Singularity formation for the incompressible Hall-MHD equations without resistivity

Chae, Dongho; Weng, Shangkun

doi:10.1016/j.anihpc.2015.03.002

Chae, Dongho ; Weng, Shangkun

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1009-1022

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Résumé

In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^{m} (R^{3})$ for any $m > \frac{7}{2}$ . Namely, either the system is locally ill-posed in $H^{m} (R^{3})$ , or it is locally well-posed, but there exists an initial data in $H^{m} (R^{3})$ , for which the $H^{m} (R^{3})$ norm of solution blows-up in finite time if $m > \frac{7}{2}$ . In the latter case we choose an axisymmetric initial data $u_{0} (x) = u_{0 r} (r, z) e_{r} + b_{0 z} (r, z) e_{z}$ and $B_{0} (x) = b_{0 θ} (r, z) e_{θ}$ , 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 $\lim \sup_{t \to t_{⁎}} \sup_{z \in R} | \partial_{z} \partial_{r} b_{θ} (r = 0, z) | = \infty$ for some $t_{⁎} > 0$ , which will also induce a singularity in the velocity field.

MR Zbl

DOI : 10.1016/j.anihpc.2015.03.002

Classification : 35Q35, 35L67, 76D09
Keywords: Inviscid/viscous Hall-MHD without resistivity, Singularity formation, Axisymmetric data

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     author = {Chae, Dongho and Weng, Shangkun},
     title = {Singularity formation for the incompressible {Hall-MHD} equations without resistivity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1009--1022},
     year = {2016},
     publisher = {Elsevier},
     volume = {33},
     number = {4},
     doi = {10.1016/j.anihpc.2015.03.002},
     mrnumber = {3519529},
     zbl = {1347.35199},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.03.002/}
}

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Chae, Dongho; Weng, Shangkun. Singularity formation for the incompressible Hall-MHD equations without resistivity. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1009-1022. doi: 10.1016/j.anihpc.2015.03.002

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