Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Gómez-Serrano, Javier; Park, Jaemin; Shi, Jia; Yao, Yao

doi:10.1007/s00220-021-04146-3

Cited by 18 publications

(24 citation statements)

References 42 publications

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“…We refer to [13,14,40,43] for single rotating patches, [26,27] for doubly connected V-states, [42,39] for corotating and counter-rotating vortex pairs and [33] for steady states. See [34] for further properties of rotating solutions and [35,36] for the case of the vortex-sheet problem. See [29,41,58] for related constructions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Traveling waves near Couette flow for the 2D Euler equation

Castro¹,

Lear²

2021

Preprint

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In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in H s , with s < 3/2, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see [7]), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the H 3 2 + neighborhoods of Couette flow (see [51]). Contents 1. Introduction and main result 1.1. Sketch of the proof 1.2. Organization 2. Formulation of the problem 2.1. The 2D Euler as an equation for the level curves of the vorticity 2.2. The equation for the traveling wave 2.3. The profile function ε,κ . 3. Bifurcation theory and Crandall-Rabinowitz 4. Functional setting and regularity 4.1. The functional setting 4.2. Hypothesis 1 and 2 5. Analysis of the linear part 5.1. Decomposition of the linear operator 5.2. Rescaling of the decomposition 5.3. One dimensionality of the kernel of the linear operator 5.4. Codimension of the image of the linear operator 5.5. The transversality property 6. Main theorem 6.1. Distance of the traveling wave to the Couette flow 6.2. Full regularity of the solution 7. Appendix References

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Traveling waves near Couette flow for the 2D Euler equation

Castro¹,

Lear²

2021

Preprint

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“…In the case of 2D Navier-Stokes, Koch-Nadirashvili-Seregin-Šverák also proved a Liouville theorem in [66]. See also [49,50] where together with Yao we proved rigidity and flexibility results for the vortex sheet problem.…”

Section: D Euler Rigidity and Construction Of Stationary Solutionsmentioning

confidence: 71%

Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy

Gómez-Serrano¹,

Park²,

Shi³

2021

Preprint

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In this paper, we prove the existence of locally non-radial solutions to the stationary 2D Euler equations with compact support but non-concentrated around one or several points. Our solutions are of patch type, have analytic boundary, finite energy and sign-changing vorticity and are new to the best of our knowledge. The proof relies on a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the Nash-Moser iteration procedure.

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“…This remarkable result indicates that solutions bifurcated from concentric circles can be constructed in the same way, and constitute the second class of vortex sheet solutions. In their follow-up work [14], they have further showed that one can not expect vortex sheets other than concentric circles when ω ≥ 0 and Ω ≤ 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…and angular velocity Ω. (Ω > 0 for counterclockwise) Recently, Gómez-Serrano et al [13] gave another vortex sheet supported on a closed curve. By Lyapunov-Schmidlt reduction and degenerate bifurcation, they obtained two curves of solution bifurcated from one trivial solution, which is a unit circle at specific angular velocity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

Cao¹,

Qin²,

Zou³

2021

Preprint

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We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.

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Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Cited by 18 publications

References 42 publications

Traveling waves near Couette flow for the 2D Euler equation

Traveling waves near Couette flow for the 2D Euler equation

Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

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