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<p>In this paper, we classify steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the entire circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows that appear in the whole space case, an additional class of steady flows exists for which the set of flow angles is either the upper or lower closed semicircle. This type of flows is proved to be the class of non-shear flows that have the least total curvature. An immediate consequence of our classification result is the structural stability of any shear flow with a convex shear profile. Our proof relies on the observation and analysis of some quantities related to the curvature of the streamlines.</p>

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On a Classification of Steady Solutions to Two-Dimensional Euler Equations

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • Changfeng Gui,
  • Chunjing Xie,
  • Huan Xu

摘要

In this paper, we classify steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the entire circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows that appear in the whole space case, an additional class of steady flows exists for which the set of flow angles is either the upper or lower closed semicircle. This type of flows is proved to be the class of non-shear flows that have the least total curvature. An immediate consequence of our classification result is the structural stability of any shear flow with a convex shear profile. Our proof relies on the observation and analysis of some quantities related to the curvature of the streamlines.