The Riemann Problem for a Two-Dimensional Hyperbolic System of Nonlinear Conservation Laws: Multiplication of Distribution Solutions

Abstract

In the papers [6-8], the author has constructed the Riemann solutions to a two-dimensional hyperbolic system of nonlinear conservation laws for any piecewise constant initial data having two discontinuity rays with origin as vertex. It has been found that, for some initial data, the Riemann solutions no longer lie in $L_{loc}^{\infty }({\mathbb{R}}^2\times {\mathbb{R}}_{+})$, and the non-classical waves (labelled as Dirac-contact waves) have arisen. But it remains open in [6-8] to verify that the non-classical solutions constructed satisfy the system considered. In the present paper, we borrow the new mathematical theory of generalized functions, chiefly initiated by J. F. Colombeau and Rosinger, to deal with the diffculty of the multiplication of distribution solutions. The non-classical Riemann solutions we constructed in [6-8] satisfy the system in the sense of association. The present paper provides a good example of applications for this new mathematical theory in powerfully handling the product of generalized functions.