Location of concentrated vortices in planar steady Euler flows

In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics. The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff–Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3}$, which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.

Keywords

concentrated vortices, steady Euler flow, Kirchhoff–Routh function

2010 Mathematics Subject Classification

35Q31, 76B03, 76B47

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G. Wang was supported by National Natural Science Foundation of China (12001135, 12071098) and China Postdoctoral Science Foundation (2019M661261, 2021T140163).

B. Zuo was supported by National Natural Science Foundation of China (12101154).

Received 9 August 2021

Accepted 8 June 2022

Published 1 February 2023