تقرير
Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability
| العنوان: | Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability |
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| المؤلفون: | Long, Yiming, Wang, Yuchen, Zeng, Chongchun |
| سنة النشر: | 2018 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | We consider concentrated vorticities for the Euler equation on a smooth domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N \omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j} \omega_j d\mu =\mu_j \ne 0, \] supported on well-separated vortical domains $\Omega_j$, $j=1, \ldots, N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with $\Omega_j$ being part of unknowns. For any given vorticities $\mu_1, \ldots, \mu_N$ and small $r_1, \ldots, r_N\in \mathbf{R}^+$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near non-degenerate critical configurations of the Kirchhoff-Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $\partial \Omega_j$, through an invariant subspace decomposition, it is also proved that the spectral/linear stability of such steady vortex patches is largely determined by that of the $2N$-dimensional linearized point vortex dynamics, while the motion is highly oscillatory in the $2N$-codim directions corresponding to the vortical domain shapes. Comment: To appear in Journal of Differential Equations |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1809.06425 |
| رقم الأكسشن: | edsarx.1809.06425 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |