关于Beta平面方程的全局稳定性
On the Global Stability of a Beta-Plane Equation
摘要：
本文研究不可压缩、无粘二维流体在旋转坐标系中的运动。在那里，流体受到科里奥利力，我们假设它线性地依赖于其中一个坐标。这是地球物理流体动力学中常见的近似，称为β平面。在涡量公式中，我们考虑的模型由欧拉方程给出，并添加了线性各向异性，非退化，色散项。这使我们能够把问题作为一个准线性色散方程，其线性解表现出衰减的时间在一个临界速率。我们的主要结果是充分小的局部解的全局稳定性和衰减到平衡点。证明的关键方面是利用“双零型“，消除空间相干波之间的相互作用和傅里叶积分算子，使我们能够控制一个强有力的加权范数引理。
英文摘要：
We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a "double null form" that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.
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