Two-dimensional Euler flows with concentrated vorticities

Abstract:

For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields $\textbf {w}_\varepsilon$ 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, \[ \Delta \psi _\varepsilon + \varepsilon ^2 e^{\psi _\varepsilon }=4\pi \alpha \delta _{p_{a,\varepsilon }}\quad \mbox {in }\Omega , \quad \psi _\varepsilon = 0 \quad \mbox {on } \partial \Omega , \] for a suitable point $p=p_{a,\varepsilon } \in \Omega$. The vorticities $\omega _\varepsilon := -\Delta \psi _\varepsilon$ concentrate in the sense that \[ \omega _\varepsilon +4 \pi \alpha \delta _{p_{a,\varepsilon }}- 8\pi \sum _{j=1}^{\alpha +1}\delta _{p_{a,\varepsilon }+a_j} \rightharpoonup 0\quad \hbox {as }\varepsilon \to 0 ,\] 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.