Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation

Cao, Daomin; Qin, Guolin; Zhan, Weicheng; Zou, Changjun

doi:10.1016/j.jde.2021.07.026

Cited by 9 publications

(6 citation statements)

References 28 publications

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“…has been treated in [5] and hence no additional difficulty arises in the proof of this case. Further more, by Taylor's formula (3.4), we have…”

Section: The Contour Dynamics Equation and Function Spacesmentioning

confidence: 96%

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Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations

Cao¹,

Qin²,

Zhan³

et al. 2021

Preprint

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By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.

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“…has been treated in [5] and hence no additional difficulty arises in the proof of this case. Further more, by Taylor's formula (3.4), we have…”

Section: The Contour Dynamics Equation and Function Spacesmentioning

confidence: 96%

“…For the case 0 < α < 1, the proof is almost the same as the case α = 0. For the case 1 ≤ α < 2, the most singular terms are treated in [5] and hence the proof of this case contains no additional difficulty.…”

Section: The Contour Dynamics Equation and Function Spacesmentioning

confidence: 99%

Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations

Cao¹,

Qin²,

Zhan³

et al. 2021

Preprint

Self Cite

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“…In the case of infinite near planar fronts, recent works [22,21] gave global-in-time solutions of small slope. Existence and construction of co-rotating and traveling pairs for the SQG equations were recently given in [6,7]. The main result in this paper provides the first construction of general fronts with low regularity, allowing unbounded curvature.…”

Section: Introductionmentioning

confidence: 93%

Well-posedness for SQG sharp fronts with unbounded curvature

Gancedo¹,

Nguyen²,

Patel³

2021

Preprint

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“…Later, Castro, Córdoba and Gómez-Serrano [6] proved the existence of V-states for the range α ∈ [1, 2) and obtained the C ∞ -regularity of their boundary for all α ∈ (0, 2), see also [7] for the real analyticity of the patch boundary. For other connected topics we refer to the papers [1,5,6,24,22,31,33,42] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

Hmidi¹,

Xue²,

Xue³

2022

Preprint

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In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α ∈ (0, 1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

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Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation

Cited by 9 publications

References 28 publications

Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations

Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations

Well-posedness for SQG sharp fronts with unbounded curvature

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

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