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Appendix A: Mathematical Tools

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Navier-Stokes Turbulence
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Abstract

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.

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References

  1. Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River, New Jersey (2015)

    Google Scholar 

  2. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge U.K (2002)

    Google Scholar 

  3. Pesin, Y.B.: Dimension Theory in Dynamical Systems, p. 60637. The University of Chicago Press, Chicago (1997)

    Google Scholar 

  4. Ruelle, D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press (1989)

    Google Scholar 

  5. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

    Google Scholar 

  6. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  7. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

  8. Kreyszig, E.: Introductory Functional Analysis with Applications. J. Wiley, New York (1989)

    Google Scholar 

  9. Werner, D.: Funktionalanalysis, 6th edn. Springer, New York (2007)

    Google Scholar 

  10. Von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig (1985)

    Book  Google Scholar 

  11. Sohr, H.S.: The Navier-Stokes Equations. Springer, Basel (2001)

    Book  Google Scholar 

  12. Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publication, Dordrecht (1988)

    Book  Google Scholar 

  13. Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  Google Scholar 

  14. Suhubi, E.: Functional Analysis. Kluwer Academic Publication, Dordrecht (2000)

    Google Scholar 

  15. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Education (Asia), Singapore (2005)

    Google Scholar 

  16. Walters, P.: An Introduction to Ergodic Theory. Springer (1982)

    Google Scholar 

  17. Dalecky, YuL, Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Space. Kluwer Academic Publ, Dordrecht (1991)

    Book  Google Scholar 

  18. Feller, M.N.: The Lévy-Laplacian. Cambridge University Press (2005)

    Google Scholar 

  19. Conway, J.: A course in Functional Analysis. Springer, New York (1990)

    Google Scholar 

  20. Bogachev, V.I.: Measure Theory, vol. 1. Springer, New York (2006)

    Google Scholar 

  21. Gray, A.: Tubes. Addison-Wesley Publication, Comp (1990)

    Google Scholar 

  22. Smirnov, V.I.: A Course of Higher Mathematics, vol. II, Pergamon Press, Oxford (1964)

    Google Scholar 

  23. Wang, X.: Volumes of generalized unit balls. Math. Mag. 78, 390–395 (2005)

    Article  Google Scholar 

  24. Folland, G.B.: How to integrate a polynomial over a sphere. Am. Math. Month. 108, 446–448 (2001)

    Article  MathSciNet  Google Scholar 

  25. Courant, R.: Differential & Integral Calculus, vol. II, Blackie & Sons Ltd, London (1962)

    Google Scholar 

  26. Guzman, A.: Derivatives and Integrals of Multivariable Functions. Birkhäuser Boston (2003)

    Google Scholar 

  27. Taylor, M.E.: Measure Theory and Integration. AMS Graduate Studies in Math., vol. 76 (2006)

    Google Scholar 

  28. Lerner, N.: A Course on Integration Theory. Birkh/"auser/Springer Basel (2014)

    Google Scholar 

  29. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)

    Google Scholar 

  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. Roman, S.: Advanced Linear Algebra. Graduate Texts in Mathematics, Springer, Berlin (2005)

    Google Scholar 

  32. Blumenson, L.E.: A derivation of n-dimensional spherical coordinates. Am. Math. Month. 67, 63–66 (1960)

    MathSciNet  Google Scholar 

  33. Miller, K.S.: Multidimensional Gaussian Distributions. Wiley, New York (1963)

    Google Scholar 

  34. Yeh, J.: Real Analysis, 2nd edn. World Scientific, New Jersey (2006)

    Google Scholar 

  35. Skorohod, A.V.: Integration in Hilbert Space. Springer, New York (1974)

    Book  Google Scholar 

  36. Egorov, A.D., Sobolevsky, P.I., Yanovich, L.A.: Functional Integrals: Approximate Evaluation and Applications. Kluwer Academic Publication, Dordrecht (1993)

    Book  Google Scholar 

  37. DeWitt-Morette, C., Cartier, P., Folacci, A.: Functional Integration. Plenum Press, New York and London (1997)

    Google Scholar 

  38. Simon, B.: Functional Integration and Quantum Physics. AMS Chelsea Publication, Providence, Rhode Island (2004)

    Google Scholar 

  39. Cartier, P., DeWitt-Morette, C.: Functional Integration: Action and Symmetries. Cambridge University Press, Cambridge U.K (2006)

    Google Scholar 

  40. Benyamini, Y., Sternfeld, Y.: Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88, 439–445 (1983)

    Article  MathSciNet  Google Scholar 

  41. Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge Texts (2001)

    Google Scholar 

  42. Klauder, J.R.: A Modern Approach to Functional Integration. Birkhaeuser/Springer, New York (2010)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  44. Bogachev, V.I.: Gaussian Measures, p. 62. American Mathematical Society, Mathematical surveys and monographs vol (1998)

    Google Scholar 

  45. Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4. Academic Press, New York (1964)

    Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  48. Kamps, O., Friedrich, R., Grauer, R.: Exact relation between Eulerian and Lagrangean velocity increment statistics. Phys. Rev. E 79, 066301 (2009)

    Article  ADS  Google Scholar 

  49. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)

    Book  Google Scholar 

  50. Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)

    Google Scholar 

  51. Hartmann, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  52. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 6(Suppl.), 279–287 (1949)

    Article  ADS  MathSciNet  Google Scholar 

  53. Eyink, G.L.: Dissipative anomalies in singular Euler flows. Physica D 237, 1956–1968 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  54. Tsinober, A.: An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer, Dordrecht (2009)

    Book  Google Scholar 

Download references

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Kollmann, W. (2019). Appendix A: Mathematical Tools. In: Navier-Stokes Turbulence. Springer, Cham. https://doi.org/10.1007/978-3-030-31869-7_23

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