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Duality of Turbulence to a solvable string theory

Alexander Migdal, School of Mathematics, IAS, Princeton
Tuesday, May 27, 2025 - 3:00pm
PAT C-421

The Navier–Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear partial differential equations (PDEs) system. Despite their fundamental importance, their behavior in turbulent regimes remains poorly understood, and their global regularity is still an open problem. Here, we reformulate the NS equations as a nonlinear equation for the momentum loop  $\vec P(\theta, t)$, effectively reducing the original three-dimensional PDE to a one-dimensional problem.  We present an explicit analytical solution—the Euler ensemble—which describes the universal asymptotic state of decaying turbulence and is supported by numerical simulations and experimental validation. Mathematically, this solution is equivalent to a string theory with a discrete target space, made of regular star polygons, with extra Ising (or fermionic) degrees of freedom on each side. The computations of observable energy spectrum in turbulence are based on Number theory distributions of fractions, plus the instanton computations of harmonic fluctuations in a nontrivial background of our string.

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