The proposal of this paper is to study the local existence of analytic solutions, and blowup of solutions in a finite time for the geophysical boundary layer problem. In contrast with the classical Prandtl boundary layer equation, the geophysical boundary layer equation has an additional integral term arising from the Coriolis force. Under the assumption that the initial velocity and outer flow velocity are analytic in the horizontal variable, we obtain the local well-posedness of the geophysical boundary layer problem by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outer flow velocity satisfy certain condition on a transversal plane, for any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem, it is proved that its \(W^{1,\infty }-\)norm blows up in a finite time. Comparing with the blowup result obtained in Kukavica et al. (Adv Math 307:288–311, 2017) for the classical Prandtl equation, we find that the integral term in the geophysical boundary layer equation triggers the formulation of singularities earlier.

中文翻译：

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地球物理边界层问题的适定性和膨胀

本文的建议是研究解析解的局部存在，并在有限时间内解决地球物理边界层问题。与经典的Prandtl边界层方程相反，地球物理边界层方程具有一个由科里奥利力引起的附加积分项。假设在水平变量中分析了初始速度和外流速度，我们在加权的Chemin-Lerner空间中使用能量方法获得了地球物理边界层问题的局部适定性。此外，当初始速度和外流速度在横向平面上满足一定条件时，对于任何光滑解在地球物理边界层问题的法向变量中呈指数衰减的情况，都证明了\（W ^ {1，\ infty}-\）规范会在有限的时间内爆发。与Kukavica等人获得的爆破结果相比。（Adv Math 307：288–311，2017）对于经典的Prandtl方程，我们发现地球物理边界层方程中的积分项更早地触发了奇点的提法。