Optimal results on TV bounds for scalar conservation laws with discontinuous flux

This paper is concerned with the total variation of the solution of scalar conservation law with discontinuous flux in one space dimension. One of the main unsettled questions concerning conservation law with discontinuous flux was the boundedness of the total variation of the solution near interface. In [1], it has been shown by a counter-example at T=1, that the total variation of the solution blows up near interface, but in that example the solution become of bounded variation after time T>1. So the natural question is what happens to the BV-ness of the solution for large time. Here we give a complete picture of the bounded variation of the solution for all time. For a uniform convex flux with only L∞ data, we obtain a natural smoothing effect in BV for all time t>T0. Also we give a counter-example (even for a BV data) to show that the assumptions which have been made are optimal.