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SUMMARY:To the theory of decaying turbulence - Alexander Migdal (New York 
 University)
DTSTART:20231107T143000Z
DTEND:20231107T153000Z
UID:TALK207811@talks.cam.ac.uk
DESCRIPTION:We have found an infinite dimensional manifold of exact soluti
 ons of the Navier-Stokes loop equation for the Wilson loop in decaying Tur
 bulence in arbitrary dimension $d >2$. This solution family is equivalent 
 to a fractal curve in complex space $\\mathbb C^d$ with random steps param
 etrized by $N$ Ising variables $\\sigma_i=\\pm 1$\, in addition to a ratio
 nal number $\\frac{p}{q}$ and an integer winding number $r$\, related by $
 \\sum \\sigma_i = q r$. This equivalence provides a dual theory describing
  a strong turbulent phase of the Navier-Stokes flow in $\\mathbb R_d$ spac
 e as a random geometry in a different space\, like ADS/CFT correspondence 
 in gauge theory. From a mathematical point of view\, this theory implement
 s a stochastic solution of the unforced Navier-Stokes equations. For a the
 oretical physicist\, this is a quantum statistical system with integer-val
 ued parameters\, satisfying some number theory constraints. Its long-range
  interaction leads to critical phenomena when its size $N \\rightarrow \\i
 nfty$ or its chemical potential $\\mu \\rightarrow 0$. The system with fix
 ed $N$ has different asymptotics at odd and even $N\\rightarrow \\infty$\,
  but the limit $\\mu \\rightarrow 0$ is well defined. The energy dissipati
 on rate is analytically calculated as a function of $\\mu$ using methods o
 f number theory. It grows as $\\nu/\\mu^2$ in the continuum limit $\\mu \\
 rightarrow 0$\, leading to anomalous dissipation at $\\mu \\propto \\sqrt{
 \\nu} \\to 0$. The same method is used to compute all the local vorticity 
 distribution\, which has no continuum limit but is renormalizable in the s
 ense that infinities can be absorbed into the redefinition of the paramete
 rs. The small perturbation of the fixed manifold satisfies the linear equa
 tion we solved in a general form. This perturbation decays as $t^{-\\lambd
 a}$\, with a continuous spectrum of indexes $\\lambda$ in the local limit 
 $\\mu \\to 0$.The spectrum is determined by a resolvent\, which is represe
 nted as an infinite product of $3\\otimes3$ matrices depending of the elem
 ent of the Euler ensemble.
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