Global weak solutions to some two-fluid models with magnetic field

TL;DR

This paper establishes the existence of global weak solutions with finite energy for certain two-fluid models with magnetic fields, extending previous methods and removing some restrictive assumptions.

Contribution

It provides a more explicit proof for magnetic bi-fluid systems with ideal gas pressure laws, removing unnecessary conditions from prior work.

Findings

Proved existence of global weak solutions for two-fluid magnetic systems.

Extended proof techniques to real magnetic bi-fluid systems.

Removed some restrictive hypotheses from earlier theorems.

Abstract

We prove the existence of global weak solutions with finite energy to some two-fluid systems with magnetic field and the results suit for corresponding two-fluid systems. The proof method is mainly inspired by Novotn\'y et al. and Vasseur et al. For $academic~magnetic~ bi\text{-}fluid~ system$, we focus on the case of pressure law with ideal gases and the new ingredient is that we can make the proof more explicit without using the preposed hypotheses and remove some unnecessary conditions of the main theorem in Novotn\'y et al. Meanwhile, the same proof method can be used for $real~magnetic~bi\text{-}fluid~system$ combined with pressure law proposed in Vasseur et al.