Nonexistence of Coherent Structures in Two-dimensional Inviscid Channel Flow

Kalisch, Henrik

doi:10.1051/mmnp/20127207

Cited by 3 publications

(1 citation statement)

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“…In a previous paper [12] (see also [13]), we considered the case of a two-dimensional strip with bounded section and the case of bounded flows in a half-plane, assuming in both cases that the flows v are tangential on the boundary and that inf |v| > 0: all streamlines are then proved to be lines which are parallel to the boundary of the domain (in other words the flow is a parallel flow). Earlier results by Kalisch [15] were concerned with flows in two-dimensional strips under the additional assumption v • e = 0, where e is the main direction of the strip. Lastly, in [14], we considered the case of the whole plane R 2 and we showed that any C 2 (R 2 ) bounded flow v is still a parallel flow under the condition inf R 2 |v| > 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Circular flows for the Euler equations in two-dimensional annular domains

Hamel¹,

Nadirashvili²

2019

Preprint

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In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs are based on the study of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function, which is shown to satisfy a semilinear elliptic equation in the whole domain. In exterior or punctured domains, the method of moving planes is applied to some almost circular domains located between some streamlines of the flow, and the radial symmetry of the stream function is shown by a limiting argument. The paper also contains two Serrin-type results in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary and the domains are then proved to be disks or annuli.

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