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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.07848 |
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| _version_ | 1866911200143474688 |
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| author | Cherevan, Pylyp |
| author_facet | Cherevan, Pylyp |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A log-free estimate for the diagonal paraproduct high $\times$ high $\to$ low in the 3D Navier-Stokes equation Cherevan, Pylyp Analysis of PDEs 35Q30, 76D05, 42B20 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. |
| title | A log-free estimate for the diagonal paraproduct high $\times$ high $\to$ low in the 3D Navier-Stokes equation |
| topic | Analysis of PDEs 35Q30, 76D05, 42B20 |
| url | https://arxiv.org/abs/2510.07848 |