AI 中文总结
该研究针对半空间中受轴对称齐次外力驱动的定常Navier--Stokes方程，证明了小切向旋度外力下唯一小自相似解的存在性，给出无旋流外力下解存在的标度参数范围，并将结果推广到实心锥情形。
AI 中文摘要
我们研究半空间中满足无滑移边界条件、由轴对称(-3)齐次外力驱动的定常Navier--Stokes方程的轴对称自相似解。若单位球面上外力的切向旋度足够小，我们证明存在唯一的小解；当外力无旋流时，解自动无旋流且唯一。对于无旋流外力$\boldsymbol{F}$，我们引入标度参数$\boldsymbol{λ}$并考虑外力为$λ\boldsymbol{F}$的系统；证明解恰好存在于包含零的开区间内的$λ$取值范围。同样的方法可推广到带无滑移边界条件的实心锥，其中张角越窄，在更大外力下仍能保证解的存在性。
英文摘要
We study axisymmetric self-similar solutions to the stationary Navier--Stokes equations in the half-space with the no-slip boundary condition, driven by an axisymmetric (-3)-homogeneous external force. If the tangential curl of the force on the unit sphere is sufficiently small, we prove the existence of a unique small solution; when the force is swirl-free, the solution is automatically swirl-free and unique. For a swirl-free external force $\boldsymbol{F}$, we introduce a scaling parameter $λ$ and consider the system with force $λ\boldsymbol{F}$; we prove that solutions exist precisely for $λ$ in an open interval containing zero. The same approach extends to solid cones with the no-slip boundary condition, where narrower opening angles allow the existence of solutions under larger external forces.
Comments23 pages; comments are welcome
