          系統識別號 U0026-2001201810450200 論文名稱(中文) 關於臨界能量的非線性波方程在四維和五維空間之Strichartz估計 論文名稱(英文) The Strichartz norm control in the four and five dimensional energy-critical nonlinear wave equations 校院名稱 成功大學 系所名稱(中) 數學系應用數學碩博士班 系所名稱(英) Department of Mathematics 學年度 105 學期 2 出版年 106 研究生(中文) 朱育嶙 研究生(英文) Yu-Lin Chu 學號 L16031085 學位類別 碩士 語文別 英文 論文頁數 43頁 口試委員 指導教授-史習偉口試委員-方永富口試委員-郭鴻文 中文關鍵字 臨界能量  四維非線性波方程  五維非線性波方程  Strichartz估計 英文關鍵字 energy-critical  nonlinear wave equations in four dimensions  nonlinear wave equations in five dimensions  Strichartz bound 學科別分類 中文摘要 這個工作致力於建立時空的控制，對於一維時間五維空間臨界能量的非線性波方程 □u=u^(3/7)， 然後藉由使用質量和能量集中的技巧與來自比較區間的估計來構築精確的Strichartz 控制 ||u||_(L_t^(58/27) L_x^(29/6) (R^(1+5)))≤C(1+CE)^CE^100。 並且我們重新考慮在一維時間四維空間臨界能量的非線性波方程 □u=u^(3)， 根據能量在大球外面的使用技巧和來自區間大小比較的估計得到精確的Strichartz 控制 ||u||_(L_t^(5/2) L_x^(20/3) (R^(1+4)))≤C(1+CE)^CE^24。 英文摘要 The work is devoted to establish the spacetime bound for the energy-critical nonlinear wave equation □u=u^(3/7) in R^(1+5).Then we build the explicit Strichartz bound ||u||_(L_t^(58/27) L_x^(29/6) (R^(1+5)))≤C(1+CE)^CE^100 for some absolute C>0 by using the techniques of the mass and energy concentration and interval-comparing estimate from . And we reconsider the energy-critical nonlinear wave equation □u=u^(3) in R^(1+4).Thus we also build the explicit Strichartz bound ||u||_(L_t^(5/2) L_x^(20/3) (R^(1+4)))≤C(1+CE)^CE^24 for some absolute C>0 by using the techniques of the energy outside tha big ball and the method of the interval-size comparing from . 論文目次 1 Introduction--- 1 2 Notation and basic inequalities ---3 3 Fundamental solutions, Strichartz estimate and inverse Sobolev inequality ---4 4 Energy controls ---14 5 Proof of theorem 1.1 ---21 6 Some novelty controls in d=4 ---29 7 Proof of theorem 1.4 ---34 References ---42 參考文獻  H. Bahouri and P. G'erard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., (19):131{175, 1999.  J. Bourgain. Global well-posedness of defocusing 3D critical NLS in the radial case. JAMS, (12):145{171, 1999.  M. Grillakis. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math., (132), 1990.  M. Grillakis. Regularity for the wave equation with a critical nonlinearity. Commun. Pure Appl. Math., (45):749{774, 1992.  M. Grillakis. A priori estimates and regularity of nonlinear waves. Proceed. Inter. Congress of Math., 1994.  G. Sta lani H. Takaoka J. Colliander, M. Keel and T. Tao. Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation in R3. Annals of Math., (167):767{865, 2008.  A. So er J. Ginibre and G. Velo. The global Cauchy problem for the critical nonlinear wave equation. Jour. Func. Anal., (110):96{130, 1992.  M. Struwe J. Shatah. Well Posedness in the energy space for semilinear wave equations with critical growth. Inter. Math. Research Not., (7):303{309, 1994.  L. Kapitanski. Global and unique weak solutions of nonlinear wave equations. Math. Res. Letters, (1):211{223, 1994.  F. Planchon N. Masmoudi. On Uniqueness for the Critical Wave Equation. Commun. Partial Di erential Equations., 31(7):1099{1107, 2006.  K. Nakanishi. Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power. Internat. Math. Res. Not., (1):31{60, 1999.  K. Nakanishi. Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity. Commun. Partial Di erential Equations., (24):185{221, 1999.  H. Pecher. Nonlinear small data scattering for the wave and Klein-Gordon equations. Math. Z., (185):261{270, 1984.  J. Rauch. The u5 Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations in Nonlinear partial di erential equations and their applications.  I.E. Segal. The global Cauchy problem for a relativistic scalar eld with power interactions. Bull. Soc. Math. France, (91):129{135, 1963.  H.W. Shih. Spacetime bounds for the energy-critical nonlinear wave equation in high spatial dimensions. Ready to submit.  C. D. Sogge. Lectures on Non-linear Wave Equations. Monographs in analysis. International Press, 2008.  M. Struwe. Globally regular solutions to the u5 Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (15):495{513, 1988.  T. Tao. Global well-posedness and scattering for the higher-dimensional energycritical nonlinear Schrodinger equation for radial data. New York J. Math., (11):57{80, 2005.  T. Tao. Nonlinear dispersive equations: local and global analysis. Number 106. American Mathematical Soc., 2006.  T. Tao. Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dynamics of PDE, (2):93{110, 2006. 論文全文使用權限 同意授權校內瀏覽/列印電子全文服務，於2019-02-24起公開。同意授權校外瀏覽/列印電子全文服務，於2019-02-24起公開。 如您有疑問，請聯絡圖書館 聯絡電話：(06)2757575#65773 聯絡E-mail：etds@email.ncku.edu.tw 