论文标题
关于旋转单元球上2D Euler方程的区域稳定解决方案
On zonal steady solutions to the 2D Euler equations on the rotating unit sphere
论文作者
论文摘要
本文研究了旋转单元球上不可压缩的欧拉方程的一组固定溶液的结构，这些溶液靠近两个基本的区域流量：沿极轴沿极轴的2层Rossby-Haurwitz溶液和Zonal刚性旋转$ Y_1^0 $。 对于任何给定的球体旋转，我们构建了一个非区域稳态解决方案的新家族，以任意接近分析规律性的Zonal Rossby-Haurwitz流函数。这表明，对于此Rossby-Haurwitz溶液附近的解决方案，都无法期望任何非线性的无粘性阻尼到区域。 另一方面，我们证明，在球体旋转方面的适当条件下，任何足够接近刚性旋转区域流量的固定溶液$ y_1^0 $本身必须是Zonal的，见证了从方程式继承的某种刚性，球体的几何形状和基本流量。然而，当有关球体旋转的条件失败时，解决方案集更为丰富，我们能够证明存在明显的静止和行驶波的非区域非区域溶液，从$ y_1^0 $分叉，与从Zonal Rossby-Haurwitz Zonal Rossby-Haurwitz solution of Leg y_1^0 $相同的精神。
The present paper studies the structure of the set of stationary solutions to the incompressible Euler equations on the rotating unit sphere that are near two basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and the zonal rigid rotation $Y_1^0$ along the polar axis. We construct a new family of non-zonal steady solutions arbitrarily close in analytic regularity to the second degree zonal Rossby-Haurwitz stream function, for any given rotation of the sphere. This shows that any non-linear inviscid damping to a zonal flow cannot be expected for solutions near this Rossby-Haurwitz solution. On the other hand, we prove that, under suitable conditions on the rotation of the sphere, any stationary solution close enough to the rigid rotation zonal flow $Y_1^0$ must itself be zonal, witnessing some sort of rigidity inherited from the equation, the geometry of the sphere and the base flow. Nevertheless, when the conditions on the rotation of the sphere fail, the set of solutions is much richer and we are able to prove the existence of both explicit stationary and travelling wave non-zonal solutions bifurcating from $Y_1^0$, in the same spirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.
