Existence, uniqueness and optimal decay rates for the 3D compressible Hall-magnetohydrodynamic system

TL;DR

This paper proves the existence, uniqueness, and optimal decay rates of strong solutions for the 3D compressible Hall-magnetohydrodynamic system near equilibrium, using Fourier analysis and energy methods.

Contribution

It establishes the global well-posedness and decay rates for solutions in critical Besov spaces, advancing understanding of this complex fluid system.

Findings

Unique global solutions near equilibrium in critical spaces

Optimal decay rates under low-frequency conditions

Application of Fourier analysis and energy functionals

Abstract

We are concerned with the study of the Cauchy problem to the 3D compressible Hall-magnetohydrodynamic system. We first establish the unique global solvability of strong solutions to the system when the initial data are close to a stable equilibrium state in critical Besov spaces. Furthermore, under a suitable additional condition involving only the low frequencies of the data and in $L^{2}$-critical regularity framework, we exhibit the optimal time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined time-weighted energy functional.