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研究生: 林冠廷
Lin, Guan-Ting
論文名稱: 關於具有徑向對稱的克萊恩-戈登-札克洛夫系統之散射問題
On small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry
指導教授: 方永富
Fang, Yung-Fu
學位類別: 碩士
Master
系所名稱: 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年: 2018
畢業學年度: 106
語文別: 英文
論文頁數: 70
中文關鍵詞: Klein-Gordon-Zakharov 系統Strichartz 估計散射理論
外文關鍵詞: Klein-Gordon-Zakharov system, Strichartz estimate, scatter theory
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  • 我們仔細研讀了由Z. Guo 、K. Nakanishi 和S. Wang 三位學者所寫的論文,其標題為“small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry”,該論文探討了Klein-Gordon-Zakharov 方程的散射問題。作者將系統進行轉換和處理,接著給出球對稱版本的Strichartz 估計,最後使用估計證明了解的存在唯一性與漸近行為。而我們詳細的給出了在他們論文中所省略的細節和計算,並針對一些打字或論述的錯誤做了修正。

    We study the paper “Small Energy Scattering with Radial Klein-Gordon-Zakharov System Symmetry(2012) ”, which was written by Zihua Guo, Kenji Nakanishi and Shuxia Wang. In that paper, they discuss the scattering problem of the Klein-Gordon-Zakharov system. First, they transform the system, then give the radial Strichartz estimate. Finally, they use the Strichartz estimate to prove the existence, uniqueness and asymptotic behavior of the solution. We give the detail of the calculation and proofs which are omitted in their papers and made some corrections for some typos.

    1 Introduction 1 2 Preliminaries 4 2.1 Notations 4 2.2 Fourier Transforms 4 2.3 Dyadic Decomposition 5 2.4 Sobolev and Besov spaces 6 2.5 Some Theorems and Inequalities 9 3 The Solution of Klein-Gordon Zakharov system 11 3.1 Transformation of equations 11 3.2 The resonance function and normal form reduction 12 4 Estimates 21 4.1 Strichartz estimates 21 4.2 Other estimates 36 5 Proof of Main Theorem 53 5.1 Contraction mapping 53 5.2 Scattering 65 References 69

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    [7] Guo, Z., Nakanishi, K. and Wang, S. Small energy scattering for the Klein–Gordon–Zakharov system with radial symmetry. arXiv:1208.1657 [math.AP]
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    [12] Ozawa, T., Tsutaya, K. and Tsutsumi, Y. Well-posedness in energy space for the Cauchy problem of the Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. Mathematische Annalen 313.1 (1999): 127-140.
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