三维纳维-斯托克斯方程的紧支有限能量定常解
Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
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中文总结 AI 辅助
本文构造了范数可任意小的三维不可压缩纳维-斯托克斯方程非零紧支定常分布解，通过引入对数Mikado剖面解决了单尺度Mikado机制在有限能量端点的代数增益消失问题，完成了定常凸积分方案的闭合。
中文摘要 AI 辅助
我们构造了无外力作用的不可压缩纳维-斯托克斯方程的非零紧支定常分布解u∈L²(R³;R³)，其L²范数可任意小。该构造解决了三维有限能量端点问题，在此问题中，通常的单尺度间歇Mikado机制对拉普拉斯算子的代数增益消失。主要新要素是对数Mikado剖面，由截断的二维调和偶极子构建并分布在对数多横向尺度上，这产生了闭合定常凸积分方案所需的端点小性。
英文摘要
We construct nonzero compactly supported stationary distributional solutions $u\in L^2(\mathbb{R}^3;\mathbb{R}^3)$ of the unforced incompressible Navier--Stokes equations, together with compactly supported pressures in $L^1$, with both norms arbitrarily small. The construction addresses the three-dimensional finite-energy endpoint at which the usual single-scale intermittent Mikado mechanism loses its algebraic gain against the Laplacian. The main new ingredient is a logarithmic Mikado profile, built from a truncated two-dimensional harmonic dipole and spread over logarithmically many transverse scales. The same perturbation yields a localized $h$-principle: nonzero stationary solutions are strongly dense in the localized $L^p$ classes for every $1\leq p<2$ and in $H^{-1}$, and weakly dense at $p=2$, while strong $L^2$ density fails because of an exact isotropic quadratic-moment constraint. We also prescribe arbitrary positive $L^2$ norms, periodize the construction to $\mathbb{T}^3$, and obtain a stationary-versus-Leray nonuniqueness mechanism for the evolutionary equations.
