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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[21]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 [21].
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 [21].
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
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[16] H.W. Shih. Spacetime bounds for the energy-critical nonlinear wave equation in high spatial dimensions. Ready to submit.
[17] C. D. Sogge. Lectures on Non-linear Wave Equations. Monographs in analysis. International Press, 2008.
[18] M. Struwe. Globally regular solutions to the u5 Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (15):495{513, 1988.
[19] T. Tao. Global well-posedness and scattering for the higher-dimensional energycritical nonlinear Schrodinger equation for radial data. New York J. Math., (11):57{80, 2005.
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[21] T. Tao. Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dynamics of PDE, (2):93{110, 2006.
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