Nonlinear Dynamics and The Limits of Local Closures in LES Models Under Critical Regularity

Abstract

This paper examines the structural limitations of finite-order local differential closures in Large Eddy Simulation (LES) near Onsager-critical regularity. The study defines the functional setting, convergence topology, filtering operator, exact subgrid stress, energy-flux functional, and restricted finite-jet class used throughout the analysis. The central claim is kept within this class and is presented as a diagnostic spectral limitation, not as a universal impossibility theorem for all LES models. The analytical framework is formulated on the three-dimensional periodic domain using Littlewood-Paley decomposition, Besov critical scaling, and Bony paraproduct decomposition.A reproducible a priori shell-flux diagnostic complements the theoretical argument. The numerical values are recalculated from the stated formulas for , ensuring consistency among the mean error, shell-wise error, maximum shell error, and spectral correlation. The results show that finite-order local closures can match mean flux indicators while failing to reproduce the spectral distribution of the exact filtered flux, supporting nonlocal, multiscale, or dynamically constrained closures assessed by spectral-flux behavior.