On the Piecewise Smooth Solutions to Non-homogeneous Scalar Conservation Laws

We study the structure and smoothness of non-homogeneous convex conservation laws. The question regarding the number of smoothness pieces is addressed. It is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves. We also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics. These estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws.