论文标题
圆盘中广义表面准斑方方程的时间周期溶液的出现
Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc
论文作者
论文摘要
在本文中，我们解决了单位光盘中通用的Inviscid sqg方程的存在时间周期解决方案，当$α\ in（0,1）$中时，具有均匀的dirichlet边界条件的存在。我们显示了来自径向斑块的可数曲线曲线的存在。与主动标量方程中的前面研究相反，绿色函数不再是显式的，我们通过将合适的分裂成奇异的显式部分（与平面一致）和由域边界引起的平滑隐式来阐明此问题。另一个问题是通过涉及贝塞尔函数及其零的离散总和来接受复杂形式的线性频率的分析。我们通过使用SNEDDON的公式来克服这一困难，从而导致频率的合适积分表示。
In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when $α\in (0,1)$. We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.
