论文标题
关于快速旋转和垂直粘度对$ 3D $原始方程的寿命的影响
On the effect of fast rotation and vertical viscosity on the lifespan of the $3D$ primitive equations
论文作者
论文摘要
我们研究了具有不可渗透且无应力的边界条件的三维原始方程（也称为静液压导航方程）对解决方案寿命的影响。首先，在短时间间隔内，与旋转率$ |ω| $无关，我们建立了解决方案的局部良好性，其初始数据在水平变量中是分析性的，而垂直变量中只有$ l^2 $。此外，只要溶液的存在，溶液在所有变量中立即在所有变量中立即成为分析性分析性的分析性。另一方面，水平变量中的分析性半径可能会随着时间而减小，但是只要它保持阳性，溶液就存在。其次，随着快速旋转（即大$ |ω| $），我们证明了解决方案的存在时间可以延长，并使用“准备好”的初始数据。最后，在两个具有$ω= 0 $的空间维度的情况下，我们确定了全局良好的度量，但前提是初始数据足够小。初始数据的较小条件取决于水平变量中的垂直粘度和分析性的初始半径。
We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation $|Ω|$, we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only $L^2$ in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large $|Ω|$, we show that the existence time of the solution can be prolonged, with "well-prepared" initial data. Finally, in the case of two spatial dimensions with $Ω=0$, we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.
