AI 中文总结
该研究针对三维定常Navier–Stokes方程的光滑解，通过改进两个Osgood型准则，得到仅涉及速度与涡量切向相互作用的Liouville型定理，允许次临界增长。
AI 中文摘要
我们证明三维欧氏空间中定常Navier–Stokes方程光滑解$(u,p)$满足有限Dirichlet能量条件与一致衰减条件时的Liouville型定理。我们的结果对近期两个Osgood型准则进行了几何改进，即基于头压$Q=\frac{1}{2}|u|^2+p$的相对衰减准则与加权可积性准则。证明利用了头压的若干性质以及$(u\times\omega)\cdot\nabla Q$的标量三重积结构，结合总涡量能量的Osgood型表示，该几何结构给出仅涉及$|Q|$超水平集上$u$与$\omega$切向相互作用的Liouville型准则。在相对衰减情形下，这些准则导出Osgood量级的小性条件，允许超出对应一致界的次临界增长；在加权可积性情形下，涉及全速度与速度梯度的先前条件被改进为仅涉及$Q$水平面上$u$与$\omega$的切向分量，以及衡量其相对取向的角因子。
英文摘要
We prove Liouville-type theorems for smooth solutions $(u,p)$ of the stationary Navier--Stokes equations in $\mathbb R^3$ satisfying the finite Dirichlet energy condition and the uniform decay condition. Our results give geometric refinements of two recent Osgood-type criteria, namely the relative decay criterion and the weighted integrability criterion formulated in terms of the head pressure $Q=\frac{1}{2}|u|^2+p$. The proof exploits several properties of the head pressure and the scalar triple-product structure of $(u\timesω)\cdot\nabla Q$. Combined with an Osgood-type representation of the total vorticity energy, this geometric structure yields Liouville-type criteria involving only the tangential interaction of $u$ and $ω$ over the superlevel sets of $|Q|$. In the relative decay setting, they lead to Osgood-scale smallness conditions that allow subcritical growth beyond the corresponding uniform bounds. In the weighted integrability setting, the previous conditions involving the full velocity and velocity gradient are refined to involve only the tangential components of $u$ and $ω$ along the level surfaces of $Q$, together with an angular factor measuring their relative orientation.
Comments10 pages
