arXiv subjects

J. C. Cantero

Publications and source records attributed to J. C. Cantero.

2 recordsLinked to original sources

arXiv2023

The regularity of the boundary of vortex patches for some non-linear transport equations

We prove the persistence of boundary smoothness of vortex patches for a non-linear transport equation in $\mathbb{R}^n$ with velocity field given by convolution of the density with an odd kernel, homogeneous of degree $-(n-1)$ and of class $C^2(\mathbb{R}^n\setminus\{0\}, \mathbb{R}^n).$ This allows the velocity field to have non-trivial divergence. The quasi-geostrophic equation in $\mathbb{R}^3$ and the Cauchy transport equation in the plane are examples.

math.AP↗

arXiv2022

$C^γ$ well-posedness of some non-linear transport equations

Given $k:\mathbb{R}^n\setminus\{0\} \to \mathbb{R}^n$ a kernel of class $C^2$ and homogeneous of degree $1-n$, we prove existence and uniqueness of Hölder regular solutions for some non-linear transport equations with velocity fields given by convolution of the density with $k$. The Aggregation, the 3D quasi geostrophic, and the 2D Euler equations can be recovered for particular choices of $k$.

math.AP↗