发表机构
School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了五维定常Navier-Stokes方程的临界Morrey刚性定理，推导了五维可去奇点准则，还通过有限能量论证证明了四维对应的三次Morrey刚性定理。
AI 中文摘要
我们证明了五维定常Navier-Stokes方程的一个临界Morrey刚性定理。更准确地说，在$\boldsymbol{\rm R}^5\backslash\{0\}$上满足$\boldsymbol{\rm \text{sup}}_{R>0}R^{-2}\boldsymbol{\rm \text{∫}}_{B_R}|u|^3\boldsymbol{\rm \text{d}}x<\boldsymbol{\rm \text{∞}}$的每个光滑解，除压力的一个可加常数外，均恒为零。这用仅依赖速度的尺度不变平均条件取代了已知高维刚性理论中的逐点I型控制，该条件允许空间集中。证明发展了一种弱头压机制，不依赖逐点压力估计或经典法迹。我们从速度重构一个正则压力，推导正头压的重标化不等式，并引入两个单调径向通量。随后利用环形能量估计、合适弱紧性以及爆破和缩尺极限来识别端点通量并确立刚性。作为应用，我们得到了五维的可去奇点准则：若合适弱解在除一点外均光滑，且其尺度不变Dirichlet能量或三次速度Morrey量在该点附近保持有界，则该奇点是可去的。因此，在孤立奇点类中，经典定常正则性准则中的小性假设被有界性取代。我们还通过不同的有限能量论证证明了四维对应的仅依赖速度的三次Morrey刚性定理。
英文摘要
We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on $\mathbb R^5\setminus\{0\}$ satisfying \[ \sup_{R>0}R^{-2}\int_{B_R}|u|^3\,dx<\infty \] is identically zero, up to an additive constant in the pressure. This replaces the pointwise Type-I control in the known higher-dimensional rigidity theory by a velocity-only, scale-invariant averaged condition that allows spatial concentration. The proof develops a weak head-pressure mechanism that does not rely on pointwise pressure estimates or classical normal traces. We reconstruct a canonical pressure from the velocity, derive a renormalized inequality for the positive head pressure, and introduce two monotone radial fluxes. Annular energy estimates, suitable-weak compactness, and blow-up and blow-down limits are then used to identify the endpoint fluxes and force rigidity. As an application, we obtain a removable-singularity criterion in dimension five: if a suitable weak solution is smooth away from one point and either its scale-invariant Dirichlet energy or its cubic velocity Morrey quantity remains bounded near that point, then the singularity is removable. Thus, within the isolated-singularity class, the smallness assumption in the classical stationary regularity criterion is replaced by boundedness. We also prove the corresponding velocity-only cubic Morrey rigidity theorem in dimension four by a different finite-energy argument.
