非線性雙曲型偏微分方程講義

內容概要

本書是以作者1986年～1987年在Lund大學三個學期授課的講義為基礎，經改寫而成的，主要論述了非線性雙曲型偏微分方程解的全局存在性或“爆破”(blowup)，以及解的奇異性傳播。書中所用的方法是基于對波方程或Yang-Mills方程的非線性攝動研究中采用的保角變換，以及對非線性方程解的余法向奇異性的傳播。     目次：常微分方程；一個空間變量的一階標量方程；多個空間變量的一階標量方程；一個空間變量的一階守恒律系統；補償列緊性；波方程的非線性攝動；Klein-Gordon方程的非線性攝動；微局部分析；擬微分算子；仿微分計算；奇異性的傳播。    讀者對象：本書可作為大學生在學習基礎的分布理論、測度論和泛函分析等課程之后，進一步學習非線性雙曲型偏微分方程的教科書。

書籍目錄

PrefaceContentsChapter Ⅰ Ordinary differential equations  1.1 Introduction  1.2 Local existence and uniqueness for the Cauchy problem  1.3 Existence of solutions in the large  1.4 Generalized solutionsChapter Ⅱ Scalar first order equations with one space variable  2.1 Introduction  2.2 The linear case  2.3 Classical solutions of Burges'equation  2.4 Weak solutions of Burgers'equation  2.5 General strictly convex conservation lawsChapter Ⅲ Scalar first order equations with several variables  3.1 Introduction  3.2 Parabolic equations  3.3 The conservation law with viscosity  3.4 The entropy solution of the conservation lawChapter Ⅳ First order systems of conservation laws with one space variable  4.1 Introduction  4.2 Generalities on first order systems  4.3 The lifespan of classical solutions  4.4 The Riemann problem  4.5 Glimm's existence theorem  4.6 Entropy pairsChapter Ⅴ Compensated compactness  5.1 Introduction  5.2 Weak convergence in Loo  5.3 Weak convergence of solutions of linear differential equations  5.5 Probability measures associated with a system of two equetions  5.6 Existence of weak solutions for a system of two equationsChapter Ⅵ Nonlinear perturbations of the wave equationChapter Ⅶ Nonlinear perturbations of the Klein-Gordon equationChapter Ⅷ Microlocal analysisChapter Ⅸ Pseudo-differential operators of type1,1Chapter Ⅹ Paradifferential calculusChapter Ⅺ Propagation of singularitiesAppendix on pseudo-Riemannian geometryReferences Index of notationIndex

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