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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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