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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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