Hydrodynamic Coherence and Vortex Solutions of the Euler–Helmholtz Equation

  • N. N. Fimin
  • V. M. Chechetkin


The form of the general solution of the steady-state Euler–Helmholtz equation (reducible to the Joyce–Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.


Joyce–Montgomery equation Euler equation vortex structures Gibbs measure statistical integral conformal mapping 


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  1. 1.
    A. E. Perry, T. T. Lim, M. S. Chong, and E. W. Tex, “The fabric of turbulence,” AIAA Paper 80, 1358 (1980).Google Scholar
  2. 2.
    M. Lesieur, Turbulence in Fluids (Kluwer Academic, Dordrecht, 1997).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. K. M. Hussain, “Role of coherent structures in turbulent shear flows,” Proc. Indian Acad. Sci. (Eng. Sci.) 4, Part 2, 129–175 (1981).Google Scholar
  4. 4.
    B. W. van de Fliert and E. van Groesen, “On variational principles for coherent vortex structures,” Appl. Sci. Res. 51, 399–403 (1993).CrossRefzbMATHGoogle Scholar
  5. 5.
    E. R. Fledderus and E. van Groesen, “Deformation of coherent structures,” Rep. Prog. Phys. 59, 511–600 (1996).CrossRefzbMATHGoogle Scholar
  6. 6.
    B. W. van de Fliert and E. van Groesen, “Monopolar vortices as relative equilibria and their dissipative decay,” Nonlinearity 5, 473–495 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. van Groesen, “Time-asymptotics and the self-organization hypothesis for 2D Navier–Stokes equations,” Phys. A 148, 312–330 (1988).CrossRefzbMATHGoogle Scholar
  8. 8.
    L. Onsager, “Statistical hydrodynamics,” Nuovo Cimento Suppl. 6, 279–289 (1949).MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Batchelor, “Steady laminar flow with closed streamlines at large Reynolds number,” J. Fluid Mech. 1, 177–190 (1957).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. K.-H. Kiessling, “Statistical mechanics of classical particles with logarithmic interactions,” Commun. Pure Appl. Math. 46, 27–56 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. L. Helms, Introduction to Potential Theory (Wiley Interscience, New York, 1969).zbMATHGoogle Scholar
  12. 12.
    M. Brelot, Éléments de la théorie classique du potentiel (Centre de Documentation Universitaire, Paris, 1961).zbMATHGoogle Scholar
  13. 13.
    D. Smets and J. V. Schaftingen, “Desingularization of vortices for the Euler equation,” Arch. Ration Mech. Anal. 198, 869–925 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    C. Neri, “Statistical mechanics of the N-point vortex system with random intensities on,” Electr. J. Differ. Equations 2005 (92), 1–26 (2005).MathSciNetzbMATHGoogle Scholar
  15. 15.
    O. Rey, “The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent,” J. Funct. Anal. 89, 1–52 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E. Caglioti, P. L. Lions, C. Marchioro, and M. Pulvirenti, “A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description,” Commun. Math. Phys. 143, 501–525 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Neri, “M canique statistique des syst mes de vortex ponctuels intensiti s al atoires sur un domaine born,” Ann. Inst. H. Poincare. Sec. A 21, 381–399 (2004).CrossRefGoogle Scholar
  18. 18.
    J. Messer and H. Spohn, “Statistical mechanics of the isothermal Lane–Emden equation,” J. Stat. Phys. 29 (3), 561–578 (1982).MathSciNetCrossRefGoogle Scholar
  19. 19.
    N. Cohen and J. V. T. Benavides, “Explicit radial Bratu solutions in dimension,” UNICAMP IMECC Report 22–07. Scholar
  20. 20.
    I. M. Gel’fand, “Some problems in the theory of quasilinear equations,” Usp. Mat. Nauk 14 (2), 87–158 (1959).zbMATHGoogle Scholar
  21. 21.
    J. Jacobsen and K. Schmitt, “The Liouville–Bratu–Gelfand problem for radial operators,” J. Differ. Equations 184, 283–298 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. S. Lin, “On the existence of positive radial solutions for nonlinear elliptic equations in annular domains,” J. Differ. Equations 81, 221–233 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    C. Bandle, Isoperimetric Inequalities and Applications (Pitman, London, 1980).zbMATHGoogle Scholar
  24. 24.
    V. H. Weston, “On the asymptotic solution of a partial differential equation with an exponential nonlinearity,” SIAM J. Math. 9 (6), 1030–1053 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    K. Nagasaki and T. Suzuki, “Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities,” Asymptotic Anal. 3, 173–188 (1990).MathSciNetzbMATHGoogle Scholar
  26. 26.
    E. J. Routh, “Some applications of conjugate functions,” Proc. London Math. Soc. 12 (170/171), 73–89 (1881).MathSciNetzbMATHGoogle Scholar
  27. 27.
    A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman and Hall/CRC, Boca Raton, 2002).zbMATHGoogle Scholar
  28. 28.
    T. Suzuki, “Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity,” Ann. Inst. Henri Poincare 9 (4), 367–397 (1992).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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