A log-free estimate for the diagonal paraproduct high $\times$ high $\to$ low in the 3D Navier-Stokes equation

Abstract

We consider the diagonal paraproduct arising in the nonlinearity $(u\cdot \nabla) u$ for the three-dimensional Navier-Stokes equations. On scale-critical windows and in the range $1/6 < \delta \le 5/8$ we obtain a log-free estimate at the level $L^2_t {\dot H}^{-1}_x$ for the projection $P_{< N^{1-\delta}} \nabla(u_N \otimes v_N)$, consistent with the critical energy scheme. The main tools are phase-geometric integration, anisotropic local estimates on cylinders, and bilinear $ell^2$ decoupling on a finite-rank surface; the narrow diagonal zone is controlled via suppression of the null form. The work is restricted to a single resonant component; extensions to the full structure $(u \cdot\nabla) u$ and to sup$_t$ versions are left for further analysis.