arXiv subjects

Marc Magaña

Publications and source records attributed to Marc Magaña.

4 recordsLinked to original sources

arXiv2026

Unified theory for regularity persistence of vortex patch boundaries

We establish a unified local theory for the persistence of Sobolev regularity of vortex patch boundaries in a family of two-dimensional active scalar equations with radial convolution kernels $K(|x-y|)$. The class includes the 2D Euler equation, the generalized SQG equation in the locally integrable range $0<β<1$, and the quasi-geostrophic shallow water equation. Under natural assumptions on $K$ (smoothness, integrability near the origin, monotonicity, and polynomial growth), we prove that if the initial boundary belongs to $H^3(\mathbb T)$ and satisfies the arc-chord condition, then the contour dynamics equation admits a unique local solution in $C([0,T];H^3(\mathbb T))$. Under a stronger integrability condition on the kernel, we also obtain local existence of $H^2$ solutions. The proof combines Sobolev energy estimates for the contour equation with quantitative control of the arc-chord quantity.

math.AP↗

arXiv2026

The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little Hölder spaces.

math.AP↗

arXiv2025

Sharp Strong Convergence in Ideal Flows

We investigate the strong convergence of weak solutions to the two-dimensional Quasi-Geostrophic Shallow-Water (QGSW) equation as the inverse Rossby radius tends to zero. In this limit, we recover the Yudovich solution of the incompressible Euler equations. We prove that the vorticity convergence holds in $L^\infty_t L^p_x$, for any finite integrability exponent $p<\infty$. This extends to the case $p=\infty$ provided that the initial vorticities are continuous and converge uniformly. We also discuss the sharpness of this limit by demonstrating that the continuity assumption on the initial data is necessary for the endpoint convergence in $L^\infty_{t,x}$. The proof of the strong convergence relies on the {\em Extrapolation Compactness} method, recently introduced by Arsénio and the first author to address similar stability questions for the Euler equations. The approach begins with establishing the convergence in a lower regularity space, at first. Then, in a later step, the convergence to Yudovich's vorticity of Euler equations in Lebesgue spaces comes as a consequence of a careful analysis of the evanescence of specific high Fourier modes of the QGSW vorticity. A central challenge arises from the absence of a velocity formulation for QGSW, which we overcome by employing advanced tools from Littlewood Paley theory in endpoint settings. The sharpness of the convergence in the endpoint $L^\infty_{t,x}$ case is obtained in the context of vortex patches, drawing insights from key findings on uniformly rotating and stationary solutions of active scalar equations.

math.AP↗

arXiv2024

Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces

We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.

math.AP↗