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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Two-dimensional Euler flows with concentrated vorticities


Authors: Manuel del Pino, Pierpaolo Esposito and Monica Musso
Journal: Trans. Amer. Math. Soc. 362 (2010), 6381-6395
MSC (2000): Primary 35J25, 35B25, 35B40
Published electronically: July 15, 2010
MathSciNet review: 2678979
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Abstract: 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,

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

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

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

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.


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Additional Information

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
Email: delpino@dim.uchile.cl

Pierpaolo Esposito
Affiliation: Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy
Email: esposito@mat.uniroma3.it

Monica Musso
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy – and – Departamento de Matematica, Pontificia Universidad Catolica de Chile, Avenida Vicuna Mackenna 4860, Macul, Santiago, Chile
Email: mmusso@mat.puc.cl

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04983-9
PII: S 0002-9947(2010)04983-9
Keywords: $2D$ Euler equations, singular Liouville equation, Liouville formula, concentrating solutions
Received by editor(s): August 12, 2008
Published electronically: July 15, 2010
Additional Notes: The first author was supported by grants Fondecyt 1070389 and Fondo Basal CMM
The second author was supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations”.
The third author was supported by grants Fondecyt 1080099 and Anillo ACT 125 CAPDE
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.