| 偏微分方程的调和分析方法简介 |
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| 引用本文： | 苗长兴. 偏微分方程的调和分析方法简介[J]. 数学进展, 2007, 36(6): 641-671 |
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| 作者姓名： | 苗长兴 |
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| 作者单位： | 北京应用物与计算数学研究所,100088,北京 |
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| 基金项目： | 国家重点基础研究发展计划(973计划);国家自然科学基金 |
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| 摘 要： | 本文致力于阐述调和分析与现代偏微分方程研究的关系,特别是奇异积分算子、拟微分算子、Fourier限制性估计、Fourier频率分解方法在椭圆边值问题、非线性发展方程研究中的重要作用.对于偏微分方程研究的各种方法进行了比较与分析,指出了偏微分方程的调和分析方法的优点与局限性.与此同时,还给出了偏微分方程的调和分析方法这一领域的最新研究进展.
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| 关 键 词： | 偏微分方程 边值问题 Cauchy问题 奇异积分算子 拟微分算子 Fourier限制性估计 Fourier频率分解 Littlewood-Paley分解 |
| 文章编号： | 1000-0917（2007）06-0641-31 |
| 收稿时间： | 2005-03-30 |
| 修稿时间： | 2005-03-30 |
The Introduction to Harmonic Analysis Method for Partial Differential Equations |
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| MIAO Changxing. The Introduction to Harmonic Analysis Method for Partial Differential Equations[J]. Advances in Mathematics(China), 2007, 36(6): 641-671 |
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| Authors: | MIAO Changxing |
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| Abstract: | This survey paper is devoted to discribe the relation between harmonic anal- ysis and PDEs.In particular,we emphasize the important role of singular integral operators, pesudo-differential operators,Fourier restriction estimates and Fourier frequency decomposition in the study of boundary value problem for elliptic equations and Cauchy problem for nonlinear evolution equations.We also give some analysis and compare to different study methods of par- tial differential equations,and point out the advantage and disadvantage of harmonic analysis method for partial differential equations.At last,we present some new progress on harmonic analysis method for partial differential equations. |
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| Keywords: | partial differential equations boundary value problem Cauchy problem singular differential operator pesduo-differential operator Fourier frequency decomposition Littlewood-Paley decomposition |
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