Science China Mathematics

, Volume 55, Issue 4, pp 727–729

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


DOI: 10.1007/s11425-012-4372-4

Cite this article as:
Li, B. Sci. China Math. (2012) 55: 727. doi:10.1007/s11425-012-4372-4


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.


countability of shocks uncountability of shocks conservation law 


35L65 35L67 

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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