Fluid Dynamics Duality and Solution of Decaying Turbulence

We present a duality in the dynamics of incompressible Navier-Stokes fluids in three dimensions, leading to a reformulation of the problem as a one-dimensional momentum loop equation. Importantly, the momentum loop equation does not admit finite-time blow-up solutions. The phenomenon of decaying turbulence emerges as a solution to this equation and can be interpreted as a string theory with a discrete target space composed of regular star polygons and Ising degrees of freedom along their edges. This string theory is solvable in the turbulent limit, which corresponds to a quasiclassical approximation in a nontrivial, calculable background. As a result, the spectrum of decay exponents is derived analytically and exhibits excellent agreement with both experimental data and numerical simulations. Notably, the spectrum includes complex conjugate pairs of exponents associated with the nontrivial zeros of the Riemann zeta function. The classical Kolmogorov scaling laws are replaced by specific functions derived from number theory, exhibiting nonlinear behavior in log-log scale. In particular, we compare the theoretically predicted effective exponent for the second moment of the velocity difference with new DNS results. This comparison reveals a remarkable agreement, with deviations well within the small DNS error margin over a broad range of the scaling variable r / sqrt(t).