Construction of generalized solution for first order divergence type equation

Abstract

We consider the Cauchy problem for a first order divergence type equation with the right hand side independent of the unknown function and with a discontinuous initial condition. This equation was first mentioned by J. M. Burgers in 1940 and it is a model equation for the system of equations describing the non-stationary motion of a gas. Various properties of the solution to this problem we studied in works by O. A. Oleinik (1957), J. Whitham (1974), S. N. Kruzhkov (1970), E. Yu. Panov (2006). The original problem is reduced to the Cauchy problem for Hamilton–Jacobi equation with a continuous initial condition. It is suggested to apply the method of singular characteristics to this problem, while this method was developed A. A. Melikyan for game problems. The effectiveness of technique is demonstrated by the example, when in the original equation the derivative w.r.t. the spatial variable is applied to a cubic polynomial of the unknown function, and boundary condition is specified as a “raising” step. The Hamiltonian in the modified problem is a third degree polynomial of a partial derivative for the unknown function, and the boundary condition is given by the piecewise linear convex function with a break in the origin. We single out the domains of the parameters for which the construction of a generalized solution is possible, and we describe the procedure of constructing the solution. It is shown that the solution involves nonsmooth singularities called the dispersal and equivocal surfaces according to the terminology of differential games. The constructing of the solution is illustrated by figures.