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Exact solutions to Euler equation and Navier–Stokes equation

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Abstract

The Lie symmetry analysis method and Bäcklund transformation method are proposed for finding similarity reduction and exact solutions to Euler equation and Navier–Stokes equation, respectively. By using symmetry reduction method, we reduce nonlinear partial differential equation to nonlinear ordinary differential equation. The infinitesimal generators and the soliton solutions to the Euler equation are obtained by Lie symmetry analysis method. Furthermore, the Bäcklund transformation of the Navier–Stokes equation is proposed to obtain the exact solution. We obtain the exact solutions to Navier–Stokes equation on background flow.

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Acknowledgements

The work is supported by the Nature Science Foundation of China (No. 11701134) and the Science and Technology Plan Project of the Educational Department of Shandong Province of China (Nos. J16LI12, J15LI54 ).

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  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Mingshuo Liu, Xinyue Li & Qiulan Zhao

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  1. Mingshuo Liu
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Correspondence to Qiulan Zhao.

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Liu, M., Li, X. & Zhao, Q. Exact solutions to Euler equation and Navier–Stokes equation. Z. Angew. Math. Phys. 70, 43 (2019). https://doi.org/10.1007/s00033-019-1088-0

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  • DOI: https://doi.org/10.1007/s00033-019-1088-0

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