Well-posedness for compressible MHD system with highly oscillating initial data

TL;DR

This paper establishes local and global well-posedness results for the compressible MHD system with highly oscillating initial data by transforming the system into Lagrange coordinates and extending the functional framework.

Contribution

It improves the Lebesgue exponent range in Besov spaces for the system and provides a lower bound for the maximal existence time, enabling global solutions with oscillatory initial data.

Findings

Extended the Lebesgue exponent range to [2, 2N) in Besov spaces.

Constructed unique local solutions for the compressible MHD system.

Obtained global solutions with highly oscillating initial velocity and density.

Abstract

In this paper, we transform compressible MHD system written in Euler coordinate to Lagrange coordinate in critical Besov space. Then we construct unique local solutions for compressible MHD system. Our results improve the range of Lebesgue exponent in Besov space from $[2, N)$ to $[2, 2N)$ with $N$ stands for dimension. In addition, we give a lower bound for the maximal existence time which is important for our construction of global solutions. Based on the local solution, we obtain a unique global solution with high oscillating initial velocity and density by using effective viscous flux and Hoff's energy methods to explore the structure of compressible MHD system.