Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

2019/04/12 by Jiu, Quansen, Suo, Xiaoxiao, Wu, Jiahong +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.06006

Abstract

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation (-Δ)^αu and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case α=1. This paper discovers that there are new phenomena with the case α<1. The approach for α=1 can not be directly extended to α<1. We establish that, for α<1, any initial data (u₀, b₀) in the inhomogeneous Besov space B^σ_2,∞(\mathbb R^d) with σ> 1+(d)/(2)-α leads to a unique local solution. For the case α≥ 1, u₀ in the homogeneous Besov space \mathring B^{1+(d)/(2)-2α}_2,1(\mathbb R^d) and b₀ in \mathring B^1+(d)/(2)-α_2,1(\mathbb R^d) guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

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