Singularity formation for the incompressible Hall-MHD equations without resistivity

By Dongho Chae, Shangkun Weng. 2013

137 citations recorded by the source corpus.

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^{m}(\mathbb{R}^3)$ for any $m>\frac{7}{2}$. Namely, either the system is locally ill-posed in $H^{m}(\mathbb{R}^3)$, or it is locally well-posed, but there exists an initial data in $H^{m}(\mathbb{R}^3)$, for which the $H^{m}(\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$. In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and $B_0(x)=b_{0\theta}(r,z)e_{\theta}$, 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 $ \lim\sup_{t\to t_*} \sup_{z\in \Bbb R} |\partial_z\partial_r b_\theta(r=0,z)|=\infty$ for some $t_*>0$.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/a8d2ab7c3212097f28fd7f9fa05a4e6264bc3783

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations (2015, background): “f scaling limits. They also provided a kinetic formulation for the Hall-MHD, and proved the existence of global weak solutions for the incompressible viscous resistive Hall-MHD system. The authors in [5] have showed unsteady Hall-MHD system without resistivity may develop ﬁnite time singularity for ”