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The Riemann Problem with Delta Initial Data for the One-Dimensional Transport Equations

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Abstract

In this paper, the Riemann problem with delta initial data for the one-dimensional transport equations is studied. The global existence of generalized solutions is obtained constructively using generalized Rankine–Hugoniot conditions and the entropy condition. Moreover, a new kind of nonclassical wave appears, namely a delta contact discontinuity, which is a Dirac delta function supported on a contact discontinuity. Furthermore, it can be found that the generalized solutions are stable by making use of the perturbation of the initial data.

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Acknowledgments

Supported by the National Natural Science Foundation of China (No. 11401508, 11461066, 11101348) and the Scientific Research Program of the Higher Education Institution of Xinjiang (No. XJEDU2014I001, XJEDU2011S02).

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Ürümqi, 830046, People’s Republic of China

    Lihui Guo & Gan Yin

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  1. Lihui Guo
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  2. Gan Yin
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Correspondence to Gan Yin.

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Communicated by Yong Zhou.

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Guo, L., Yin, G. The Riemann Problem with Delta Initial Data for the One-Dimensional Transport Equations. Bull. Malays. Math. Sci. Soc. 38, 219–230 (2015). https://doi.org/10.1007/s40840-014-0015-y

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  • DOI: https://doi.org/10.1007/s40840-014-0015-y

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