arXiv subjects

Edison Cuba

Publications and source records attributed to Edison Cuba.

6 recordsLinked to original sources

arXiv2025

Existence of nested polygonal vortex patches for the generalized SQG equation

This paper investigates time-periodic solutions of both the surface quasi-geostrophic (SQG) equation and its generalized form (gSQG) within the more singular regime, focusing on the evolution of patch-type structures. Assuming the underlying point vortex equilibrium satisfies a natural nondegeneracy condition, we employ an implicit function argument to construct families of co-rotating nested polygonal vortex patch solutions. These configurations provide precise asymptotic descriptions of the geometry of the evolving patch boundaries. Our results contribute to the broader understanding of coherent rotating structures arising in active scalar equations with singular velocity coupling.

math.AP↗

arXiv2024

Existence of Vortex Patch Equilibria for Active Scalars Equations

In this paper, we investigate the existence of a finite number of vortex patches for the generalized surface quasi-geostrophic (gSQG) equations with $\alpha \in [1,2)$, focusing on configurations that may rotate uniformly, translate, or remain stationary. Using a desingularization technique, we reformulate the problem to resolve singularities arising in the point vortex limit. Assuming a nondegenerate equilibrium of the point vortices, we apply the implicit function theorem to construct time-periodic solutions to the gSQG equations, offering asymptotic descriptions of the vortex patch boundaries.

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arXiv2024

Existence of stationary vortex patches for the gSQG in bounded domains

In this paper we show the existence of time-periodic vortex patches for the generalized surface quasi-geostrophic equation within a bounded domain. This construction is carried out for values of $\gamma$ in the range of $(1,2)$. The resulting vortex patches possess a fixed vorticity and total flux, and they are located in the neighborhood of critical points that are non-degenerate for the Kirchhoff--Routh equation. The proof is accomplished through a combination of analyzing the linearization of the contour dynamics equation and employing the implicit function theorem as well as carefully selected function spaces.

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arXiv2022

Existence of asymmetric vortex patch for the generalized SQG equations

This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $\alpha\in[1,2)$ in the whole plane, where $\alpha=1$ corresponds to the surface quasi-geostrophic equations (SQG). More precisely, we construct non-trivial simply connected co-rotating and traveling patches with unequal vorticity magnitudes. The proof is carried out by means of a combination of a desingularization argument with the implicit function theorem on the linearization of contour dynamics equation. Our results extend recent ones in the range $\alpha\in[0,1)$ by Hassainia-Hmidi (DCDS-A, 2021) and Hassainia-Wheeler (SIAM J. Math. Anal., 2022) to more singular velocities, filling an open gap in the range of $\alpha$.

math.AP↗

arXiv2022

Existence of vortex sheet solutions to the SQG Equation

This paper addresses the existence of vortex sheets for the SQG equation. More precisely, we construct a family of stationary vortex sheet solutions that are concentrated on curves that are small perturbations of circles centered on a given non-degenerate critical point. To the best of our knowledge, this is a novel result on the existence of vortex sheets to the SQG equation.

math.AP↗

arXiv2019

Radiative Transfer with long-range interactions in the half-space

We study the well-posedness and regularity theory for the Radiative Transfer equation in the peaked regime posed in the half-space. An average lemma for the transport equation in the half-space is stablished and used to generate interior regularity for solutions of the model. The averaging also shows a fractional regularization gain up to the boundary for the spatial derivatives.

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