任意高维数下稳态Navier-Stokes方程的正则解
Regular Solutions of the Stationary Navier-Stokes Equations in Arbitrarily High Dimensions
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- School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
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中文总结 AI 辅助
本文通过结合平均带符号压力势恒等式与L^2-BMO估计，证明了任意高维数下稳态Navier-Stokes方程正则解的存在性，并构造了周期解与全空间解。
中文摘要 AI 辅助
最近，Li-Yang（Comm. Math. Phys., 2022）建立了在$\mathbb R^n$上具有有界紧支撑力$f$的稳态不可压缩Navier-Stokes方程在$5\le n\le15$时正则解$(u,p)$的存在性。然而，对于维数$n>15$时是否存在正则解仍是未知的。本文回答了这个问题，并获得了任意高维数下稳态Navier-Stokes方程正则解的存在性。关键创新在于将平均带符号压力势恒等式与Bernoulli源通量的$L^2$-BMO估计相结合，得到了一个与表示漂移的斜系数大小无关的Bernoulli上确界界。结合比较解的能量估计，这给出了正Bernoulli函数的Newton势的次线性界，从而去除了早期存在性论证中的上维数限制。我们还为$L^\infty\cap L^{2n/(n+2)}$中的力构造了周期解和全空间解。对于紧支撑力，我们记录了两种远场轮廓，它们的速度、梯度和压力余项，以及相应的消去准则。
英文摘要
Recently, the existence of regular solutions $(u,p)$ to the stationary incompressible Navier--Stokes equations on $\mathbb R^n$ with bounded compactly supported forces $f$ for $5\le n\le15$ was established by Li--Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for dimensions $n>15$. Here we answer this question and obtain the existence of regular solutions to the stationary Navier--Stokes equations in arbitrarily high dimensions. The key innovation is to combine an averaged signed pressure-potential identity with $L^2$-BMO estimates for the Bernoulli source flux, yielding a Bernoulli supremum bound independent of the size of the skew coefficient representing the drift. Together with the energy estimate for the comparison solution, this yields a sublinear bound for the Newtonian potential of the positive Bernoulli function, removing the upper dimension restriction in the earlier existence argument. We also construct periodic solutions and whole-space solutions for forces in $L^\infty\cap L^{2n/(n+2)}$. For compactly supported forces, we record two far-field profiles, their velocity, gradient and pressure remainders, and the corresponding cancellation criteria.
