Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

Ao, Weiwei; Dávila, Juan; Pino, Manuel del; Musso, Monica; Wei, Juncheng

doi:10.48550/arxiv.2008.12911

Cited by 5 publications

(6 citation statements)

References 31 publications

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“…In spite of this, in [15,16] smooth rotating solutions were constructed for SGQ equation and for 2D Euler equation, respectively. See also [2] for the construction of a different type of smooth rotating solutions for SQG and [37] for the existence of traveling waves also for SQG. The existence of non-smooth rotating vortices with non-uniform densities can be found in [32] for 2D Euler.…”

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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“…Remark 1.11. The proof can be easily adapted to the rotating vortex polygon (with γ 0 = γ 2 = 0 in (1.9)) and which has been studied in [1,24,48,22]. This remains equally true for the body-centered polygonal configurations (γ 2 = 0), treated in [49], as well as for the nested polygons without a central patch (γ 0 = 0).…”

mentioning

confidence: 82%

“…Implementing the same approach, Keady [37] proved the existence of translating pairs of symmetric patches and Wan [49] studied the existence and stability of desingularizations of a general system of rotating point vortices. Very recently, Godard-Cadillac, Gravejat and Smets [24] extended Turkington's result for the gSQG equations, while Ao, Dávila, Del Pino, Musso and Wei [1] have obtained related families of smooth solutions via gluing techniques. See [39,40,41,46,50] for additional references on multiply connected patches.…”

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confidence: 87%

Multipole vortex patch equilibria for active scalar equations

Hassainia¹,

Wheeler²

2021

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We study how a general configuration of finitely-many point vortices, in a state of uniform rotation or translation with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. Using a technique first introduced by Hmidi and Mateu for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular, at which point classical techniques such as Lyapunov-Schmidt reduction can be used. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we prove the existence of various families of solutions with patches arranged in asymmetric pairs, regular polygons, body-centered polygons, and nested regular polygons. These configurations are degenerate due to their additional symmetries, but this difficulty can be overcome by integrating the appropriate symmetries into the formulation of the problem.

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“…Corotating patch solutions with two patches [14] and N patches forming an N -fold symmetrical pattern [9] were recently exhibited with bifurcation argument. The C 1 analogous of these solutions has also been investigated independently recently in [1]. Another recent independent result [12] also build corotating solutions with N patches using variational argument.…”

Section: The Inviscid Surface Quasi-geostrophic Equationmentioning

confidence: 94%

“…Lemma 3.2 (Reformulation of Theorem 2.7). Denote by Y i (t) := y ij (t) j =i the solution to (31) 1) . Assume that for all i ∈ {1 • • • N } and for all T > 0 and ρ > 0 the set…”

Section: The Modified Systemmentioning

confidence: 99%

Vortex collapses for the Euler and Quasi-Geostrophic Models

Godard-Cadillac

2021

Preprint

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This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains three main results with several links between each other. In the first part, we provide two uniform bounds on the trajectories for Euler and quasi-geostrophic vortices related to the non-neutral cluster hypothesis. In a second part we focus on the Euler point-vortex model and under the non-neutral cluster hypothesis we prove a convergence result. The third part is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

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Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

Cited by 5 publications

References 31 publications

Traveling waves near Couette flow for the 2D Euler equation

Traveling waves near Couette flow for the 2D Euler equation

Multipole vortex patch equilibria for active scalar equations

Vortex collapses for the Euler and Quasi-Geostrophic Models

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