Development of global analysis for solutions of nonlinear partial differential equations

• Project/Area Number: 17K05317
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kyoto University
• Principal Investigator: NISHIDA TAKAAKI 京都大学, 情報学研究科, 名誉教授 (70026110)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 非線形偏微分方程式 / Navier-Stokes equation / 圧縮性粘性流体運動方程式 / 大域的解析 / 計算機援用解析 / 自由表面問題 / 熱対流問題 / Navier-Stokes 方程式 / 圧縮性粘性流体方程式 / 分岐理論 / 解析学 / 流体運動方程式系 / 特異摂動極限 / 非線形偏微分方程式系 / 特異極限

Outline of Final Research Achievements

(1) Analysis on asymptotic behaviors and their stability of incompressible viscous fluids is our subject. To avoid the constraint of incompressibility Chorin (1976)
introduced an artificial compressibility in the mass conservation law. We examined its applicability to the pattern formations and their stability in heat convection problems. We justified the method analytically for the limit of compressiblity parameter tending to zero. (2) We proved that periodic traveling waves bifurcate from the instability of the plane Poiseuille flow of compressible viscous fluids for much smaller Reynolds number compared with incompressible case. (3) We proved that the solutions of free surface flow of linearized Navier-Stokes equations in the infinite horizontal layer decay as an algebraic rate.

Academic Significance and Societal Importance of the Research Achievements

流体運動方程式の解の挙動を大域的に解析することを目指した。その偏微分方程式系が非線形であり物理的な無次元化などのパラメーターをいくつか含む。それらの解の挙動、安定性を解明したいが、系の非線形性やパラメーターの値に依存して解は複雑に変化する。特別の場合を除いて解が陽に解析的に求められないために、パラメーターによる特異摂動や分岐理論などによる解析とともに計算機援用解析が必要になる。熱対流問題、非圧縮性粘性流体運動などでのパターン形成とそれらの安定性を調べるのに人工的圧縮性を導入し、数値的に調べるとともに、解析的にもその正当性を示した。

Research Products

• [Journal Article] Decay of solutions of the Stokes system arising in free surface flow on an infinite layer (2020)
  • Author(s): J. Thomas Beale, Takaaki Nishida and Yoshiaki Teramoto
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B 77 Pages: 1-20
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Traveling Waves Bifurcating from Plane Poiseuille Flow of the Compressible Navier-Stokes Equation (2019)
  • Author(s): Yoshiyuki Kagei and Takaaki Nishida
  • Journal Title: Arch.Rational Mech. Anal. Volume: 231 Issue: 1 Pages: 1-44
  • DOI: 10.1007/s00205-018-1269-6
  • Peer Reviewed
• [Journal Article] On the spectrum for the artificial compressible system (2018)
  • Author(s): Yoshiyuki Kagei, Takaaki Nishida, Yuka Teramoto
  • Journal Title: Journal of Differential Equations Volume: 264 Issue: 2 Pages: 897-928
  • DOI: 10.1016/j.jde.2017.09.026
  • Peer Reviewed
• [Presentation] Navier-Stokes 方程式の水平領域での自由表面問題 (2019)
  • Author(s): 西田 孝明
  • Organizer: 偏微分方程式の臨界現象と正則性理論および漸近解析、京都大学数理解析研究所
  • Invited
• [Presentation] Decay of Surface Waves of Navier-Stokes Equations (2018)
  • Author(s): Takaaki NIshida
  • Organizer: 岐阜数理科学研究会
  • Invited
• [Presentation] Artificial Compressible Systems for the Incompressible Viscous Fluids (2018)
  • Author(s): Takaaki Nishida
  • Organizer: 成功大学、機械工学セミナー、台南（台湾）
  • Invited
• [Presentation] Decay of Surface Waves of Navier-Stokes Equations (2018)
  • Author(s): Takaaki NIshida
  • Organizer: Colloquium, Department of Mathematics, Cheng-Kung University, Tainan(Taiwan)
  • Invited
• [Presentation] Heat Convection Problems of Compressible Atmosphere and Its Incompressible Limit (2018)
  • Author(s): Takaaki Nishida
  • Organizer: Colloquium, National Taiwan University
  • Invited
• [Presentation] On Chorin's Method, Artificial Compressible System for the Incompressible Viscous Fluids (2018)
  • Author(s): Takaaki Nishida
  • Organizer: NCTS PDE Seminar
  • Invited