A Solution of the Cauchy Problem for the Loop Equation in Turbulence

TL;DR

This paper presents a mathematical solution to the Cauchy problem for the loop equation in turbulence, utilizing a smearing procedure to handle the functional Laplacian under specific conditions.

Contribution

It introduces a novel approach to solving the loop equation in turbulence using smearing of the functional Laplacian, considering initial data and random force potentials.

Findings

Solution depends on initial data and functional derivatives

Method applies under specific viscosity and contour conditions

Addresses turbulence with Gaussian random forces

Abstract

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.