Log-free estimate of the full nonlinearity in the three-dimensional Navier-Stokes equations outside the diagonal regime

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.