On steady states with concentrated vorticity of the 2-dim Euler equation on bounded domains

On a smooth bounded domain $\Omega \subset \mathbf{R}^2$, we consider steady solutions of the incompressible Euler equation with concentrated vorticity. More precisely, with prescribed integer $n>0$, vortical domain sizes $r_1, \ldots, r_n >0$, and vorticity strengths $\mu_1, \ldots, \mu_n\ne 0$, we seek steady vorticity distributions in the form of $\omega= \sum_{j=1}^n \omega_j (x)$ where 1.) the vortical domains satisfy $\Omega_j \triangleq supp(\omega_j) \subset B(p_j, 2r_j \epsilon)$, $|\Omega_j| = \pi r_j^2\epsilon^2$, with $0< |\epsilon| <<1$ and distinct $p_1, \ldots, p_n \in \Omega$; and 2.) $\mu_j = \int \omega_j dx$. Since the dynamics of localized vorticity is approximated by the point vortex dynamics, we take $\{p_1, \ldots, p_n\}$ close to a non-degenerate steady configuration of the point vortex system in $\Omega$ with parameters $\mu_1, \ldots, \mu_n$. Through a perturbation method applied to $\Omega_j$ parametrized by conformal mappings, we obtained two types of steady solutions with smooth $\partial \Omea_j$ being $O(\epsilon^2)$ perturbations to $\partial B(p_j, r_j\epsilon)$: a.) infinitely many piecewise smooth solutions $\omega \in C^{0,1} (\Omega)$; and b.) a unique steady vortex patch with piecewise constant vorticity, i.e. $\omega_j=\frac{\mu_j}{\pi r_j^2 \epsilon^2} \chi (\Omega_j)$. Moreover, the spectral and evolutionary properties (stability, exponential trichotomy, etc.) of the linearized vortex patch dynamics at the latter is determined by those of the linearized point vortex dynamics at the steady configuration $\{p_1, \ldots, p_n\}$. This is a joint work with Yiming Long and Yuchen Wang at Nankai University.