Research of Harmonic analysis and Partial differential equations

• Project/Area Number: 15K20919
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis; Mathematical analysis
• Research Institution: Shinshu University
• Principal Investigator: Yohei Tsutsui 信州大学, 学術研究院理学系, 助教 (40722773)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: Sparse bound / Navier-Stokes equations / Navier-Stokes 方程式 / Navier-Stokes方程式 / Besov空間 / Wiener amalgam 空間 / 重みつきHardy空間 / 非圧縮Navier-Stokes方程式 / BMO / Local smoothing / Fourier restriction

Outline of Final Research Achievements

The result in Harmonic analysis we obtained is a sparse bound for an operator related to the maximal Riesz means. The maximal operator is deeply connected to the Kakeya conjecture. The method of sparse bound is recently developed.
The results in PDEs we obtained mainly are related to incompressible Navier-Stokes equations. Using techniques in Harmonic analysis, we gave estimates of the solutions. Moreover, we analyzed the Stokes operator on domains that do not satisfy a condition.

Academic Significance and Societal Importance of the Research Achievements

実解析的研究で得られた結果については、さらなる発展が不可欠ではあるが、sparse bound を用いた Kakeya 予想への貢献への第1歩であると考えている。
偏微分方程式での結果について： Naiver-Stokes 方程式は大気などの動きを記述する方程式であるため、その解の様々な情報は、風力発電、天気予報などの分野で有益である。

Research Products

• [Journal Article] Equivalence of $BMO$-type norms with applications to the heat and Stokes semigroup (2017)
  • Author(s): M. Bolkart, Y. Giga, T. Suzuki and Y. Tsutsui
  • Journal Title: Potential Analysis Volume: First Online Issue: 1 Pages: 1-26
  • DOI: 10.1007/s11118-017-9650-x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bounded global solutions to a Keller-Segel system with non-diffusive chemical in Rn (2017)
  • Author(s): Yohei Tsutsui
  • Journal Title: Journal of Evolution Equations Volume: 未定 Issue: 2 Pages: 627-640
  • DOI: 10.1007/s00028-016-0330-x
  • Peer Reviewed
• [Journal Article] Time periodic strong solutions to the incompressible Navier-Stokes equations with external forces of non-divergence form (2017)
  • Author(s): Takahiro Okabe and Yohei Tsutsui
  • Journal Title: Journal of Differential Equations Volume: 263 Pages: 8229-8263
  • Peer Reviewed
• [Journal Article] On analyticity of the L^p-Stokes semigroup for some non-Helmholtz domains (2017)
  • Author(s): M. Bolkart, Y. Giga, T.-H. Miura, T. Suzuki and Y. Tsutsui
  • Journal Title: Math. Nachr. Volume: 印刷中 Issue: 16 Pages: 2524-2546
  • DOI: 10.1002/mana.201600016
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Navier-Stokes flow in the weighted Hardy space with applications to time decay problem, (2016)
  • Author(s): Takahiro Okabe and Yohei Tsutsui
  • Journal Title: J. Differential Equations, Volume: 261 Pages: 1712-1755
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Div-curl lemma with critical power weights in dimension three (2016)
  • Author(s): Yohei Tsutsui
  • Journal Title: Math. Nachr. Volume: 289 Issue: 2-3 Pages: 360-376
  • DOI: 10.1002/mana.201400291
  • Peer Reviewed
• [Journal Article] Trace operators on Wiener amalgam spaces (2016)
  • Author(s): Jayson Cunanan and Yohei Tsutsui
  • Journal Title: Journal of Function Spaces Volume: 2016 Pages: 1-6
  • DOI: 10.1155/2016/1710260
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] A sparse bound for an integral operator with wave propagator (2019)
  • Author(s): 筒井容平
  • Organizer: Work- shop on Analysis in Kagurazaka 2019
  • Invited
• [Presentation] A sparse bound for a time-integral operator with wave propagator (2019)
  • Author(s): 筒井容平
  • Organizer: 調和解析セミナー
• [Presentation] Sparse bounds for local smoothing operators (2018)
  • Author(s): Yohei Tsutsui
  • Organizer: RIMS「調和解析と非線形偏微分方程式」
  • Int'l Joint Research / Invited
• [Presentation] A sparse bound for local smoothing operators (2018)
  • Author(s): Yohei Tsutsui
  • Organizer: Conference on the occasion of the 60th birthday of Professor Masao Yamazaki Harmonic analysis and the Navier-Stokes equations
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic stability of the stationary Navier-Stokes flows in Besov spaces (2018)
  • Author(s): Yohei Tsutsui
  • Organizer: Workshop on the Navier-Stokes Flow
  • Int'l Joint Research / Invited
• [Presentation] A sparse bound for local smoothing operators (2018)
  • Author(s): 筒井容平
  • Organizer: Linear and Nonlinear Waves
  • Invited
• [Presentation] Sparse bounds for local smoothing operators (2018)
  • Author(s): Yohei Tsutsui
  • Organizer: The 6th East Asian Conference in Harmonic Analysis and Applications
• [Presentation] A sparse bound for local smoothing operators (2018)
  • Author(s): 筒井容平
  • Organizer: 日本数学会2017 年度 秋季総合分科会, 実函数論分科会
• [Presentation] Remark on the strong solvability of the Navier-Stokes equations in the weak L n space (2018)
  • Author(s): Okabe Takahiro
  • Organizer: 日本数学会2018年度年会 函数方程式論分科会
• [Presentation] Critical estimates and asymptotic stability of steady incompressible Navier-Stokes flows in Besov spaces (2017)
  • Author(s): 筒井容平
  • Organizer: 第９回名古屋微分方程式研究集会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2017-03-21
  • Invited
• [Presentation] Three rearrangements and medians (2017)
  • Author(s): 筒井容平
  • Organizer: 調和解析セミナー
  • Place of Presentation: 石川県政記念しいのき迎賓館 (石川県金沢市)
  • Year and Date: 2017-03-03
• [Presentation] Estimates for integral equations with real interpolation technique (2017)
  • Author(s): 筒井容平
  • Organizer: RIMS 研究集会「関数空間の構造とその周辺」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2017-02-06
  • Invited
• [Presentation] Asymptotic stability of the stationary Navier- Stokes ows in Besov spaces (2017)
  • Author(s): 筒井容平
  • Organizer: 京都大学NLPDE セミナー
  • Invited
• [Presentation] Asymptotic stability of stationary so- lutions to the Navier-Stokes equations in Besov spaces (2017)
  • Author(s): 筒井容平
  • Organizer: 日本数学会2017 年度 秋季総合分科会, 函数方程式論分科会
• [Presentation] Asymptotic stability of the stationary Navier-Stokes ows in Besov spaces (2017)
  • Author(s): Yohei Tsutsui
  • Organizer: Harmonic Analysis and its Applications in Tokyo 2017,
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic stability of stationary solutions to the Navier-Stokes equations in Besov spaces (2017)
  • Author(s): Yohei Tsutsui
  • Organizer: The 5th East Asian Conference in Harmonic Anal- ysis and Applications
  • Int'l Joint Research
• [Presentation] Asymptotic stability of the stationary solution to the incompressible Navier-Stokes equation in Besov spaces (2017)
  • Author(s): Yohei Tsutsui
  • Organizer: Conference "New perspec- tives in the theory of function spaces and their applications"
  • Int'l Joint Research
• [Presentation] Time periodic strong solutions to the incompressible Navier-Stokes equations with external forces of non-divergence form, (2016)
  • Author(s): 筒井容平
  • Organizer: 第14 回浜松偏微分方程式研究集会
  • Place of Presentation: 静岡大学工学部
  • Year and Date: 2016-12-22
  • Invited
• [Presentation] Time periodic strong solutions to the incompressible Navier-Stokes equations with external force of the non-divergence form, (2016)
  • Author(s): 筒井容平
  • Organizer: 代数解析奈良研究集会
  • Place of Presentation: 奈良女子大学
  • Year and Date: 2016-11-25
  • Invited
• [Presentation] Navier-Stokes flow in the weighted Hardy space with applications to time decay problem, (2016)
  • Author(s): 筒井容平
  • Organizer: 談話会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2016-10-28
  • Invited
• [Presentation] Navier-Stokes flow in the weighted Hardy space with applications to time decay problem, (2016)
  • Author(s): Yohei Tsutsui
  • Organizer: Harmonic Analysis and its Applications in Beijing 2016
  • Place of Presentation: Beihang University, China
  • Year and Date: 2016-09-29
  • Int'l Joint Research / Invited
• [Presentation] Div - curl estimate with critical power weight, (2016)
  • Author(s): Yohei Tsutsui
  • Organizer: Special session SS119 "Geometric functional inequalities and application to PDEs" of The 11th AIMS Conference on Dynamical Systems, Differential Equations and Ap- plications
  • Place of Presentation: Orlando, U.S.
  • Year and Date: 2016-07-01
  • Int'l Joint Research / Invited
• [Presentation] Navier-Stokes flow in the weighted Hardy space with applications to time decay problem, (2016)
  • Author(s): 筒井容平
  • Organizer: 応用解析セミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2016-04-28
  • Invited
• [Presentation] Applications of weighted Hardy spaces to the incompressible Navier-Stokes equations (2016)
  • Author(s): Yohei Tsutsui
  • Organizer: 12th Japanese-German international workshop on Mathematical Fluid Dynamics
  • Place of Presentation: 早稲田大学
  • Year and Date: 2016-03-01
  • Int'l Joint Research / Invited
• [Presentation] Applications of weighted Hardy spaces to the incompressible Navier-Stokes equations (2015)
  • Author(s): 筒井容平
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 東京大学
  • Year and Date: 2015-12-19
  • Invited
• [Presentation] Div-curl lemma with critical power weights (2015)
  • Author(s): 筒井容平
  • Organizer: 第五回弘前非線形方程式研究会
  • Place of Presentation: 弘前大学
  • Year and Date: 2015-12-12
  • Invited
• [Presentation] Div-curl lemma with critical power weights (2015)
  • Author(s): Yohei Tsutsui
  • Organizer: Harmonic Analysis and its Applications in Tokyo 2015
  • Place of Presentation: 早稲田大学
  • Year and Date: 2015-11-27
  • Int'l Joint Research / Invited
• [Presentation] Div-curl lemma with critical power weights (2015)
  • Author(s): Yohei Tsutsui
  • Organizer: RIMS研究集会「非圧縮粘性流体の数理解析」
  • Place of Presentation: 京都大学、数理解析研究所
  • Year and Date: 2015-11-16
  • Int'l Joint Research / Invited
• [Presentation] Div-curl lemma with critical power weights (2015)
  • Author(s): 筒井容平
  • Organizer: 名古屋大学微分方程式セミナー
  • Place of Presentation: 名古屋大学
  • Year and Date: 2015-11-09
  • Invited
• [Presentation] Navier-Stokes flows in the weighted Hardy space with applications to time decay problem (2015)
  • Author(s): 岡部孝宏, 筒井容平
  • Organizer: 2015秋季総合分科会 (日本数学会)
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Div-curl lemma with critical power weights in dimension three (2015)
  • Author(s): 筒井容平
  • Organizer: 金沢解析研究会
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2015-08-27
  • Invited
• [Presentation] Div - curl lemma with critical power weights in dimension three (2015)
  • Author(s): Yohei Tsutsui
  • Organizer: 3rd East Asia Conference in Harmonic Analysis and Applications,
  • Place of Presentation: The University of Tokyo
  • Year and Date: 2015-08-10
  • Int'l Joint Research
• [Presentation] Div - curl lemma with critical power weights in dimension three (2015)
  • Author(s): Yohei Tsutsui
  • Organizer: 10th International ISAAC Congress
  • Place of Presentation: Macau, China
  • Year and Date: 2015-08-03
  • Int'l Joint Research / Invited
• [Presentation] Div - curl lemma with critical power weights in dimension three (2015)
  • Author(s): 筒井容平
  • Organizer: 微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2015-06-19
  • Invited
• [Presentation] Div - curl lemma with critical power weights in dimension three (2015)
  • Author(s): 筒井容平
  • Organizer: 金曜セミナー
  • Place of Presentation: 茨城大学
  • Year and Date: 2015-05-01
  • Invited
• [Presentation] Div - curl lemma with critical power weights in dimension three (2015)
  • Author(s): 筒井容平
  • Organizer: 談話会
  • Place of Presentation: 信州大学
  • Year and Date: 2015-04-16
  • Invited
• [Funded Workshop] Special session SS119 "Geometric functional inequalities and application to PDEs" of The 11th AIMS Conference on Dynamical Systems, Differential Equations and Ap- plications, (2016)
  • Place of Presentation: Orlando, U.S.
  • Year and Date: 2016-07-01