发表机构
Rutgers University–New Brunswick（罗格斯大学新布朗斯维克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将定常Navier-Stokes方程齐次解的存在定理推广到n≥4维，结合次临界压力恒等式与二进估计完成证明，还简化了有界紧支力情形的存在定理证明并建立小力下的唯一性。
AI 中文摘要
对于每个整数n≥4，我们证明了在ℝⁿ∖{0}上，当存在任意局部Lipschitz的(-3)次齐次力时，定常不可压缩Navier-Stokes方程存在(-1)次齐次解，其压力为(-2)次齐次，且解在原点外正则。这将Bang、Gui、Liu、Wang和Xie的齐次存在定理推广到了16维以上的情形。主要估计结合了次临界局域压力恒等式与合适有限Lebesgue空间中辅助切向量场的初等二进估计，在齐次消去指数处测试伯努利方程，随后控制其正部在L^((n-2)(n-1)/(2(n-3)))(S^{n-1})空间中，所得估计在每个固定维数下与能量不等式闭合。我们还对充分小的力，在经典齐次类中建立了唯一性。利用相同思路，我们还对ℝⁿ（n≥5）上有界紧支力的已知存在定理给出了简化证明：带无散漂移的标量方程的一般有限指数估计，结合二进核估计的欧氏版本，在不使用BMO估计或迭代以获取其上确界范数的情况下，对正伯努利函数进行了估计。
英文摘要
For every integer $n\ge4$, we prove the existence of a $(-1)$-homogeneous solution of the stationary incompressible Navier-Stokes equations on $\mathbb{R}^n\setminus\{0\}$ with an arbitrary locally Lipschitz $(-3)$-homogeneous force. The pressure is $(-2)$-homogeneous, and the solution is regular away from the origin. This extends the homogeneous existence theorem of Bang, Gui, Liu, Wang, and Xie beyond dimension sixteen. The principal estimate combines a subcritical localized pressure identity with an elementary dyadic estimate for an auxiliary tangential vector field in a suitable finite Lebesgue space. Testing the Bernoulli equation at the homogeneous cancellation exponent then controls its positive part in $L^{\frac{(n-2)(n-1)}{2(n-3)}}(S^{n-1})$. The resulting estimate closes with the energy inequality in every fixed dimension. We also establish uniqueness throughout the classical homogeneous class for sufficiently small force. Using the same ideas, we also give a simplified proof of the known existence theorem for bounded compactly supported forces on $\mathbb{R}^n$, $n\ge5$. A general finite-exponent estimate for a scalar equation with divergence-free drift, together with a Euclidean version of the dyadic kernel estimate, bounds the positive Bernoulli function without BMO estimates or an iteration to obtain its supremum norm.
