Low regularity solutions for the Cauchy problem of the ideal incompressible Magnetohydrodynamics equations

TL;DR

This paper proves local well-posedness for low regularity solutions of the ideal incompressible MHD equations in Lagrangian coordinates, lowering the regularity threshold compared to classical results.

Contribution

It reformulates the MHD system into a null-structured wave-elliptic system and establishes well-posedness with less regular initial data than previously known.

Findings

Local well-posedness for initial velocity in H^s with s > (n+1)/2

Lowered regularity requirement by half a derivative compared to classical results

Utilized null structure and bilinear estimates for the analysis

Abstract

In Lagrangian coordinates, the local well-posedness of low regularity solutions is established for an ideal incompressible magnetohydrodynamic (MHD) system subject to a homogeneous background magnetic field. First, the MHD system is reformulated into a degenerate wave-elliptic system with a particular null structure. By introducing a suitably defined solution space, several refined product estimates are derived. Next, using the inherent null structure, a Klainerman-Machedon type bilinear estimate is obtained for the nonlinear terms. These nice structures and estimates yield the local well-posedness of the ideal incompressible MHD equations in Lagrangian coordinates for initial velocity fields $\bv_0 \in H^{s}(\mathbb{R}^n)$ with $s > \frac{n+1}{2}$ $(n=2,3,4)$. Moreover, the regularity requirement is lowered by half a derivative compared with the classical exponent $s > \frac{n}{2}+1$.