Dyadic models for ideal MHD

TL;DR

This paper investigates two dyadic models of incompressible ideal MHD, establishing existence, well-posedness, blow-up phenomena, fixed points, and stability analysis to deepen understanding of energy cascades in such systems.

Contribution

It introduces and analyzes two new dyadic models for ideal MHD, revealing their solution behaviors, fixed points, and stability properties, which were previously unexplored.

Findings

Global weak solutions and local well-posedness are established for both models.

Finite-time blow-up occurs for solutions with positive initial data in the uni-directional cascade model.

Fixed points and their linear instability are identified for each model.

Abstract

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.