5 v 3 1 A pr 1 99 7 A Solution of the Cauchy Problem for the Loop Equation in Turbulence

Under certain conditions, imposed on the viscosity of the fluid, initial data and the class of contours under consideration, the Cauchy problem with finite values of time for the loop equation in turbulence with Gaussian random forces is solved by making use of the smearing procedure for the loop space functional Laplacian. The solution obtained depends on the initial data and its functional derivatives and on the potential of the random forces. Nowadays there exist various field theoretical approaches to the problem of turbulence. The main one, which exploits the methods of conformal field theory 1 , was proposed in Ref. 2 and developed in Ref. 3. In Ref. 4 the operator product expansion method was applied to investigation of the so-called Burgers' turbulence, i.e. one-dimensional turbulence without pressure (see also Ref. 5, where alternative approaches to this problem such as the instanton approach and replica method were suggested). Field theoretical methods were also used in Ref. 6 in order to calculate the probability distribution and develop Feynman diagrammatic technique in the theory of wave turbulence. An approach to the problem of turbulence, based on the loop calculus 7,8 (for a review see Ref. 9) was suggested in Ref. 10 and developed in Ref. 11 (see also Ref. 12, where its relation to the generalized Hamiltonian dynamics and the Gibbs-Boltzmann statistics was established). Within this approach one deals with loop functionals of the Stokes type Ψ(C) = exp i ν C dr α v α f , where v is the velocity of the fluid, ν is its viscosity, and f 's are external Gaussian random forces, whose bilocal correlator reads