Appendix A: Mathematical Tools

  • Wolfgang KollmannEmail author


The derivation and analysis of the basic equation describing turbulence generated by solutions of the Navier–Stokes equations require several mathematical tools collected and briefly discussed and relevant references are given in the present appendix.


  1. 1.
    Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River, New Jersey (2015)Google Scholar
  2. 2.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge U.K (2002)Google Scholar
  3. 3.
    Pesin, Y.B.: Dimension Theory in Dynamical Systems, p. 60637. The University of Chicago Press, Chicago (1997)Google Scholar
  4. 4.
    Ruelle, D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press (1989)Google Scholar
  5. 5.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)Google Scholar
  6. 6.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)Google Scholar
  8. 8.
    Kreyszig, E.: Introductory Functional Analysis with Applications. J. Wiley, New York (1989)Google Scholar
  9. 9.
    Werner, D.: Funktionalanalysis, 6th edn. Springer, New York (2007)Google Scholar
  10. 10.
    Von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig (1985)CrossRefGoogle Scholar
  11. 11.
    Sohr, H.S.: The Navier-Stokes Equations. Springer, Basel (2001)CrossRefGoogle Scholar
  12. 12.
    Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publication, Dordrecht (1988)CrossRefGoogle Scholar
  13. 13.
    Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)CrossRefGoogle Scholar
  14. 14.
    Suhubi, E.: Functional Analysis. Kluwer Academic Publication, Dordrecht (2000)Google Scholar
  15. 15.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Education (Asia), Singapore (2005)Google Scholar
  16. 16.
    Walters, P.: An Introduction to Ergodic Theory. Springer (1982)Google Scholar
  17. 17.
    Dalecky, YuL, Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Space. Kluwer Academic Publ, Dordrecht (1991)CrossRefGoogle Scholar
  18. 18.
    Feller, M.N.: The Lévy-Laplacian. Cambridge University Press (2005)Google Scholar
  19. 19.
    Conway, J.: A course in Functional Analysis. Springer, New York (1990)Google Scholar
  20. 20.
    Bogachev, V.I.: Measure Theory, vol. 1. Springer, New York (2006)Google Scholar
  21. 21.
    Gray, A.: Tubes. Addison-Wesley Publication, Comp (1990)Google Scholar
  22. 22.
    Smirnov, V.I.: A Course of Higher Mathematics, vol. II, Pergamon Press, Oxford (1964)Google Scholar
  23. 23.
    Wang, X.: Volumes of generalized unit balls. Math. Mag. 78, 390–395 (2005)CrossRefGoogle Scholar
  24. 24.
    Folland, G.B.: How to integrate a polynomial over a sphere. Am. Math. Month. 108, 446–448 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Courant, R.: Differential & Integral Calculus, vol. II, Blackie & Sons Ltd, London (1962)Google Scholar
  26. 26.
    Guzman, A.: Derivatives and Integrals of Multivariable Functions. Birkhäuser Boston (2003)Google Scholar
  27. 27.
    Taylor, M.E.: Measure Theory and Integration. AMS Graduate Studies in Math., vol. 76 (2006)Google Scholar
  28. 28.
    Lerner, N.: A Course on Integration Theory. Birkh/"auser/Springer Basel (2014)Google Scholar
  29. 29.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)Google Scholar
  30. 30.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, U.K. (2010)Google Scholar
  31. 31.
    Roman, S.: Advanced Linear Algebra. Graduate Texts in Mathematics, Springer, Berlin (2005)Google Scholar
  32. 32.
    Blumenson, L.E.: A derivation of n-dimensional spherical coordinates. Am. Math. Month. 67, 63–66 (1960)MathSciNetGoogle Scholar
  33. 33.
    Miller, K.S.: Multidimensional Gaussian Distributions. Wiley, New York (1963)Google Scholar
  34. 34.
    Yeh, J.: Real Analysis, 2nd edn. World Scientific, New Jersey (2006)Google Scholar
  35. 35.
    Skorohod, A.V.: Integration in Hilbert Space. Springer, New York (1974)CrossRefGoogle Scholar
  36. 36.
    Egorov, A.D., Sobolevsky, P.I., Yanovich, L.A.: Functional Integrals: Approximate Evaluation and Applications. Kluwer Academic Publication, Dordrecht (1993)CrossRefGoogle Scholar
  37. 37.
    DeWitt-Morette, C., Cartier, P., Folacci, A.: Functional Integration. Plenum Press, New York and London (1997)Google Scholar
  38. 38.
    Simon, B.: Functional Integration and Quantum Physics. AMS Chelsea Publication, Providence, Rhode Island (2004)Google Scholar
  39. 39.
    Cartier, P., DeWitt-Morette, C.: Functional Integration: Action and Symmetries. Cambridge University Press, Cambridge U.K (2006)Google Scholar
  40. 40.
    Benyamini, Y., Sternfeld, Y.: Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88, 439–445 (1983)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge Texts (2001)Google Scholar
  42. 42.
    Klauder, J.R.: A Modern Approach to Functional Integration. Birkhaeuser/Springer, New York (2010)Google Scholar
  43. 43.
    Hunt, B.R., Sauer, T., Yorke, J.A.: Prevalence: a translation-invariant almost every on infinite-dimensional spaces. Bull. Amer. Math. Soc. 27, 217–238 (1992)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Bogachev, V.I.: Gaussian Measures, p. 62. American Mathematical Society, Mathematical surveys and monographs vol (1998)Google Scholar
  45. 45.
    Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4. Academic Press, New York (1964)Google Scholar
  46. 46.
    Lumley, J.L.: The Mathematical Nature of the Problem of Relating Lagrangian and Eulerian Statistical Functions in Turbulence. Mécanique de la Turbulence, CNRS no. 108, Marseille, France (1962)Google Scholar
  47. 47.
    Homann, H., Kamps, O., Friedrich, R., Grauer, R.: Bridging from Eulerian to Lagrangian statistics in 3D hydro- and magnetohydrodynamic turbulent flow. New J. Phys. 11, 073020 (2009)ADSCrossRefGoogle Scholar
  48. 48.
    Kamps, O., Friedrich, R., Grauer, R.: Exact relation between Eulerian and Lagrangean velocity increment statistics. Phys. Rev. E 79, 066301 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)CrossRefGoogle Scholar
  50. 50.
    Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)Google Scholar
  51. 51.
    Hartmann, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  52. 52.
    Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 6(Suppl.), 279–287 (1949)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Eyink, G.L.: Dissipative anomalies in singular Euler flows. Physica D 237, 1956–1968 (2008)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Tsinober, A.: An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer, Dordrecht (2009)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

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