Abstract
This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane \({\mathbb{R}^2}\). We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at-most-logarithmic growth of the argument of the flow in large balls. These estimates lead to the conclusion that the streamlines of the flow are all parallel lines.
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Communicated by V. Šverák
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This work has been carried out in the framework of Archimède Labex (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR). The research leading to these results has also received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement No. 321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modelling and from the ANR NONLOCAL project (ANR-14-CE25-0013).
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Hamel, F., Nadirashvili, N. A Liouville Theorem for the Euler Equations in the Plane. Arch Rational Mech Anal 233, 599–642 (2019). https://doi.org/10.1007/s00205-019-01364-x
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DOI: https://doi.org/10.1007/s00205-019-01364-x