The Riemann problem for a two-dimensional hyperbolic system of nonlinear conservation laws: Multiplication of distribution solutions
type: Item
instance of: scholarly article
title: The Riemann problem for a two-dimensional hyperbolic system of nonlinear conservation laws: Multiplication of distribution solutions
zbMATH Open document ID: 0913.35084
DOI: 10.4171/ZAA/848
author: Jiaxin Hu
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-05-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 1211550
zbMATH Keywords: two discontinuity rays
zbMATH Keywords: non-classical waves
zbMATH Keywords: Dirac-contact waves
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/848
OpenAlex ID: W2091800553
description: scientific article; zbMATH DE number 1211550