Concentrated steady vorticities of the Euler equation on 2d domains and their linear stability 
 
Authors:  Yiming Long Yuchen Wang Chongchun Zeng 
 
Institution:  1. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China;2. Chern Institute of Mathematics, Nankai University, Tianjin 300071, China;3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States 
 
Abstract:  We consider concentrated vorticities for the Euler equation on a smooth domain $\mathrm{\Omega}?{\mathbf{R}}^{2}$ in the form of $\omega =\sum _{j=1}^{N}{\omega}_{j}{\chi}_{{\mathrm{\Omega}}_{j}},\phantom{\rule{1em}{0ex}}{\mathrm{\Omega}}_{j}=\pi {r}_{j}^{2},\phantom{\rule{1em}{0ex}}\underset{{\mathrm{\Omega}}_{j}}{\int}{\omega}_{j}d\mu ={\mu}_{j}\ne 0,$ supported on wellseparated vortical domains ${\mathrm{\Omega}}_{j}$, $j=1,\dots ,N$, of small diameters $O({r}_{j})$. A conformal mapping framework is set up to study this free boundary problem with ${\mathrm{\Omega}}_{j}$ being part of unknowns. For any given vorticities ${\mu}_{1},\dots ,{\mu}_{N}$ and small ${r}_{1},\dots ,{r}_{N}\in {\mathbf{R}}^{+}$, through a perturbation approach, we obtain such piecewise constant steady vortex patches as well as piecewise smooth Lipschitz steady vorticities, both concentrated near nondegenerate critical configurations of the Kirchhoff–Routh Hamiltonian function. When vortex patch evolution is considered as the boundary dynamics of $?{\mathrm{\Omega}}_{j}$, 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 2Ndimensional linearized point vortex dynamics, while the motion is highly oscillatory in the 2Ncodim directions corresponding to the vortical domain shapes. 
 
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