The geometry of a vorticity model equation
The geometry of a vorticity model equation
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DOI：
10.3934/cpaa.2012.11.1407
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发表时间：
2010-10
影响因子：
1
通讯作者：
J. Escher;B. Kolev;Marcus Wunsch
中科院分区：
文献类型：
--
作者：
J. Escher;B. Kolev;Marcus Wunsch
我们证明了修正后的Constantin-Lax-Majda方程模拟涡旋和准地转动力学[27]可以被改写为子群$\mathrm{Diff}_{1}^{\infty}上的测地线流（\mathbb{S}）$的保向同构$\varphi \in \mathrm{Diff}^{\infty}（\mathbb{S}）$使得$\varphi（1）= 1$配备有由齐次Sobolev范数$\dot H^{1/2}$诱导的右不变度量。在Sobolev类$H^{k}$具有$k\ge 2$的复同态的扩展群上，这诱导出一个弱黎曼结构。我们建立了测地线喷雾是光滑的，我们得到了局部存在唯一的测地线。
We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.