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arXiv 2304.08367math.AP
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全平面上的二维稳态Navier--Stokes方程的适定性缺失

Ill-posedness of the two-dimensional stationary Navier--Stokes equations on the whole plane

Mikihiro Fujii

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27 pages
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AI总结：

本文针对全平面二维稳态Navier--Stokes方程这一长期公开问题，提出基于反证论证的新方法，证明了在基于L^p(R^2)（1≤p≤2）的标度临界Besov空间中解映射不连续且小解不存在，从而确立了不适定性。

摘要：

我们考虑全平面$\mathbb{R}^2$上的二维稳态Navier--Stokes方程。在$n \geqslant 3$的高维情形$\mathbb{R}^n$中，标度临界空间中的适定性与不适定性已被大量论文充分研究。然而，尽管受到许多研究者的关注，二维全平面情形下的相应问题由于二维分析的内在困难，一直是一个长期悬而未决的公开问题。本文旨在解决这一问题，并证明对于所有$1 \leqslant p \leqslant 2$，基于$L^p(\mathbb{R}^2)$的标度临界Besov空间中解映射的不连续性以及小解的不存在性意义上的不适定性。为克服这一困难，我们提出了一种基于反证论证的新方法，将问题归结为对相应的非稳态Navier--Stokes方程的分析，并证明：如果我们反设稳态问题是适定的，则存在具有奇异大时间行为的非稳态解。

英文摘要：

We consider the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with $n \geqslant 3$, the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, despite the attention of many researchers, the corresponding problem in the two-dimensional whole plane case was a long-standing open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and prove the ill-posedness in the scaling critical Besov spaces based on $L^p(\mathbb{R}^2)$ for all $1 \leqslant p \leqslant2$ in the sense of the discontinuity of the solution map and the non-existence of small solutions. To overcome the difficulty, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier--Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed.

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