圆盘上度量形变下稳态涡层的延拓与约化

中文摘要 AI 辅助

我们研究在单位圆盘上共形度量 $g_s= e^{2\sigma_s}g_{\mathrm{e}}$ 的给定形变下，稳态涡层的局部延拓。通过法向图表示移动的涡层，我们将流线条件和伯努利条件表述为一个非线性残差，其两个分量具有不同的Sobolev阶。如果参考涡层的两个单侧切向速度不同时为零，则未归一化残差的导数是指标为二的Fredholm算子。固定平均法向位移和总环量得到一个指标为零的问题；当归一化微分在未归一化核上的限制是到 $\mathbb{R}^2$ 的满射时，我们得到一个依赖于参数的Lyapunov-Schmidt约化。在非退化情形下，这产生一个局部唯一的归一化分支；在退化情形下，给出显式的一阶和二阶必要条件。对于旋转对称的参考度量和同心圆涡层，线性化问题分解为Fourier块，且只有有限多个模态可能共振。在常曲率圆盘族中，每个模态 $k\geq2$ 都是非共振的，而第一个模态产生一个二维约化问题。两个第一模态核轮廓是环境Killing场的法向迹，但它们不产生固定边界对称性，也不单独产生非同心分支。

英文摘要

We study the local continuation of a steady vortex sheet under a prescribed deformation $g_s= e^{2σ_s}g_{\mathrm{e}}$ of a conformal metric on the unit disk. Representing the moving sheet by a normal graph, we formulate the streamline and Bernoulli conditions as a nonlinear residual whose two components have different Sobolev orders. If the two one-sided tangential velocities of the reference sheet do not vanish simultaneously, the derivative of the unnormalized residual is Fredholm of index two. Fixing the mean normal displacement and the total circulation gives an index-zero problem; when the normalization differential restricted to the unnormalized kernel is onto $\mathbb{R}^2$, we obtain a parameter-dependent Lyapunov-Schmidt reduction. This yields a locally unique normalized branch in the nondegenerate case and explicit first- and second-order necessary conditions in the degenerate case. For a rotationally symmetric reference metric and a concentric circular sheet, the linearized problem separates into Fourier blocks and only finitely many modes can be resonant. In the constant-curvature disk family, every mode $k\geq2$ is nonresonant, whereas the first mode produces a two-dimensional reduced problem. The two first-mode kernel profiles are normal traces of ambient Killing fields, but they do not generate fixed-boundary symmetries or, by themselves, a nonconcentric branch.

发表机构

• Faculty of Science, Academic Assembly, University of Toyama（富山大学理学部学术会）

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