Existence or non-existence for the solutions of nonlinear wave equations

• Project/Area Number: 16K05191
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Saitama University
• Principal Investigator: Machihara Shuji 埼玉大学, 理工学研究科, 教授 (20346373)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Discontinued (Fiscal Year 2020)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 初期値問題適切性 / 初期値問題非適切性 / ノルムインフレーション / 非線形偏微分方程式 / 非適切性 / Chern-Simons-Dirac / レリッヒの不等式 / Thirring model / Dirac-Klein-Gordon / well-posedness / ill-posedness / trace theorem / 非線形波動方程式

Outline of Final Research Achievements

We have considered the time local well-posedness for the Cauchy problems of the Dirac-Klein-Gordon equations, generalized Thirring model of nonlinear Dirac equation, Chern-Simons-Dirac system and the dimension-reduced Chern-Simons-Dirac equations. Especially for the Dirac-Klein-Gordon equation with the Yukawa type nonlinearity, we have finished the classification of the regularities for which this problem is well-posed or ill-posed.
We have also studied some kinds of inequalities, Hardy's inequality, Rellich inequality, and trace theorems.

Academic Significance and Societal Importance of the Research Achievements

偏微分方程式を研究対象としたときに解の存在定理がその重要性を持つことは明らかであろう。この研究期間中に行った研究主題は非線形分散型・双曲型偏微分方程式の初期値問題の適切性となる。つまり非線形項を含む形の偏微分方程式を考え、初期値から解を求めるという初期値問題を取り扱った。そして適切性とは解の存在を含み、そこへ加えて解の一意性と初期値への連続依存性を要求した性質となる。これら三つの条件を満たすとき初期値問題は適切であるといい、一つでも欠如した場合は初期値問題は非適切であるという。問題が適切であるか非適切であるかは解を扱う関数空間の正則度に依存する。正則度による場合分けが完成すれば問題は終了する。

Research Products

• [Journal Article] Well-posedness for the dimension-reduced Chern-Simons-Dirac system (2017)
  • Author(s): Shuji Machihara and Mamoru Okamoto
  • Journal Title: Journal of Evolution Equations Volume: 印刷中 Issue: 3 Pages: 1031-1048
  • DOI: 10.1007/s00028-016-0371-1
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Global wellposedness for a one-dimensional Chern-Simons-Dirac system in L^p (2017)
  • Author(s): Machihara, Shuji and Ogawa, Takayoshi
  • Journal Title: Comm. Partial Differential Equations Volume: 42 Issue: 8 Pages: 1175-1198
  • DOI: 10.1080/03605302.2017.1330339
  • Peer Reviewed
• [Journal Article] Remarks on the Rellich inequality (2017)
  • Author(s): S. Machihara, T. Ozawa, H. Wadade
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 3-4 Pages: 1367-1373
  • DOI: 10.1007/s00209-016-1805-8
  • NAID: 120005982988
  • Peer Reviewed
• [Journal Article] Remarks on ill-posedness for the Dirac-Klein-Gordon system (2016)
  • Author(s): S. Machihara, M. Okamoto
  • Journal Title: Dyn. Partial Differ. Equ. Volume: 13 Issue: 3 Pages: 179-190
  • DOI: 10.4310/dpde.2016.v13.n3.a1
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Sharp well-posedness and ill-posedness the Chern-Simons-Dirac system in one dimension (2016)
  • Author(s): Shuji Machihara and Mamoru Okamoto
  • Journal Title: Int. Math. Res. Not. Volume: 2016 (6) Issue: 6 Pages: 1640-1694
  • DOI: 10.1093/imrn/rnv160
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model (2016)
  • Author(s): Huh, Hyungjin and Shuji Machihara and Mamoru Okamoto
  • Journal Title: Differential and Integral Equations Volume: 29
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Extremisers for the trace theorem on the sphere (2016)
  • Author(s): N. Bez, S. Machihara, M. Sugimoto
  • Journal Title: Math. Res. Lett. Volume: 23 Issue: 3 Pages: 633-647
  • DOI: 10.4310/mrl.2016.v23.n3.a3
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Spherical Harmonics and Hardy’s equality (2020)
  • Author(s): Machihara Shuji
  • Organizer: Recent Progress on Dispersive and Wave Equations
• [Presentation] 非線形シュレディンガー方程式の有限時間爆発解に対する注意 (2019)
  • Author(s): 町原 秀二
  • Organizer: 反応拡散方程式と非線形分散型方程式の解の挙動
  • Invited
• [Presentation] The range of Sobolev regularity for the well-posedness of the Dirac-Klein-Gordon system in 1d (2019)
  • Author(s): 町原 秀二
  • Organizer: 第11回名古屋微分方程式研究集会
  • Invited
• [Presentation] The range of Sobolev regularity for the well-posedness of the Dirac-Klein-Gordon system in 1d (2018)
  • Author(s): 町原 秀二
  • Organizer: Workshop on nonlinear wave equations and related topics in Kobe
  • Invited
• [Presentation] Global well-posedness for one dimensional Chern-Simons-Dirac system in Lp (2017)
  • Author(s): Shuji Machihara
  • Organizer: Nonlinear PDE for Future Applications -Hyperbolic and Dispersive PDE-
  • Int'l Joint Research / Invited
• [Presentation] 空間１次元Dirac-Klein-Gordon方程式の初期値問題の適切性について (2017)
  • Author(s): 町原 秀二
  • Organizer: 日本数学会実函数論分科会, 函数解析学分科会共催 第56回実函数論・函数解析学合同シンポジウム
  • Invited
• [Presentation] Remarks on the ill-posedness for the Dirac-Klein-Gordon system (2017)
  • Author(s): Shuji Machihara
  • Organizer: International Workshop on ``Fundamental Problems in Mathematical and Theoretical Physics”
  • Int'l Joint Research / Invited
• [Presentation] 空間1次元 Dirac-Klein-Gordon 方程式初期値問題の非適切性の議論 (2016)
  • Author(s): 町原 秀二
  • Organizer: 第 10 回 実解析と函数解析による偏微分方程式論 研究集会
  • Place of Presentation: 鳴子観光ホテル（宮城県大崎市）
  • Year and Date: 2016-12-25
• [Presentation] Ill-posedness results for the 1d Dirac-Klein-Gordon system (2016)
  • Author(s): Shuji Machihara
  • Organizer: Interactions Between Harmonic and Geometric Analysis
  • Place of Presentation: 埼玉大学東京ステーションカレッジ（東京都千代田区）
  • Year and Date: 2016-11-28