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Two-dimensional Euler flows with concentrated vorticities. (English) Zbl 1205.35217

Summary: For a planar model of Euler flows proposed by A. Tur and V. Yanovsky [Phys. Fluids 16, No. 8, Paper No. 2877 (2004; Zbl 1186.76541)], we construct a family of velocity fields \({\mathbf w}_\varepsilon\) for a fluid in a bounded region \(\Omega\), with concentrated vortices \(\omega_\varepsilon\) 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
\[ \Delta \psi_\varepsilon+ \varepsilon^2 e^{\psi_\varepsilon}= 4\pi \alpha \delta_{p_{a,\varepsilon}} \quad\text{in }\Omega, \qquad \psi_\varepsilon=0 \quad\text{on }\partial\Omega, \]
for a suitable point \(p=p_{a,\varepsilon}\in \Omega\). The vortices \(\omega_\varepsilon:= -\Delta \psi_\varepsilon\) concentrate in the sense that
\[ \omega_\varepsilon+4 \pi\alpha\delta_{p_{a,\varepsilon}}- 8\pi \sum_{j=1}^{\alpha+1} \delta_{p_{a,\varepsilon}+a_j} \rightharpoonup 0\quad\text{as }\varepsilon \to 0, \]
where the satellites \( a_1,\dots, a_{\alpha+1}\) 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 \(\leq \alpha+1\) of the regular part of Green’s function of the domain.

MSC:

35Q31 Euler equations
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J08 Green’s functions for elliptic equations
76M40 Complex variables methods applied to problems in fluid mechanics

Citations:

Zbl 1186.76541
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References:

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