Mathematical analysis of nonlinear partial differential equations describing interaction of several fields

• Project/Area Number: 21540190
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto University
• Principal Investigator: WADA Takeshi 熊本大学, 大学院・自然科学研究科, 准教授 (70294139)
• Co-Investigator(Kenkyū-buntansha): NAKAMURA Makoto 東北大学, 大学院・理学研究科, 准教授 (70312634)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 非線形偏微分方程式 / 適切性 / 相互作用 / 函数方程式論 / 非線形Schrodinger方程式 / 分散型方程式 / 波動方程式

Research Abstract

We studied the well-posedness of nonlinear partial differential equations in mathematical physics and their systems. A given problem for a partial differential equation with initial or boundary conditions is called well-posed if the problem has a unique solution, and if the solution depends continuously on the data given in the problem. This is an important step to ensure that the equation correctly describes the phenomenon. In this research we proved the well-posedness of nonlinear Schrodinger equations and the Maxwell-Schrodinger system under appropriate conditions.

Research Products

• [Journal Article] Continuous dependence for nonlinear Schrodinger equation in H^s (2012)
  • Author(s): H. Uchizono and T. Wada
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] On wellposedness for nonlinear Schrodinger equations with power nonlinearity in fractional order Sobolev spaces (2012)
  • Author(s): H. Uchizono and T. Wada
  • Journal Title: J. Math. Anal. Appl Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Smoothing effects for Schrodinger equations with electromagnetic potentials and applications to the Maxwell-Schrodinger equations (2012)
  • Author(s): T. Wada
  • Journal Title: J. Funct. Anal Volume: 263 Pages: 1-24
  • Peer Reviewed
• [Journal Article] Global solutions for nonlinear wave equations with localized dissipations in exterior domains (2012)
  • Author(s): M. Nakamura
  • Journal Title: J. Differential Equations Volume: 252 Pages: 4742-4785
  • Peer Reviewed
• [Journal Article] Scattering theory for the Dirac Equation of Hartree type and the semirelativistic Hartree equation (2012)
  • Author(s): M. Nakamura and K. Tsutaya
  • Journal Title: Nonlinear Anal. Series A : Theory, Methods and Applications Volume: 75 Pages: 3531-3542
  • Peer Reviewed
• [Journal Article] Smoothing effects for Schrodinger equations with electro-magnetic potentials and applications to the Maxwell-Schrodinger Equations (2012)
  • Author(s): Takeshi Wada
  • Journal Title: Journal of Functional Analysis Volume: (印刷中) Issue: 1 Pages: 1-24
  • DOI: 10.1016/j.jfa.2012.04.010
  • Peer Reviewed
• [Journal Article] Scattering theory for the Dirac Equation of Hartree type and the semir elativistic Hartree equation (2012)
  • Author(s): Makoto Nakamura, Kimitoshi Tsutaya
  • Journal Title: Nonlinear Analysis Series A : Theory, Methods and Applications Volume: 75 Pages: 3531-3542
  • Peer Reviewed
• [Journal Article] Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces (2011)
  • Author(s): M. Nakamura
  • Journal Title: Rev. Math. Phys Volume: 23 Pages: 903-931
  • Peer Reviewed
• [Journal Article] Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher-order wave equations (2011)
  • Author(s): M. Nakamura
  • Journal Title: Differential Integral Equations Volume: 24 Pages: 519-540
  • Peer Reviewed
• [Journal Article] Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces (2011)
  • Author(s): Makoto Nakamura
  • Journal Title: Reviews in Mathematical Physics Volume: 23 Pages: 903-931
  • Peer Reviewed
• [Journal Article] Global solutions for nonlinear wave equations with localized dissipations in exterior domains (2011)
  • Author(s): Makoto Nakamura
  • Journal Title: Journal of Differential Equations Volume: 252 Pages: 4742-4785
  • Peer Reviewed
• [Journal Article] Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher order wave equations
  • Author(s): Nakamura, Makota
  • Journal Title: Differential and Integral Equations Volume: (未定 掲載確定)
  • Peer Reviewed
• [Presentation] 非線形Schrodinger方程式の解の初期条件に関する連続依存性について (2011)
  • Author(s): 内園晴典, 和田健志
  • Organizer: 第125回日本数学会九州支部例会
  • Place of Presentation: 熊本大学
  • Year and Date: 2011-10-22
• [Presentation] Smoothing effects for Schrodinger equations with electromagnetic potentials and applications to the Maxwell-Schrodinger Equations (2011)
  • Author(s): T. Wada
  • Organizer: 第7回非線型の諸問題
  • Place of Presentation: 熊本大学
  • Year and Date: 2011-09-25
• [Presentation] Smoothing effects for Schrodinger equations with electro-magnetic potentials and applications to the Maxwell-Schrodinger Equations (2011)
  • Author(s): 和田健志
  • Organizer: 非線型の諸問題
  • Place of Presentation: 熊本大学(招待講演)
  • Year and Date: 2011-09-24
• [Presentation] 電磁ポテンシャルを含むSchrodinger方程式の平滑化効果とMaxwell-Schrodinger方程式への応用 (2010)
  • Author(s): 和田健志
  • Organizer: 夏の偏微分方程式セミナー
  • Place of Presentation: 神戸大学瀧川記念会館
  • Year and Date: 2010-08-25
• [Presentation] 空間2次元におけるMaxwell-Schrodinger方程式 (2009)
  • Author(s): 和田健志
  • Organizer: 夏の偏微分方程式セミナー
  • Place of Presentation: 龍谷大学セミナーハウスともいき荘(京都)
  • Year and Date: 2009-08-28