It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In Vasseur and Vishik (2020) [17] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated.

中文翻译：

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不可压缩欧拉方程轴对称爆破解的不稳定性

目前尚不清楚不可压缩欧拉方程的一个解是否具有平滑的初始值，是否可以在有限时间内爆破。在 Vasseur 和 Vishik (2020) [17] 中，已经表明，如果存在，这样的解在接近爆破时间时会变得线性不稳定。在本文中，我们表明即使在更刚性的轴对称情况下，同样的现象也成立。为了获得这个结果，我们首先证明了一个仅涉及涡量的环形分量的爆破准则。还研究了爆破剖面的不稳定性。