论文标题
在非排水部分的扰动下，单位球上的两射线kolmogorov类型的非线性稳定性
Nonlinear stability of the two-jet Kolmogorov type flow on the unit sphere under a perturbation with nondissipative part
论文作者
论文摘要
我们考虑了二维单位球体上Navier-Stokes方程的涡度形式，并研究了两极kolmogorov型流的非线性稳定性，这是由第二级的Zonal球形谐波功能给出的固定溶液。特别是，我们假设一个扰动包含一个非隔离部分，该部分是由一级的球形谐波的线性组合给出的，并研究了非隔离部分对通过对流术语扰动的长时间行为的影响。我们表明，对于所有初始数据，可以及时保留针对非线性稳定性问题的弱解决方案的非隔离部分。此外，我们证明，弱解的耗散部分会及时地呈指数收敛，朝着平衡，这是根据初始数据的非隔离部分明确表示的，并且通常不会消失。特别是，事实证明，弱解的渐近行为最终取决于线性的普通微分方程系统。为了证明这些结果，我们利用在歧管上杀死向量场的属性。我们还考虑了旋转球体的情况。
We consider the vorticity form of the Navier-Stokes equations on the two-dimensional unit sphere and study the nonlinear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. In particular, we assume that a perturbation contains a nondissipative part given by a linear combination of the spherical harmonics of degree one and investigate the effect of the nondissipative part on the long-time behavior of the perturbation through the convection term. We show that the nondissipative part of a weak solution to the nonlinear stability problem is preserved in time for all initial data. Moreover, we prove that the dissipative part of the weak solution converges exponentially in time towards an equilibrium which is expressed explicitly in terms of the nondissipative part of the initial data and does not vanish in general. In particular, it turns out that the asymptotic behavior of the weak solution is finally determined by a system of linear ordinary differential equations. To prove these results, we make use of properties of Killing vector fields on a manifold. We also consider the case of a rotating sphere.
