# zbMATH — the first resource for mathematics

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

fluid mechanics
Full Text:
##### References:
 [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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.