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A steady Euler flow with compact support

  • A. V. GavrilovEmail author
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Abstract

A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

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Notes

Acknowledgements

The author would like to thank the anonymous referee for pointing out the interest-ing work of Khesin, Kuksin, and Peralta-Salas [KKP14] as well as some properties of the given solution.

References

  1. AK99.
    V.I. Arnold and B.A. Khesin. Topological Methods in Hydrodynamics. Springer, Berlin (1999).Google Scholar
  2. AS89.
    Ambrosetti A., Struwe M.: Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Anal. 108, 97–109 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. CC15.
    Chae D., Constantin P.: Remarks on a Liouville-type theorem for Beltrami flows. Int. Math. Res. Not. 2015, 10012–10016 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. CS14.
    Choffrut A., Szekelyhidi L.: Weak solutions to the stationary incompressible Euler equations. SIAM J. Math. Anal. 46, 4060–4074 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Hil97.
    E. Hille. Ordinary differential Equations in the Complex Domain. Dover Publications, Mineloa (1997).Google Scholar
  6. JX09.
    Jiu Q., Xin Z.: Smooth approximations and exact solutions of the 3D steady axisymmetric Euler equations. Commun. Math. Phys. 287, 323–350 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. KKP14.
    Khesin B., Kuksin S., Peralta-Salas D.: KAM theory and the 3D Euler equation. Adv. Math. 267, 498–522 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Nad14.
    Nadirashvili N.: Liouville theorem for Beltrami flow. Geom. Funct. Anal. 24, 916–921 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. NV17.
    Nadirashvili N., Vladut S.: Integral geometry of Euler equations. Arnold Math. J. 3, 397–421 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

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