论文标题
旋转的无粘性原始方程的索博莱夫空间中的有限时间爆炸和不稳定性
Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation
论文作者
论文摘要
海洋和大气的大规模动力学受原始方程（PES）的控制。众所周知，三维粘性PES在Sobolev空间中全球范围很好。另一方面，已知无旋转的无粘性PE在Sobolev的空间中被遗化，其平滑溶液可以在有限的时间内形成奇异性。在本文中，我们在旋转的情况下扩展了上述结果。首先，我们将有限的时间爆炸解决方案构建为旋转的无粘性PE，并确定旋转的无粘性PE在Sobolev空间中被置于不足的意义上，其围绕某个稳态背景流的扰动是线性性和非线性在Sobolev空间中均呈线性和非线性的。它的线性不稳定性是开尔文 - 赫尔莫尔兹（Kelvin-Helmholtz）类型类似于涡流板问题的上下文中出现的类型。这意味着Inviscid PES在Gevrey类的订单类别$ s> 1 $中也是线性不良的，并表明适合良好的空间是Gevrey类别的订单$ S = 1 $，这正是分析功能的空间。
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. First, we construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order $s > 1$, and suggests that a suitable space for the well-posedness is Gevrey class of order $s = 1$, which is exactly the space of analytic functions.
