Local And Global Analysis of Nonlinear Dispersive And Wave Equations

出版者:American Mathematical Society

作者:Terence Tao

页数:373

出版时间:2006-7-1

价格:USD 57.00

装帧:Paperback

isbn号码:9780821841433

丛书系列:Conference Board of the Mathematical Sciences

图书标签:PDE数学nonlinear色散方程偏微分方程7monographmathMath

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Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrödinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations.

Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.

As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.

读后感

" My personal philosophy that to truly understand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to an...

" My personal philosophy that to truly understand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to an...

" My personal philosophy that to truly understand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to an...

" My personal philosophy that to truly understand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to an...

" My personal philosophy that to truly understand a mathematical result one must view it from as many perspectives as possible (including both rigorous arguments and informal heuristics), and must also be able to translate easily from one perspective to an...

用户评价

陶哲轩神作

因为想了解nonlinear wave选了这本书，但不是个明智的选择，尤其在没学过调和分析的情况下。但Tao很有洞见，尤其第一章所表述的内容和之后章节的联系。细节太少，typo多到后来不想去看勘误表，过多的例子和证明放在了练习。初学者可以参考Tadahiro Oh "Nonlinear Schrödinger Equations"这门课的笔记和references。 E. L. 认为最后一章关于wave maps的内容过于陈旧。

PDE的代数（对称，守恒律，显示表达）和分析（渐进，存在和唯一）；方程的几何物理直觉和严格分析之间平衡。自治方程=时间平移不变量=嵌入；第一章一阶拟线性方程组的柯西问题关键概念是解析性；诺特定理本质联系了对称和守恒；

PDE的代数（对称，守恒律，显示表达）和分析（渐进，存在和唯一）；方程的几何物理直觉和严格分析之间平衡。自治方程=时间平移不变量=嵌入；第一章一阶拟线性方程组的柯西问题关键概念是解析性；诺特定理本质联系了对称和守恒；

PDE的代数（对称，守恒律，显示表达）和分析（渐进，存在和唯一）；方程的几何物理直觉和严格分析之间平衡。自治方程=时间平移不变量=嵌入；第一章一阶拟线性方程组的柯西问题关键概念是解析性；诺特定理本质联系了对称和守恒；