三维定常Navier-Stokes方程具有对数奇异射线行为的局部(-1)次齐次轴对称解的分类
Classification of local $(-1)$-homogeneous axisymmetric solutions of the $3$D stationary Navier-Stokes equations with logarithmic singular-ray behavior
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中文总结 AI 辅助
该研究分类三维定常Navier-Stokes方程的局部(-1)次齐次轴对称解，构造含非零旋度II型解的四参数族，明确其特性并确定跨奇异射线的奇异力。
中文摘要 AI 辅助
我们研究三维不可压定常Navier-Stokes方程在奇异射线附近的局部(-1)次齐次轴对称解(u, p)的存在性与分类，聚焦于满足0<limsup_{x∈S², x→P}|u|/|ln dist(x, P)|<∞的II型行为解，其中P为南极或北极。除Landau解外，这类解是最不奇异的非平凡(-1)次齐次轴对称解。我们构造了一个四参数族的局部(-1)次齐次轴对称解，其在奇异射线附近至多具有对数增长，并将其表示为收敛级数，系数由递归方式确定。反之，我们证明：当x在S²上趋近于P时，满足|u|=o(dist(x, P)⁻¹)的每个局部(-1)次齐次轴对称解都属于该族。特别地，该族包含非零旋度的II型解，这与ℝ³\{x'=0}中无旋度的全局(-1)次齐次轴对称II型解形成对比。我们还建立了所构造解的导数估计，并在分布意义上确定了这些解跨奇异射线产生的奇异力。
英文摘要
We study the existence and classification of local $(-1)$-homogeneous axisymmetric solutions $(u, p)$ of the three-dimensional incompressible stationary Navier-Stokes equations near a singular ray. We focus on such solutions with Type II behavior, which satisfy $0<\limsup_{x\in\bS^2, x\to P}|u|/|\ln \text{dist}(x, P)|<\infty$, where $P$ is the south or north pole. Apart from the Landau solutions, these are the least singular nontrivial $(-1)$-homogeneous axisymmetric solutions. We construct a four-parameter family of local $(-1)$-homogeneous axisymmetric solutions with at most logarithmic growth near the singular ray and represent them as convergent series whose coefficients are determined recursively. Conversely, we prove that every local $(-1)$-homogeneous axisymmetric solution satisfying $|u|=o(\text{dist} (x, P)^{-1})$ as $x\to P$ on $\bS^2$ belongs to this family. In particular, this family contains Type II solutions with nonzero swirl, in contrast to the global $(-1)$-homogeneous axisymmetric Type II solutions in $\R^3\setminus\{x'=0\}$, which have no swirl. We also establish derivative estimates for the constructed solutions and identify the singular force generated by these solutions across the singular ray in the sense of distributions.
