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Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

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Abstract

The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, 100037, Peoples Republic of China

    Jiequan Li

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Yuxi Zheng

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  1. Jiequan Li
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  2. Yuxi Zheng
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Correspondence to Yuxi Zheng.

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Communicated by P. Constantin

Research partially supported by the Key Program from Beijing Educational Commission (KZ200910028002), 973 project (2006CB805902) and PHR(IHLB) and NSFC (10971142).

Research partially supported by NSF-DMS-0603859, 0908207.

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Li, J., Zheng, Y. Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations. Commun. Math. Phys. 296, 303–321 (2010). https://doi.org/10.1007/s00220-010-1019-6

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  • DOI: https://doi.org/10.1007/s00220-010-1019-6

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