GLOBAL STRUCTURE OF SHOCK WAVES
摘要
In this paper, we study global properties of the solution of a conservation law with any bounded L measurable function as initial data. We prove that for any shock curve the tangent lines exist everywhere except at most a countable set, and the tangent slopes form a function with locally bounded variation, that the shock set is a Borel set with zero measure, and that there exists even C~∞ initial data for which the number of shocks is uncountable. This shows that assertion about the number of shocks being at most countable is wrong. In addition, we have found the topological feature of shock set on the upper half-plane, and its every connected component except at most two is rcepectively contained in a region bounded by the real axis and two parallel characteristics. In such a region every shock curve is in possession of the middle line of this region as its asymptote.