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Smooth Approximations and Exact Solutions of the 3D Steady Axisymmetric Euler Equations

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Abstract

In this paper, we prove that a class of C 1-smooth approximate solutions {u ε, p ε} to the 3D steady axisymmetric Euler equations will converge strongly to 0 in \({L^2_{loc}(R^3)}\) . The main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields. We also show a Liouville type theorem that there are no non-trivial C 1-smooth exact solutions with finite energy and uniform constant state at far fields.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, PRC

    Quansen Jiu

  2. IMS and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T, Hong Kong

    Zhouping Xin

  3. Center for Nonlinear Studies, Northwest University, Xi’an, 710069, PRC

    Zhouping Xin

Authors
  1. Quansen Jiu
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  2. Zhouping Xin
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Correspondence to Quansen Jiu.

Additional information

Communicated by P. Constantin

The research is partially supported by National Natural Sciences Foundation of China (No. 10871133 & No. 10771177).

The research is partially supported by Zheng Ge Ru Funds, Hong Kong RGC Emarked Research Grant CUHK4028/04P and CUHK4040/06P, RGC Central Allocation Grant CA 05/06. SC01, and a grant from Northwest University, Xi’an, PRC.

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Jiu, Q., Xin, Z. Smooth Approximations and Exact Solutions of the 3D Steady Axisymmetric Euler Equations. Commun. Math. Phys. 287, 323–349 (2009). https://doi.org/10.1007/s00220-008-0687-y

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  • DOI: https://doi.org/10.1007/s00220-008-0687-y

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