The reason of Hopf’s and Oleinik’s proofs for countability of shocks being wrong  

作　　者：LI BangHe Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China 

出　　处：《Science China Mathematics》2012年第4期727-729,共3页中国科学：数学（英文版）

基　　金：partially supported by National Basic Research Program of China (Grant No.2011CB302400);National Natural Science Foundation of China (Grant No. 10771206)

摘　　要：For the number of complete shock curves of a conservation law with one space variable,Hopf in 1950 for the Burger equation,and Oleinik in 1956 for the general,stated that it is at most countable.In 1979,the present author published an example to show that the statement of Hopf and Oleinik is wrong.But after so long time,the wrong statement for countability still appeared in some publications,which is at least partly due to that some ones felt difficult to understand Hopf and Oleinik’s proofs being wrong.So,pointing out where they went wrong becomes very necessary.For the number of complete shock curves of a conservation law with one space variable, Hopf in 1950 for the Burger equation, and Oleinik in 1956 for the general, stated that it is at most countable. In 1979, the present author published an example to show that the statement of Hopf and Oleinik is wrong. But after so long time, the wrong statement for countability still appeared in some publications, which is at least partly due to that some ones felt difficult to understand Hopf and Oleinik＇s proofs being wrong. So, pointing out where they went wrong becomes very necessary.

关 键 词：countability of shocks uncountability of shocks conservation law 

分 类 号：O174.12[理学—数学]