Log-free estimate for the resonant paraproduct in the 3D Navier-Stokes equations

Abstract

We consider the resonant paraproduct (high-high $\to$ low regime) in the nonlinearity $(u\cdot\nabla)u$ for the three-dimensional Navier-Stokes equations. For sufficiently smooth, divergence-free u, we establish the a priori estimate without logarithmic loss $$\|R_N(u)\|{\dot H^{-1}} \lesssim N^{-1}\,\|u\|{\dot H^{1/2}}\,\|u\|_{\dot H^{1}},$$ with a constant independent of the dyadic frequency $N$. The proof combines phase-geometric integration by parts along an adapted frame, wave-packet discretization at scale $N^{-1/2}$, and an anisotropic Strichartz estimate on time windows of length $N^{-1/2}$. In the wide angular region we apply bilinear decoupling on a rank-3 phase surface; in the geometry at hand the minimal curvature yields a gain of order $N^{-1/6+o(1)}$ (with $o(1)\to 0$ as $N\to\infty$), which suffices to remove the logarithmic loss. The contribution from the narrow region is handled separately by an energy argument in $\dot H^{-1}$ using null-form suppression near the interaction diagonal. The resulting bound is scale-consistent and requires no smallness assumptions, only the divergence-free condition on $u$. The analysis is restricted to a single resonant component of the paraproduct; potential extensions are discussed.