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

Pylyp Cherevan

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 $1/6 < δ\le 5/8$ we obtain a log-free estimate at the level $L^2_t {\dot H}^{-1}_x$ for the projection $P_{< N^{1-δ}} \nabla(u_N \otimes v_N)$, 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.

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 for the three-dimensional Navier-Stokes equations. On scale-critical windows and in the range we obtain a log-free estimate at the level for the projection , consistent with the critical energy scheme. The main tools are phase-geometric integration, anisotropic local estimates on cylinders, and bilinear 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 and to sup versions are left for further analysis.

Paper Structure

This paper contains 100 sections, 24 theorems, 183 equations, 2 tables.

Table of Contents

  1. Briefly about Appendix \ref{['app:G']}.
  2. Problem setup and notation
  3. Homogeneous spaces, cutoffs, and angular decomposition
  4. Homogeneous Sobolev norms.
  5. Dyadic frequency projections.
  6. Angular localization.
  7. Output channel high $\times$ high $\to$ low.
  8. Scale dictionary (we fix the notation $\lambda$).
  9. Resonant paraproduct and frequency blocks
  10. Main theorem
  11. Brief proof outline.
  12. Phase–geometric integration
  13. Structure of the phase
  14. Resonant zone.
  15. Gradients.
  16. ...and 85 more sections

Key Result

Theorem 1.1

Let $u,v\colon\mathbb R\times\mathbb R^{3}\to\mathbb R^{3}$ be divergence-free vector fields, and let $u_{\lambda},v_{\lambda}$ be defined via the dyadic projections $P_{\lambda}$. Then for any $\delta\in(\tfrac{1}{6},\tfrac{5}{8}]$ and all $\lambda\gg1$ one has where $C_{\delta}>0$ depends only on $\delta$.

Theorems & Definitions (87)

  • Remark 1: How to read on a first pass
  • Theorem 1.1: log-free estimate of the diagonal paraproduct
  • Remark 1.2
  • Remark 1.3: On the relation between \ref{['eq:logfree-goal']} and Theorem \ref{['eq:local-L2-final']}
  • Remark 2.1: Matching of calibrations: time windows and $\rho$–IBP
  • Remark 2.2: Road sign: heat vs Schr and the size of $|\Phi|^{-1}$
  • Proposition 2.3: sevenfold IBP
  • proof : Idea of the proof
  • Remark 2.4
  • Lemma 2.5: lower Hessian bound
  • ...and 77 more