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Turbulence theory and functional integration. I, II. (English) Zbl 0097.41401

Keywords:
fluid mechanics
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[1] For excellent accounts of the historical and logical developments in Navier-Stokes fluid turbulence, see: G. K. Batchelor,Homogeneous Turbulence(Cambridge University Press, New York, 1953);
[2] A. A. Townsend,Structure of Turbulent Shear Flow(Cambridge University Press, New York, 1956). · Zbl 0070.43002
[3] Kraichnan, Phys. Rev. 109 pp 1407– (1958)
[4] Kraichnan 111 pp 1747– (1958)
[5] Kraichnan, Phys. Fluids 1 pp 358– (1958)
[6] Hopf, J. Ratl. Mech. Anal. 1 pp 87– (1952)
[7] Hopf, J. Ratl. Mech. Anal. 2 pp 587– (1953)
[8] The Fourier representation in (6) converges in the sense of adistribution; for example, see M. J. Lighthill,Introduction to Fourier Analysis and Generalized Functions(Cambridge University Press, New York, 1958).
[9] V. Volterra,Theory of Functionals(Dover Publications New York, 1959) pp. 22–31.
[10] K. O. Friedrichs, H. N. Shapiro, et al., ”Integration of functionals” (lecture notes, New York University, 1957). · Zbl 0089.31601
[11] Misner, Revs. Modern Phys. 29 pp 497– (1957)
[12] Segal, J. Ind. Math. Soc. 4 pp 105– (1949)
[13] Matthews, Nuovo Cimento 2 pp 120– (1955)
[14] J. A. Wheeler, ”Fields and particles” (mimeographed lecture notes, Princeton University, 1955).
[15] See reference 7, p. I-9.
[16] Edwards, Proc. Roy. Soc. (London) A224 pp 24– (1954)
[17] Our simplified formalism makesCan infinite constant. This is a trivial divergence, easily rectified by putting aCinto the definition (16). Moreover, with an appropriate renormalization at a late stage in the calculation, it is justifiable to ignore the fact that C = .
[18] According to (38b), the six functions B\(\alpha\)\(\beta\) exist foralmost all\(\eta\);
[19] the exceptional \(\eta\) cause the determinant of the integral in (38b) to vanish at certainxandt. The latter condition defines a set of measure zero with respect to the \(\eta\) integration in (37) and is therefore of no consequence.
[20] Polkinghorne, Proc. Roy. Soc. (London) A230 pp 272– (1955)
[21] Since admissible \(\eta\) vanish at t = t, it follows from (38) that B\(\alpha\)\(\mu\)(x,t)A\(\mu\)(x,t) diverges like (t-t)-1 for admissible \(\eta\) which are analytic intin the neighborhood of t. Thus it is not possible to work directly with B\(\alpha\)\(\mu\)(x,t)A\(\mu\)(x,t).
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