Two-dimensional Euler flows with concentrated vorticities

For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields ${\bf w}_\varepsilon$ for a fluid in a bounded region $\Omega$, with concentrated vorticities $\omega_\varepsilon$ for $\varepsilon>0$ small. More precisely, given a positive integer $\alpha$ and a sufficiently small complex number $a$, we find a family of stream functions $\psi_\varepsilon$ which solve the Liouville equation with Dirac mass source,

$$\Delta \psi_\varepsilon + \varepsilon^2 e^{\psi_\varepsilon}=4\pi \alpha \delta_{p_{a,\varepsilon}}$$    in $$\Omega, \quad \psi_\varepsilon = 0$$    on $$\partial\Omega,$$

for a suitable point $p=p_{a,\varepsilon} \in \Omega$. The vorticities $\omega_\varepsilon:= -\Delta \psi_\varepsilon$ concentrate in the sense that

$$\omega_\varepsilon+4 \pi \alpha \delta_{p_{a,\varepsilon}}- 8\pi ... ...lta_{p_{a,\varepsilon}+a_j} \rightharpoonup 0\quad\hbox{as }\varepsilon \to 0 ,$$

where the satellites $a_1,\ldots, a_{\alpha+1}$ denote the complex ($\alpha+1$)-roots of $a$. The point $p_{a,\varepsilon}$ lies close to a zero point of a vector field explicitly built upon derivatives of order $\le \alpha+1$ of the regular part of Green's function of the domain.