In this paper, we deal with the Kakutani–Matsuuchi model which describes the surface elevation $\eta$ of the water-waves under the effect of viscosity. We first derive the decay rate of weak solutions. This can be used to obtain the decay rate of ${\Vert \eta \left(t\right)\Vert }_{{\dot{H}}^1}$ when initial data is sufficiently small in ${\dot{H}}^1$. We next show the existence, uniqueness, Gevrey regularity and decay rates of $\eta$ with sufficiently small initial data in $B_{2,1}^1$. To do so, we derive a commutator estimate involving Gevrey operator. We then apply our method to the supercritical quasi-geostrophic equations. We finally show the formation of singularities of smooth solutions in finite time for a certain class of initial data.

在本文中，我们处理描述表面高程的 Kakutani-Matsuuchi 模型 $\eta$粘度作用下的水波。我们首先推导出弱解的衰减率。这可用于获得衰减率${\Vert \eta \left(吨\right)\Vert }_{{\dot{H}}^1}$ 当初始数据足够小时 ${\dot{H}}^1$. 我们接下来展示了存在性、唯一性、Gevrey 规律性和衰减率$\eta$ 具有足够小的初始数据 ${乙}_{2,1}^1$. 为此，我们导出了一个涉及 Gevrey 算子的换向器估计。然后我们将我们的方法应用于超临界准地转方程。我们最终展示了对于某类初始数据在有限时间内平滑解奇点的形成。