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 $L_t^2{\dot{H}}_x^{-1}$ 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 $el{l}^2$ 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.

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