A regularity result of Cauchy problem of the ideal incompressible Magnetohydrodynamics equations

2024/07/30 by Zhang, Huali
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.20531

Abstract

Under a homogeneous magnetic field, we establish the local well-posedness of low-regularity solutions for an ideal incompressible Magnetohydrodynamics (MHD) system in Lagrangian coordinates. Firstly, we reduce the MHD system to a degenerate wave-elliptic system inherent with a specific null form. After constructing a new solution space, we can prove some good product estimates. Combined with the inside null structure, a bilinear estimate of the Klainerman-Machedon's type for nonlinear terms can be obtained. These lead us to prove the local well-posedness of ideal incompressible MHD equations in Lagrangian coordinates if the initial velocity \bv₀ ∈ H^s(ℝⁿ), s> (n+1)/(2) (n=2,3). So our result lowers \frac12-order regularity comparing with the classical exponent s>1+(n)/(2). Moreover, to the author's knowledge, this is the first result concerning low-regularity solutions of the ideal MHD equations.

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