三维环面 $\mathbb T^3$ 上 Navier-Stokes 方程的 Hartman-Grobman 定理

中文摘要 AI 辅助

我们在三维环面上，在 $H^s$ 强拓扑中（对每个 $s>7/2$），证明了无外力 Navier-Stokes 方程在零点处的局部 Hartman-Grobman 定理。更精确地说，我们构造了一个零点的开正不变邻域之间的同胚，将 Navier-Stokes 半流共轭到热半群。我们发展了一个建立在嵌套精确渐近动力学因子层级上的无穷维边界-时间格式。其纤维几何给出具有自治有限维因子系统的 Sobolev 分块坐标，在每一有限层次上可构造边界-时间拓扑共轭，然后在公共邻域上组装成所需的 $H^s$ 共轭。无穷维组装依赖于对无散度输运结构的新谱估计，该估计控制导数损失以及有限层次共轭及其逆的分块尾部，无需额外的谱间隙假设。

英文摘要

We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of $H^s$ for every $s>7/2$. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired $H^s$ conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.

发表机构

• School of Mathematics, Sichuan University（四川大学数学学院）
• School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）

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