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 / 4階全変動流方程式 / 特異極限 / 小林・ワレン・カーターエネルギー / 有界平均振動2乗可積分関数 / ヘルムホルツ分解 / 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：特異拡散方程式の高階での例として、4階の全変動流方程式を空間全体で考え、適切な解概念を構築した。
2. 弱解の安定性と長時間挙動：輸送方程式、ハミルトン・ヤコビ方程式の不連続解を扱う枠組みについて、過去の重要な結果をまとめ、わかりやすい解説書を出版した。
3. 多粒界モデル：小林・ワレン・カーターエネルギーについては、従来空間一次元の場合に研究されてきた特異極限問題を高次元の場合に扱った。
4. 異常拡散現象：熱方程式で、境界付近だけ、比熱が異常に大きい場合の特異極限の特徴づけと収束問題を論じた。
5. ナヴィエ・ストークス方程式：ナヴィエ・ストークス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2乗可積分関数の空間に対して、摂動半空間から出発して、一般の一様に滑らかな必ずしも有界でない領域の場合に拡張した。

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。
また、本研究課題開始時から始めてきた、著書を出版することができた。

Strategy for Future Research Activity

課題ごとに成果をまとめ、論文という形にまとめ、発表していく。また、国際研究集会等を通して、その成果を広く学界に知らせていく。

Research Products

• [Int'l Joint Research] ワルシャワ大学/ポーランド科学アカデミー(ポーランド)
• [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] The Helmholtz decomposition of a BMO type vector field in general unbounded domains (2024)
  • Author(s): Giga Yoshikazu、Gu Zhongyang
  • Journal Title: Advances in Differential Equations Volume: 29 Issue: 5/6 Pages: 389-436
  • DOI: 10.57262/ade029-0506-389
  • Peer Reviewed
• [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 fourth-order total variation flow in R^n (2023)
  • Author(s): Giga Yoshikazu、Kuroda Hirotoshi、Lasica Michal
  • Journal Title: Mathematics in Engineering Volume: 5 Issue: 6 Pages: 1-45
  • DOI: 10.3934/mine.2023091
  • Peer Reviewed / Open Access
• [Journal Article] The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space (2023)
  • Author(s): Giga Yoshikazu、Gu Zhongyang
  • Journal Title: Journal of Mathematical Fluid Mechanics Volume: 25 Issue: 2
  • DOI: 10.1007/s00021-023-00781-z
  • Peer Reviewed / Open Access
• [Journal Article] The heat equation with the dynamic boundary condition as a singular limit of problems degenerating at the boundary (2023)
  • Author(s): Giga Yoshikazu、Lasica Michal、Rybka Piotr
  • Journal Title: Asymptotic Analysis Volume: 135 Issue: 3-4 Pages: 463-508
  • DOI: 10.3233/asy-231862
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain and its computation (2023)
  • Author(s): Eto Tokuhiro、Giga Yoshikazu
  • Journal Title: Annali di Matematica Pura ed Applicata (1923 -) Volume: - Issue: 3 Pages: 1195-1221
  • DOI: 10.1007/s10231-023-01398-9
  • 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] 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: 386 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] 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] Spatially discrete total variation map flow and the time discrete approximation (2023)
  • Author(s): Yoshikazu Giga
  • Organizer: Frontiers In Hamilton Jacobi Equation And Its Applications
  • Int'l Joint Research / Invited
• [Presentation] On the Helmholtz decomposition of BMO spaces of vector fields (2023)
  • Author(s): Yoshikazu Giga
  • Organizer: Days on Diffraction 2023
  • Int'l Joint Research / Invited
• [Presentation] On analyticity of the L^p-Stokes semigroup for some non-Helmholtz domains (2023)
  • Author(s): Yoshikazu Giga
  • Organizer: Trends in Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] The fourth-order total variation flow in R^n (2023)
  • Author(s): Yoshikazu Giga
  • Organizer: Differential Equations and Dynamical Systems
  • Int'l Joint Research / Invited
• [Presentation] 連続関数の空間での偏微分方程式 (2023)
  • Author(s): 儀我 美一
  • Organizer: 第2回日本数学会賞小平邦彦賞受賞講演会
  • Invited
• [Presentation] Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system (2023)
  • Author(s): Yoshikazu Giga
  • Organizer: Euro-Japanese Conference on Nonlinear Diffusions
  • Int'l Joint Research / Invited
• [Presentation] データ分離、クラスタリングへの応用を目指した小林・ワレン・カーター系の研究 (2023)
  • Author(s): 儀我 美一
  • Organizer: アリスマー株式会社講演会
  • Invited
• [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
• [Book] A basic guide to uniqueness problems for evolutionary differential equations (2023)
  • Author(s): M.-H. Giga and Y. Giga
  • Total Pages: 165
  • Publisher: Birkhauser
• [Remarks] Laboratory of Yoshikazu GIGA
  • URL: https://www.ms.u-tokyo.ac.jp/~ygiga/
• [Remarks] Laboratory of Yoshikazu GIGA
  • URL: https://www.ms.u-tokyo.ac.jp/~labgiga/
• [Funded Workshop] Mathematical aspects for interfaces and free boundaries (2023)
• [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)