Global Smooth Solutions for the Inviscid SQG
                    Equation

Castro, Ángel; Córdoba, Diego; Gómez-Serrano, Javier

doi:10.1090/memo/1292

Cited by 48 publications

(60 citation statements)

References 48 publications

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“…The main interest of this hypothesis relies on the preservation of the center of vorticity property (7). Under the non-neutral clusters hypothesis, the center of vorticity is well defined, not only for the whole system but also for any subset of vortices.…”

Section: 11mentioning

confidence: 99%

“…Exhibiting global smooth solutions or patch solutions is a challenging issue as there is no equivalent of the Yudovitch theorem [30]. A first example was recently provided in [7] by developing a bifurcation argument from a specific radially symmetric function. The variational construction of an alternative example in the form of a smooth traveling-wave solution was completed in [15,13].…”

Section: Introductionmentioning

confidence: 99%

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Vortex collapses for the Euler and Quasi-Geostrophic models

Godard-Cadillac

2022

DCDS

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<p style='text-indent:20px;'>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 two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result 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.</p>

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Section: 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vortex collapses for the Euler and Quasi-Geostrophic models

Godard-Cadillac

2022

DCDS

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“…There is another computer-assisted approach to establish the stability by tracking the spectrum of a given operator and quantifying the spectral gap; see, e.g., [2].…”

Section: Energy Estimates With Computer Assistancementioning

confidence: 99%

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

Chen

Hou

Huang

2021

Comm Pure Appl Math

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We present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We also prove self‐similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on ℝ or S1 for Hölder‐continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self‐similar singularity into the problem of establishing the nonlinear stability of an approximate self‐similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self‐similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well‐posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations. © 2021 Wiley Periodicals LLC.

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“…These equations has been intensely investigated since the work of Constantin, Majda and Tabak [4] on the case s = 1 2 where they pointed out the mathematical links that arises between (SQG- 1 2 ) and the Euler equation in dimension 3. Besides stationary solution, given by a radially symmetric rearrangement on the active scalar, the only two known examples of global smooth solutions where built by Castro, Córdoba and Gómez-Serrano [3] on the one hand and by Gravejat and Smets [7] on the other hand with two different techniques. The article of Castro, Córdoba and Gómez-Serrano also provides a wide bibliography related on SQG and its Cauchy problem.…”

Section: The Quasi-geostrophic Equationsmentioning

confidence: 99%

Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations

Godard-Cadillac¹

2021

Comptes Rendus. Mathématique

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In a recent article by Gravejat and Smets [7], it is built smooth solutions to the inviscid surface quasigeostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide similar solutions to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian. Funding. The author acknowledges grants from the Agence nationale de la Recherche, for project "Ondes Dispersives Aléatoires" (ANR-18-CE40-0020-01).

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Global Smooth Solutions for the Inviscid SQG Equation

Cited by 48 publications

References 48 publications

Vortex collapses for the Euler and Quasi-Geostrophic models

Vortex collapses for the Euler and Quasi-Geostrophic models

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations

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