Singularity formation for the incompressible Hall-MHD equations without resistivity

2013/12/19 by Chae, Dongho, Weng, Shangkun · 3 citations
#2010: 35Q30 #35L67 #35Q35 #76D05 #76D09 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1312.5519

Abstract

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space H^m(ℝ³) for any m>(7)/(2). Namely, either the system is locally ill-posed in H^m(ℝ³), or it is locally well-posed, but there exists an initial data in H^m(ℝ³), for which the H^m(ℝ³) norm of solution blows-up in finite time if m>7/2. In the latter case we choose an axisymmetric initial data u₀(x)=u_0r(r,z)e_r+ b_0z(r,z)e_z and B₀(x)=b_0θ(r,z)e_θ, and reduce the system to the axisymmetric setting. If the convection term survives sufficiently long time, then the Hall term generates the singularity on the axis of symmetry and we have limsup_{t→ t_*} sup_{z∈ \Bbb R} |∂_z∂_r b_θ(r=0,z)|=∞ for some t_*>0.

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• Lifespan of the Non-resistive Hall-MHD System with Small Magnetic Gradient
• The incompressible limit of the Euler-Maxwell two-fluid system
• Non-uniqueness of weak solutions to the 3D Hall-MHD equations on the plane

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