Log-free estimate of the full nonlinearity in the three-dimensional Navier-Stokes equations outside the diagonal regime
Pylyp Cherevan
TL;DR
This work develops a log-free a priori bound for the full nonlinearity of the 3D Navier–Stokes equations outside the diagonal interaction zone. By combining six-fold integration by parts of the oscillatory phase with an anisotropic Strichartz analysis on cylinders, epsilon-free rank-4 decoupling in folded geometry, and a null-form suppression mechanism in a narrow corona, the authors obtain a scale-invariant bound in $L^1_t\dot H^{-1}_x$ for the off-diagonal part across dyadic scales $N$ with $\delta\in(\tfrac{1}{3},\tfrac{5}{8}]$. The construction yields a local balance unit $N^{-21/4}$ (or $N^{-19/6}$ in the heat line) that, after time–angle patching and dyadic summation, converges without logarithmic losses, establishing a log-free energy-type control for the off-diagonal nonlinearity. While this advances understanding of the nonlinear Navier–Stokes dynamics in the off-diagonal regime, the diagonal zone remains a separate challenge requiring deeper endpoint Strichartz or higher-rank decoupling techniques. The results contribute to scale-invariant energy control and provide methodological building blocks toward global well-posedness insights for small data and further analysis of dispersive-dissipative interactions in fluids.
Abstract
We investigate the contribution of the full nonlinearity outside the narrow diagonal zone in the three-dimensional Navier-Stokes equations. We consider the off-diagonal components, including lh, hl, as well as part of the resonant block hh -> l for |xi + eta| >= N^(1-delta). The proof relies on three main elements: (i) six-fold integration by parts in the phase Phi(t,x,xi,eta) = x*(xi + eta) + 4trho1rho2 with respect to (t,rho1,rho2); on the window |t| <= N^(-1/2) the phase Hessian A = nabla^2_(t,rho1,rho2) Phi is non-degenerate and provides a reserve |det A| ~ N^(3/2 - delta); (ii) local Strichartz estimates on cylinders of scale N^(-1/2); in Sec. 4 a strengthened version is used to combine with the decoupling scheme, while the unconditional framework is based on heat reduction (App. D) and globalization (App. E); and (iii) bilinear epsilon-free decoupling in folded geometry of rank 4 (Appendix B), yielding a gain of N^(-1/4) for angular tiles of width N^(-1/2). For the narrow corona, suppression of the null-form type symbol is realized when delta > 1/2; for the block hh -> h with output projection P_N this mechanism is not required and is accounted for separately (see App. E.6). The combined count yields an a priori estimate without logarithmic losses in the norm L^1_t H^-1_x over the whole zone |xi + eta| >= N^(1 - delta) for delta in (1/3, 5/8]; the upper bound is imposed by the stability of the phase reserve |det A| ~ N^(3/2 - delta) >> 1 on the window |t| <= N^(-1/2). The full scheme and navigation through the sections are given in the text.