Partial Differential Equations in China PDF
Preview Partial Differential Equations in China
Partial Differential Equations in China Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 288 Partial Differential Equations in China edited by ChaohaoGu Fudan University, Shanglwi, China Xiaxi Ding Institute of Applied Mathematics, Academia SinicQ, Beijing. China and Chung-Chun Yang Dcptlrtmeflt 0/ Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay. Kow/oon, Hong Kong SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A CJ.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-4524-7 ISBN 978-94-011-1198-0 (eBook) DOI 10.1007/978-94-011-1198-0 Printed on acid-free paper All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS Preface vii Contributors ix Part One: Survey Papers Degenerate Parnbolic and Elliptic Equations 1 Yazhe Chen Nonlinear Hyperbolic Conservation Laws 19 Xiaxi Ding & Tong Zhang (Tung Chang) Elliptic and Parabolic Equations 30 Guangcang Dong Viscosity Solutions of Fully Nonlinear Elliptic and Parabolic Equations 42 Guangcang Dong & Baojun Bian Some Developments of the Theory of Mixed PDEs 50 Chaohao Gu & Jiaxing Hong Free Boundary Problems 67 Lishang Jiang The Generalized Riemann Problem for Quasilinear Hyperbolic Systems of Conservation Laws 80 Ta-tsien Li Minimal Surfaces in Riemannian Manifolds 104 Guangyin Wang Microlocal Analysis 111 Rouhuai Wang & Shuxing Chen Nonlinear Partial Differential Equations in Physics and Mechanics 127 Yulin Zhou & Boling Guo v Part Two: Short ComDluuieations Solitons and Exactly Solvable Nonlinear Evolution Equations 160 Yi Cheng & Yi-shen U The Systems of Second Order Partial Differential Equations with Constant Coefficients 173 Wei Un & Ciquian Wu PREFACE Before the founding of the People's Republic of China, there were only a few people who had researched partial differential equations. In 1936, during Professor Hadamard's visit to China, he delivered a series of lectures on the theory of second order linear partial differential equations. In 1951, Professor Sing-Mo 00, who had studied ultra-hyperbolic equations under Professor Hadamard, retwned to China and headed the division of differential equations at the Institute of Mathematics, Academia Sinica in 1953. Then in 1954, the Ministry of High Education of P.R.C. organized a summer school on partial differential equations and Professor Ou and Professor Huanwu Peng were each in charge of one course. Hundreds of young university or college teachers from all over China participated in the school and were very attracted by the lectures. Since then the subject of partial differential equations has been set up and research work on PDEs has spread to almost all universities in China Meanwhile a number of young scholars were sent to the Soviet Union for further training. In 1956, among China's national scientific programmes, the subject of PDEs was placed as one of the most important subjects in mathematics, causing Chinese mathematicians to pay more attention to it During that period, Professor Petrovski visited China and delivered lectures on PDEs. In 1957, Professor Bitsadze came to China to visit the Institute of Mathematics, Academia Sinica, and a number of young Chinese scholars gathered to listen to his lectures; some of them accomplished some research work under his guidance. After that the researchers of PDEs were directed to study some problems related to economic developments. Subjects included fluid mechanics, elastic mechanics, and other problems relating to nonlinear PDEs, such as the theory, the structure of solutions, and approximate analysis. A great deal of theoretical and applied work was done, and the level of research was raised significantly . Valuable results to some important problems were thus obtained. However, in 1966 the Cultural Revolution began and many research programmes were severely impeded. The situation did not change until 1976. Since that time, China has opened her doors to the rest of the world Academic research work received renewed attention and has recovered rapidly in China In the past few years China has invited many western scholars to share their know-how with their Chinese colleagues. They have also sent many bright young scholars or students abroad for advanced studies. As a result of this, scientific research in China has risen to a new state of vigour. In 1980 an International Conference on Differential Geometry and Differential Equations initiated by Professor S.S. Chern was held in Beijing, China Many leading world-famous foreign mathematicians were invited and presented systematic lectures. Chinese mathematicians also shared their results with the visitors at the conference. This conference also played a role in summarizing the results of the past and shaped some future research trends. Since then international academic exchange has continued to develop rapidly. These activities have enabled Chinese scholars to broaden their methods and shrupen their research ability, and in tum, many foreign researchers have come to learn more about the achievements of Chinese scholars. However, we don't vii viii feel that the exchange is sufficient. For example, many results of Chinese scholars were published in Chinese lournals, in the Chinese Janguage which are difficult to be studied or understood by western readers. The goal of this collection is to summarize and introduce the historical progress of the development of PDEs in China. As the research areas are very wide in PDEs the results covered in this book are mainly those that were published before the early 1980s. We would like to express our thanks to those who contributed to the book in different ways; particularly we appreciate Professors Lishang liang, Ta-Tsien Li, and Tong Zhang greatly for their indispensable efforts and help rendered in the preparation of the book. We want to thank: the publisher and mathematics ediur of Kluwer Academic Publishers for their support and endorsement of the project. Finally we want to express our deep gratitude to the late Professor Sing-Mo Ou for his great effM of promoting the research of PDEs in China and his endorsement during the early stage of the development of the collection. Chaohao Gu ~.* Xiaxi Ding T):PI! Chung-Chun Yang ~.Ufl Contributors U1 Baojun Bian ~~ Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027 ff Shuxing Chen ~!i Department of Mathematics, Fudan University, Shanghai 200433 Yazhe Chen ~]E~ Department of Mathematics, Beijing University, Beijing 100871 m Yi Cheng 1.; Department of Mathematics University of Science and Technology of China, Hefei, Anhui 230026 T I am Xiaxi Ding Institute of Applied Mathematics, Academia Sinica, Beijing 100080 Guangcang Dong jijt ~ Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027 S if Chaohao Gu ~ Department of Mathematics, Fudan University, Shanghai 200433 Boling Guo $IS fa ]}t Centre for Nonlinear Studies Institute of Applied Physics and Computational Mathematics Beijing, P.O.Box 8009, Beijing 100088. Jiaxing Hong m~~ Department of Mathematics, Fudan University, Shanghai 200433 li1 L fl6 Lishang Jiang Department of Mathematics, Suzhou University, Suzhou, Jiangsu 215006 **tf Ta-tsien Li Department of Mathematics, Fudan University, Shanghai 200433 * Yi-shen Li jfj ;pfI Department of Mathematics University of Science and Technology of China, Hefei, Anhui 230026 WeiLin# ~ Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275 ix Guangyin Wang £:}'t;J( Institute of Mathematics, Academia Sinica, Beijing 100080 Rouhuai Wang £~ ~ Department of Mathematics, Jilin University, Changchun, Jilin 130023 m Wi Ciquian Wu ~ Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275 PIJ "., Yulin Zhou Centre for Nonlinear Studies Institute of Applied"Physics and Computational Mathematics Beijing, P.O.Box 8009, Beijing 100088 Tong Zhang ~ fr!iJ Institute of Mathematics, Academia Sinica, Beijing 100080 x
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