论文标题
关于$ c^{1，α} $的稳定性和不稳定，对3D Euler和2D Boussinesq方程的单数解决方案
On stability and instability of $C^{1,α}$ singular solutions to the 3D Euler and 2D Boussinesq equations
论文作者
论文摘要
已知3D不可压缩的Euler方程的奇异性形成是极具挑战性的。在[18]中，Elgindi证明了没有漩涡的3D轴对称Euler方程，$ C^{1，α} $初始速度会产生有限的时间奇异性。受Elgindi的作品的启发，我们证明了具有$ C^{1，α} $初始速度和边界的3D轴对称Euler和2D BoussinesQ方程，并在渐近（或近似）自相似的有限时光上发展出稳定的稳定的有限时间[8]。另一方面，[35,52]的作者最近表明，3D Euler方程的爆炸解决方案在流体动力学上是不稳定的。在[35,52]中获得的不稳定性结果需要在初始数据上进行一些强有力的规律性假设，这对$ c^{1，α} $速度字段不满足。在本文中，我们概括了[8,18,35,52]的分析，以表明3D Euler和2D BoussinesQ方程的爆炸解决方案在[35,52]中引入的稳定性下，具有$ C^{1，α} $速度的boussinesq方程是不稳定的。这两个看似矛盾的结果反映了研究3D Euler爆炸解决方案稳定性的两种方法的差异。在[8,18]中获得的爆炸解决方案的稳定性分析是基于在自然界非线性的时空中动态重塑爆炸曲线的稳定性。 [35,52]中的线性稳定性分析是通过直接在原始变量中的爆炸溶液周围直接线性化的3D Euler方程来执行的。它没有考虑到爆炸时间的变化，扰动爆炸曲线的重新恢复速率的动态变化以及原始3D Euler方程的爆炸指数在有近似的自我相似的爆炸轮廓时使用扰动的初始条件。在[8,18,19]中建立爆炸配置文件的非线性稳定性时，已经使用了此类信息。
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,α}$ initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,α}$ initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in [35,52] require some strong regularity assumption on the initial data, which is not satisfied by the $C^{1,α}$ velocity field. In this paper, we generalize the analysis of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with $C^{1,α}$ velocity are unstable under the notion of stability introduced in [35,52]. These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. The stability analysis of the blowup solution obtained in [8,18] is based on the stability of a dynamically rescaled blowup profile in space and time, which is nonlinear in nature. The linear stability analysis in [35,52] is performed by directly linearizing the 3D Euler equations around a blowup solution in the original variables. It does not take into account the changes in the blowup time, the dynamic changes of the rescaling rate of the perturbed blowup profile and the blowup exponent of the original 3D Euler equations using a perturbed initial condition when there is an approximate self-similar blowup profile. Such information has been used in an essential way in establishing the nonlinear stability of the blowup profile in [8,18,19].
