A generalized method of characteristics in the theory of Hamilton–Jacobi equations and conservation laws

In the paper two types of generalized solutions of the Cauchy problem are presented for the Hamilton–Jacobi–Bellman equation and for the scalar conservation law. The connections between the generalized solutions are obtained. A description of the structure of the the singular set is provided for the minimax/viscosity solution of the Hamilton–Jacobi equation. The representative formula is suggested for the conservation law in terms of classical characteristics.