Axi-symmetric solutions for active vector models generalizing 3D Euler and electron--MHD equations

TL;DR

This paper investigates axi-symmetric solutions to a family of vector models that interpolate between 3D Euler and electron-MHD equations, establishing local well-posedness and traveling wave existence for certain parameter ranges.

Contribution

It introduces a generalized axi-symmetric framework for these models, proving well-posedness and traveling wave solutions, extending classical results for 3D Euler.

Findings

Proved local well-posedness of Lipschitz solutions

Established existence of traveling wave solutions

Generalized classical results to a broader family of models

Abstract

We study systems interpolating between the 3D incompressible Euler and electron--MHD equations, given by \begin{equation*} \partial_t B + V \cdot \nabla B = B\cdot \nabla V, \qquad V = -\nabla\times (-\Delta)^{-a} B, \qquad \nabla\cdot B = 0, \end{equation*} where $B$ is a time-dependent vector field in $\mathbb{R}^3$. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range $1/2<a<1$. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.