Almost all points on the real axis can be original points of shock waves

Abstract

For a conservation law with convex condition and initial data in L ^∞(R), it had been commonly believed that the number of discontinuity lines (or shock waves) of the solution is at most countable since Theorem 1 in Oleinik’s seminal paper published in 1956 asserted this fact. In 1977, the author gave an example to show that there is an initial data in C ^∞(R) ∩ L ^∞(R) such that the number of shock waves is uncountable. And in 1980, he gave an example to show that there is an initial data in C(R)∩L ^∞(R) such that the measure of original points of shock waves on the real axis is positive. In this paper, he proves further that the set consisting of initial data in C(R) ∩ L ^∞(R) with the property: almost all points on the real axis are original points of shock waves, is dense in C(R) ∩ L ^∞(R). All these results show that Oleinik’s assertion on the countability of discontinuity lines is wrong.