Title: Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations Show Chinese title

Title: 非常弱解和定常Navier-Stokes方程Leray解的渐近行为

Abstract: Let $\bfu$ be a Leray solution to the Navier-Stokes boundary-value problem in an exterior domain, vanishing at infinity and satisfying the generalized energy inequality. We show that if there exist $R>0$ and ${\sf s}\ge \frac23 q$, $q>6$, such that the $L^{\sf s}-$norm of $\bfu$ on the spherical surface of radius $R$ divided by $R$ is less than a constant depending only on {\sf s} and $q$, then $\bfu(x)$ must decay as $|x|^{-1}$ for $|x|\to\infty$. This result is proved with an approach based on a new theory of very weak solutions in exterior domains which, as such, is of independent interest. Show Chinese abstract

Abstract: 设$\bfu$为外区域中Navier-Stokes边值问题的Leray解，在无穷远处消失并满足广义能量不等式。 我们证明，如果存在 $R>0$ 和 ${\sf s}\ge \frac23 q$，$q>6$，使得半径为 $R$ 的球面上海量 $\bfu$ 的 $L^{\sf s}-$范数除以 $R$小于仅依赖于 {\sf s} 和 $q$的常数，则 $\bfu(x)$必须按 $|x|^{-1}$ 衰减，对于 $|x|\to\infty$。 这个结果是通过一种基于外部区域中非常弱解的新理论的方法证明的，该理论本身具有独立的兴趣。