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Some rigorous remarks on Migdal's momentum loop equation

Bruè Elia, Camillo De Lellis

Abstract

We give a rigorous mathematical treatment of some portions of the theory developed by Alexander Migdal on the momentum loop equation.

Some rigorous remarks on Migdal's momentum loop equation

Abstract

We give a rigorous mathematical treatment of some portions of the theory developed by Alexander Migdal on the momentum loop equation.

Paper Structure

This paper contains 39 sections, 29 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. The momentum loop equation and the Euler ensemble
  3. Discretizations
  4. Notation
  5. Acknowledgments
  6. Probability measures on trajectories of Navier-Stokes
  7. Proof of Theorem \ref{['t:prob']}
  8. Gaussian Measures on Tn
  9. Gaussian Measures on Rn
  10. Loop Functional
  11. Liquid Loop Functional
  12. Area derivative and loop equation
  13. Area Derivative and Vorticity
  14. Navier-Stokes in Vorticity Formulation
  15. Loop Equation
  16. ...and 24 more sections

Key Result

Lemma 2.2

A function $u \in C_w(I, L^2_\sigma)$ if and only if the following two properties hold: Moreover, for every $t_0 \in I$, the time average converges weakly in $L^2$ as $r \to 0$. This defines a well-posed trace of $u$ on $\Omega \times \{t\}$, which coincides with the original $u$ for almost every $(x,t)$. Furthermore, if this trace is used in the pointwise definition of $\Phi$ in e:def-Phi, then

Theorems & Definitions (65)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4: Probabilistic Solutions
  • Remark 2.5: Restriction of Probabilistic Solutions
  • Definition 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 2.9: Gaussian Initial Velocity
  • Proposition 2.10
  • ...and 55 more