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Global Existence of Weak Solutions to the Incompressible Axisymmetric Euler Equations Without Swirl

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Abstract

In this paper, we consider solutions to the incompressible axisymmetric Euler equations without swirl. The main result is to prove the global existence of weak solutions if the initial vorticity \(w_0^\theta \) satisfies that \(\frac{w_0^\theta }{r}\in L^1\cap L^p({\mathbb {R}}^3)\) for some \(p>1\). It is not required that the initial energy is finite, that is, the initial velocity \(u_0\) belongs to \(L^2({\mathbb {R}}^3)\) here. We construct the approximate solutions by regularizing the initial data and show that the concentrations of energy do not occur in this case. The key ingredient in the proof lies in establishing the \(L_{\mathrm{loc}}^{2+\alpha }({\mathbb {R}}^3)\) estimates of velocity fields for some \(\alpha >0\), which is new to the best of our knowledge.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions in improving original manuscript. This work was started when the second author was doing his postdoctoral research supported by the CNPq Grant \(\sharp \) 501376/2013-1 at Federal University of Rio de Janeiro (UFRJ) of Brazil, and he would like to thank Prof. Milton C. Lopes Filho and Prof. Helena J. Nussenzveig Lopes for their hosting and hospitality. The second author also would like to thank Prof. Edriss S. Titi for his valuable suggestions about this problem when he was visiting UFRJ. Q. Jiu is supported by National Natural Science Foundation of China (Nos. 12061003, 11931010), Beijing Natural Science Foundation (No. 1192001) and key research project of the Academy for Multidisciplinary Studies of Capital Normal University. J. Liu is supported by National Natural Science Foundation of China (No. 11801018), Beijing Natural Science Foundation (No. 1192001), Youth Backbone Individual Program of the organization department of Beijing (No. 2017000020124G052) and Beijing University of Technology (No.006000514121518). D. Niu is supported by National Natural Science Foundation of China (Nos. 11471220, 11871046, 11931010), and key research project of the Academy for Multidisciplinary Studies of Capital Normal University.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Quansen Jiu & Dongjuan Niu

  2. College of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Jitao Liu

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  1. Quansen Jiu
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  2. Jitao Liu
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Correspondence to Jitao Liu.

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Communicated by Edriss S. Titi.

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Jiu, Q., Liu, J. & Niu, D. Global Existence of Weak Solutions to the Incompressible Axisymmetric Euler Equations Without Swirl. J Nonlinear Sci 31, 36 (2021). https://doi.org/10.1007/s00332-021-09687-4

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