Advertisement

Archive for Rational Mechanics and Analysis

, Volume 230, Issue 1, pp 397–426 | Cite as

Non-degeneracy, Mean Field Equations and the Onsager Theory of 2D Turbulence

  • Daniele Bartolucci
  • Aleks Jevnikar
  • Youngae Lee
  • Wen Yang
Article

Abstract

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence seem to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non-degeneracy assumptions on the associated m-vortex Hamiltonian, the m-point bubbling solutions of the mean field equation are non-degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, T.: Nonlinear analysis on Manifolds Monge-Ampére equations. Grundlehren der Mathematischen Wissenschaften 252, Springer, New York 1982Google Scholar
  2. 2.
    Bandle, C.: Isoperimetric inequalities and applications. Pitmann, London (1980)MATHGoogle Scholar
  3. 3.
    Baraket, S., Pacard, F.: Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Differential Equations 6(1), 1–38 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bartolucci, D.: Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence, arXiv:1609.04139
  5. 5.
    Bartolucci, D., De Marchis, F.: Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence, Arch. Rat. Mech. Anal. 217/2 525-570,  https://doi.org/10.1007/s00205-014-0836-8, 2015
  6. 6.
    Bartolucci, D., De Marchis, F., Malchiodi, A.: Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. (24):5625–5643  https://doi.org/10.1093/imrn/rnq285, 2011
  7. 7.
    Bartolucci, D., Lin, C.S.: Uniqueness Results for Mean Field Equations with Singular Data, Comm. in P. D. E. 34(7–9), 676–702 2009Google Scholar
  8. 8.
    Bartolucci, D., Lin, C.S.: Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter, Math. Ann. 359, 1–44,  https://doi.org/10.1007/s00208-013-0990-6 2014
  9. 9.
    Bartolucci, D., Jevnikar, A., Yang, W., Lee, Y.: Uniqueness of bubbling solutions of mean field equations, arXiv:1704.02354
  10. 10.
    Bartolucci, D., Tarantello, G.: Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Comm. Math. Phys. 229, 3–47 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bartolucci, D., Tarantello, G.: Asymptotic blow-up analysis for singular Liouville type equations with applications. J. D. E. 262, 3887–3931 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bavaud, F.: Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation. Rev. Mod. Phys. 63(1), 129–149 (1991)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory, A. M. S. 83, Springer, New York 1989Google Scholar
  14. 14.
    Brezis, H., Merle, F.: Uniform estimates and blow-up behaviour for solutions of \(-\Delta u = V(x)e^{u}\) in two dimensions, Comm. in P.D.E. 16(8–9), 1223–1253 1991Google Scholar
  15. 15.
    Buffoni, B., Dancer, E.N., Toland, J.F.: The sub-harmonic bifurcation of Stokes waves. Arch. Rat. Mech. Anal. 152(3), 24–271 (2000)MathSciNetMATHGoogle Scholar
  16. 16.
    Buffoni, B., Toland, J.: Analytic Theory of Global Bifurcation. Princeton Univ, Press (2003)CrossRefMATHGoogle Scholar
  17. 17.
    Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 143, 501–525 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description. II. Comm. Math. Phys. 174, 229–260 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Chang, S.Y.A., Chen, C.C., Lin, C.S.: Extremal functions for a mean field equation in two dimension, in: Lecture on Partial Differential Equations, New Stud. Adv. Math. 2 Int. Press, Somerville, MA, 61–93 2003Google Scholar
  20. 20.
    Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chen, C.C., Lin, C.S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surface. Comm. Pure Appl. Math. 55, 728–771 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chen, C.C., Lin, C.S.: Topological Degree for a mean field equation on Riemann surface. Commun. Pure Appl. Math. 56, 1667–1727 (2003)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Chen, C.C., Lin, C.S.: Mean Field Equation of Liouville Type with Singular Data: Topological Degree. Comm. Pure Appl. Math. 68(6), 887–947 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chen, C.C., Lin, C.S., Wang, G.: Concentration phenomena of two-vortex solutions in a Chern-Simons model. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3(2), 367–397 2004Google Scholar
  25. 25.
    De Marchis, F.: Generic multiplicity for a scalar field equation on compact surfaces. J. Funct. An. 259(8), 2165–2192 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Esposito, P., Grossi, M., Pistoia, A.: On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(2), 227–257 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Eyink, G.L., Spohn, H.: Negative temperature states and large-scale, long-lived vortices in two dimensional turbulence. J. Stat. Phys. 70(3–4), 87–135 (1993)MathSciNetMATHGoogle Scholar
  28. 28.
    Eyink, G.L., Sreenivasan, K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 833–886 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Fang, H., Lai, M.: On curvature pinching of conic 2-spheres, Calc. Var. P.D.E. 55, 118, 2016Google Scholar
  30. 30.
    Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Amer. Math. Soc. Transl. 29(2), 295–381 (1963)MathSciNetGoogle Scholar
  31. 31.
    Gladiali, F., Grossi, M.: Some Results for the Gelfand's Problem, Comm. P.D.E. 29(9-10), 1335–1364 2004Google Scholar
  32. 32.
    Grossi, M., Ohtsuka, H., Suzuki, T.: Asymptotic non-degeneracy of the multiple blow-up solutions of the Gel'fand problem in two space dimensions. Adv. Diff. Eq. 16(1–2), 145–164 (2011)MathSciNetMATHGoogle Scholar
  33. 33.
    Gui, C., Moradifam, A.: The Sphere Covering Inequality and Its Applications, Invent. Math., to appearGoogle Scholar
  34. 34.
    Gustafsson, B.: On the convexity of a solution of Liouville's equation equation. Duke Math. J. 60(2), 303–311 (1990)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kiessling, M.K.H.: Statistical mechanics of classical particles with logarithmic interaction. Comm. Pure Appl. Math. 46, 27–56 (1993)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kiessling, M.K.H., Lebowitz, J.L.: The Micro-Canonical Point Vortex Ensemble: Beyond Equivalence. Lett. Math. Phys. 42, 43–56 (1997)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kowalczyk, M., Musso, M., del Pino, M.: Singular limits in Liouville-type equations, Calc. Var. P.D.E. 24(1), 47–81 2005Google Scholar
  38. 38.
    Li, Y.Y.: Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200, 421–444 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Lin, C.S., Yan, S.: On the Chern-Simons-Higgs equation: Part II, local uniqueness and exact number of solutions, preprintGoogle Scholar
  40. 40.
    Lin, C.S., Wang, C.L.: Elliptic functions, Green functions and the mean field equations on tori. Ann. Math. 172(2), 911–954 (2010)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Malchiodi, A.: Topological methods for an elliptic equation with exponential nonlinearities. Discr. Cont. Dyn. Syst. 21, 277–294 (2008)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Malchiodi, A.: Morse theory and a scalar field equation on compact surfaces. Adv. Diff. Eq. 13, 1109–1129 (2008)MathSciNetMATHGoogle Scholar
  43. 43.
    Newton, P.K.: The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci. 145, Springer-Verlag, New York 2001Google Scholar
  44. 44.
    Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 6(2), 279–287 (1949)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Suzuki, T.: Global analysis for a two-dimensional elliptic eiqenvalue problem with the exponential nonlinearly. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(4), 367–398 (1992)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Suzuki, T.: Free Energy and Self-Interacting Particles, PNLDE 62. Birkhauser, Boston (2005)CrossRefGoogle Scholar
  47. 47.
    Suzuki, T.: Some remarks about singular perturbed solutions for Emden-Fowler equation with exponential nonlinearity. In: Functional Analysis and Related Topics. 1991, Kyoto. Lecture Notes in Math., 1540. Berlin: Springer 1993Google Scholar
  48. 48.
    Tarantello, G.: Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys. 37, 3769–3796 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Tarantello, G.: Self-Dual Gauge Field Vortices: An Analytical Approach, PNLDE 72. Birkhäuser Boston Inc, Boston, MA (2007)Google Scholar
  50. 50.
    Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324, 793–821 (1991)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Yang, Y.: Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics 146. Springer, New York (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Rome “Tor Vergata”RomaItaly
  2. 2.Department of Mathematics Education, Teachers CollegeKyungpook National UniversityDaeguSouth Korea
  3. 3.Wuhan Institute of Physics and Mathematics, Chinese Academy of SciencesWuhanPeople’s Republic of China

Personalised recommendations