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Planar vortex patch problem in incompressible steady flow. (English) Zbl 06380404
Summary: In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain \(\varOmega\) of \(\mathbb{R}^2\). Let \( k\) be a positive integer and let \(\kappa_j\) be a positive constant, \(j = 1, \dots, k\). For any given non-degenerate critical point \(\mathbf{x}_0 = (x_{0, 1}, \dots, x_{0, k})\) of the Kirchhoff-Routh function defined on \(\varOmega^k\) corresponding to \((\kappa_1, \dots, \kappa_k)\), we prove the existence of a planar flow, such that the vorticity \(\omega\) of this flow equals a large given positive constant \(\lambda\) in each small neighborhood of \(x_{0, j}\), \(j = 1, \dots, k\), and \(\omega = 0\) elsewhere. Moreover, as \(\lambda \to + \infty\), the vorticity set \(\{y : \omega(y) = \lambda \}\) shrinks to \(\bigcup_{j = 1}^k \{x_{0, j} \}\), and the local vorticity strength near each \(x_{0, j}\) approaches \(\kappa_j\), \(j = 1, \dots, k\).

MSC:
35Q31 Euler equations
37 Dynamical systems and ergodic theory
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