论文标题
3D不可压缩的Euler方程的稳定涡旋环的独特性和稳定性
Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation
论文作者
论文摘要
在本文中，我们关注3D Euler方程的涡旋环的独特性和非线性稳定性。通过利用Arnold的变异原理来实现P. L. Lions引入的Euler方程的稳态和浓缩的紧凑方法，我们首先建立了重排类别中涡旋环的一般稳定性标准，这使我们可以减少对某些涡旋环的稳定性分析，以解决其独特性问题。随后，我们证明了具有较小的横截面和多项式分布函数的特殊涡旋环的独特性。这些涡旋环与3D Euler方程的全球古典解决方案相对应，并被许多庆祝作品证明存在。该证明是通过研究涡旋环的仔细渐近行为来实现的，因为它们倾向于圆形细丝并应用局部pohozaev身份。因此，我们为3D Euler方程提供了第一个非线性稳定的经典涡旋环解决方案。
In this paper, we are concerned with the uniqueness and nonlinear stability of vortex rings for the 3D Euler equation. By utilizing Arnold 's variational principle for steady states of Euler equations and concentrated compactness method introduced by P. L. Lions, we first establish a general stability criteria for vortex rings in rearrangement classes, which allows us to reduce the stability analysis of certain vortex rings to the problem of their uniqueness. Subsequently, we prove the uniqueness of a special family of vortex rings with a small cross-section and polynomial type distribution function. These vortex rings correspond to global classical solutions to the 3D Euler equation and have been shown to exist by many celebrate works. The proof is achieved by studying carefully asymptotic behaviors of vortex rings as they tend to a circular filament and applying local Pohozaev identities. Consequently, we provide the first family of nonlinear stable classical vortex ring solutions to the 3D Euler equation.
