Formation of Shocks for a Single Conservation Law

The initial value problem for an equation of scalar conservation law in several space dimensions is considered. By the method of characteristics, the solution of this problem with C^∞-initial datum is concretely constructed. Generally, this solution becomes multivalued in finite time. By virtue of the theory of singularities of C^∞-mappings, its structure as a multivalued function is completely revealed. The entropy solution is constructed by making it single-valued. In this process, shocks occur. Shock surfaces are constructed by using the stable manifold theory. Thus propagation of shocks is described.