichi
  • Journal Title: Journal of Differential Equations Volume: 316 Pages: 364-386
  • DOI: 10.1016/j.jde.2022.01.057
  • NAID: 120007145351
  • Peer Reviewed / Open Access
• [Journal Article] A finer singular limit of a single-well Modica?Mortola functional and its applications to the Kobayashi?Warren?Carter energy (2021)
  • Author(s): Giga Yoshikazu、Okamoto Jun、Uesaka Masaaki
  • Journal Title: Advances in Calculus of Variations Volume: － Issue: 1 Pages: 163-182
  • DOI: 10.1515/acv-2020-0113
  • Peer Reviewed / Open Access
• [Journal Article] The hydrostatic approximation for the primitive equations by the scaled Navier?Stokes equations under the no-slip boundary condition (2021)
  • Author(s): Furukawa Ken、Giga Yoshikazu、Kashiwabara Takahito
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 3 Pages: 3331-3373
  • DOI: 10.1007/s00028-021-00674-6
  • Peer Reviewed / Open Access
• [Journal Article] The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$ (2021)
  • Author(s): Giga Yoshikazu、Gries Mathis、Hieber Matthias、Hussein Amru、Kashiwabara Takahito
  • Journal Title: Journal of Evolution Equations Volume: 21 Issue: 4 Pages: 4145-4169
  • DOI: 10.1007/s00028-021-00716-z
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Discrete schemes for time-fractional fully nonlinear evolution equations and their convergence (2020)
  • Author(s): Y. Giga, Q. Liu, H. Mitake
  • Journal Title: Asymptot. Anal. Volume: accepted Issue: 1-2 Pages: 1-12
  • DOI: 10.3233/asy-191583
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Analyticity of solusions to the primitive equations (2020)
  • Author(s): Y. Giga, M. Gries, M. Hieber, A. Hussein and T. Kashiwabara
  • Journal Title: Math. Nachr. Volume: 293 Issue: 2 Pages: 284-304
  • DOI: 10.1002/mana.201700401
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Numerical computations of split Bregman method for fourth order total variation flow (2020)
  • Author(s): Y. Giga and Y. Ueda
  • Journal Title: J. Comput. Phys. Volume: 405 Pages: 109114-109114
  • DOI: 10.1016/j.jcp.2019.109114
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the continuity of the solutions to the Navier-Stokes equations with initial data in critical Besov spaces (2019)
  • Author(s): R. Farwig, Y. Giga and P.-Y. Hsu
  • Journal Title: Ann. Mat. Pura Appl. Volume: 198 Issue: 5 Pages: 1495-1511
  • DOI: 10.1007/s10231-019-00824-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A duality based approach to the minimizing total variation flow in the space $H^{-s}$ (2019)
  • Author(s): Y. Giga, M. Muszkieta and P. Rybka
  • Journal Title: Jpn. J. Ind. Appl. Math. Volume: 36 Issue: 1 Pages: 261-286
  • DOI: 10.1007/s13160-018-00340-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Strong time-periodic solutions to the bidomain equations with arbitrary large forces (2019)
  • Author(s): Y. Giga, N. Kajiwara and K. Kress
  • Journal Title: Nonlinear Anal. Real World Appl. Volume: 47 Pages: 398-413
  • DOI: 10.1016/j.nonrwa.2018.11.006
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On obstacle problem for mean curvature flow with driving force (2019)
  • Author(s): Giga Yoshikazu、Tran Hung V.、Zhang Longjie
  • Journal Title: Geometric Flows Volume: 4 Issue: 1 Pages: 9-29
  • DOI: 10.1515/geofl-2019-0002
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Continuous alignment of vorticity direction prevents the blow-up of the Navier-Stokes flow under the no-slip boundary condition (2019)
  • Author(s): Y. Giga, Z. Gu and P.-Y. Hsu
  • Journal Title: Nonlinear Anal. Volume: 189 Pages: 111579-111579
  • DOI: 10.1016/j.na.2019.111579
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms (2019)
  • Author(s): Y. Giga, H. Mitake and H. V. Tran
  • Journal Title: Discrete Contin. Dyn. Syst. Ser. B Volume: Online First Issue: 10 Pages: 3983-3999
  • DOI: 10.3934/dcdsb.2019228
  • NAID: 120006643619
  • Peer Reviewed / Int'l Joint Research
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2022)
  • Author(s): Giga Yoshikazu
  • Organizer: ESI workshop: Free Boundary Problems and related Evolution Equations
  • Int'l Joint Research / Invited
• [Presentation] Motion by crystalline-like mean curvature (2022)
  • Author(s): Giga Yoshikazu
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2022)
  • Author(s): Y. Giga
  • Organizer: ESI workshop: Free Boundary Problems and related Evolution Equations
  • Int'l Joint Research / Invited
• [Presentation] Motion by crystalline-like mean curvature (2022)
  • Author(s): Y. Giga
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] 有界領域上の有界平均振動ベクトル場のヘルムホルツ分解 (2021)
  • Author(s): 儀我 美一
  • Organizer: 応用解析研究会
  • Invited
• [Presentation] On the Helmholtz decomposition of BMO space of vector fields (2021)
  • Author(s): Giga Yoshikazu
  • Organizer: Geometric and Functional Inequalities and Applications
  • Int'l Joint Research / Invited
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2021)
  • Author(s): Giga Yoshikazu
  • Organizer: Nonlinear Analysis Seminar Series
  • Int'l Joint Research / Invited
• [Presentation] 相転移を伴う熱流体現象の数理モデリング 第1話 - 第10話 (2021)
  • Author(s): 儀我 美一
  • Organizer: 社会連携講座「冷媒熱流体の数理」
  • Invited
• [Presentation] On the Helmholtz decomposition of $BMO$ space of vector fields (2021)
  • Author(s): Y. Giga
  • Organizer: Geometric and Functional Inequalities and Applications
  • Int'l Joint Research / Invited
• [Presentation] 熱交換器 1 - 6 (2021)
  • Author(s): 儀我 美一
  • Organizer: 社会連携講座「冷媒熱流体の数理」
  • Invited
• [Presentation] On a singular limit of a single-well Modica-Mortola functional and its applications (2021)
  • Author(s): Y. Giga
  • Organizer: Nonlinear Analysis Seminar Series
  • Int'l Joint Research / Invited
• [Presentation] On total variation flow under dynamic boundary conditions (2020)
  • Author(s): Y. Giga
  • Organizer: Qualitative Properties for Nonlinear Diffusion Equations
  • Int'l Joint Research / Invited
• [Presentation] On total variation flow type equations (2019)
  • Author(s): Y. Giga
  • Organizer: PDE seminar series
  • Int'l Joint Research / Invited
• [Presentation] On the large time behavior of solutions to birth and spread type equations (2019)
  • Author(s): Y. Giga
  • Organizer: Dynamics, Equations and Applications (DEA 2019)
  • Int'l Joint Research / Invited
• [Presentation] On total variation flow type equations (2019)
  • Author(s): Y. Giga
  • Organizer: AustMS 2019
  • Int'l Joint Research / Invited
• [Remarks] Laboratory of Yoshikazu GIGA
  • URL: https://www.ms.u-tokyo.ac.jp/~labgiga/
• [Funded Workshop] Differential Equations for Data Science 2023 (2023)
• [Funded Workshop] Interface and Free Boundaries (2022)
• [Funded Workshop] Preliminary meeting for Interfaces and Free Boundaries (2021)
• [Funded Workshop] Recent topics on well-posedness and stability of incompressible fluid and related topics (2019)
• [Funded Workshop] 第44回偏微分方程式論札幌シンポジウム (2019)
• [Funded Workshop] Mathematical Aspects of Surface and Interface Dynamics 18 (2019)
• [Funded Workshop] Geometric Analysis and General Relativity (2019)
• [Funded Workshop] UTokyo-NTU Joint Symposium in Mathematics (2019)