Local strong solution to the compressible magnetohydrodynamic flow with large data

TL;DR

This paper proves the local existence and uniqueness of strong solutions for three-dimensional compressible MHD flows with large initial data, despite the challenges posed by zero magnetic diffusivity.

Contribution

It introduces novel techniques and estimates to establish local strong solutions with weaker regularity in the zero magnetic diffusivity case.

Findings

Existence and uniqueness of local strong solutions are proven.

The solutions accommodate large initial data.

New analytical methods are developed for the zero magnetic diffusivity scenario.

Abstract

The three-dimensional compressible magnetohydrodynamic (MHD) isentropic flow with zero magnetic diffusivity is studied. The vanishing magnetic diffusivity causes significant difficulties due to the loss of dissipation of the magnetic field. The existence and uniqueness of local in time strong solution with large initial data is established. The strong solution has weaker regularity than the classical solution. A generalized Lax-Milgram theorem and a Schauder-Tychonoff-type fixed point argument are applied with novel techniques and estimates for the strong solution.