The Riemann problem for a two-dimensional hyperbolic system of nonlinear conservation laws: Multiplication of distribution solutions
Summary: In the papers [Acta Math. Sci. (Engl. Ed.) 17, No. 4, 361-375 (1997; Zbl 0899.35060)] and [Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 1, 81-94 (1998; Zbl 0893.35071)], 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 \(L^\infty_{\text{loc}}(\mathbb{R}^2\times \mathbb{R}_+)\), and the non-classical waves (labelled as Dirac-contact waves) have arisen. But it remains open 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 difficulty of the multiplication of distribution solutions. The non-classical Riemann solutions 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.
scientific article; zbMATH DE number 1211550
scientific article