Conservation Laws with Time Dependent Discontinuous Coefficients

We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal locations. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1 , and that the limits are entropy solutions. Then, using the definition of an entropy solution taken from [K. H. Karlsen, N. H. Risebro, and J. D. Towers, Skr. K. Nor. Vidensk. Selsk., 3 (2003), pp. 1–49], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [S. N. Kruˇzkov, Math. USSR-Sb., 10 (1970), pp. 217–243] and also partially those from Karlsen, Risebro, and Towers