Appears in collection : Vorticity, rotation and symmetry (III) - approaching limiting cases of fluid flows / Vorticité, rotation et symétrie (III) – analyse des situations limites en théorie des fluides

In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.

$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$ $\nabla \cdot u =0, \nabla \cdot B =0, $ $\partial_tB - \nabla \times \left (u \times B\right ) + \nabla \times \left (\left (\nabla \times B\right )\times B \right ) = \mu \nabla B,$ $u\left (x,0 \right )=u_0\left (x\right ) ; B\left (x,0 \right )=B_0\left (x\right ).$

Here $u=\left (u_1, u_2, u_3 \right ) = u \left (x,t \right ) $ is the velocity of the charged fluid, $B=\left (B_1, B_2, B_3 \right ) $ the magnetic field induced by the motion of the charged fluid, $p=p \left (x,t \right )$ the pressure of the fluid. The positive constants $\nu$ and $\mu$ are the viscosity and the resistivity coefficients. Compared with the usual viscous incompressible MHD system, the above system contains the extra term $\nabla \times \left (\left (\nabla \times B\right )\times B \right ) $ , which is the so called Hall term. This term is important when the magnetic shear is large, where the magnetic reconnection happens. On the other hand, in the case of laminar ows where the shear is weak, one ignores the Hall term, and the system reduces to the usual MHD. Compared to the case of the usual MHD the history of the fully rigorous mathematical study of the Cauchy problem for the Hall-MHD system is very short. The global existence of weak solutions in the periodic domain is done in [1] by a Galerkin approximation. The global existence in the whole domain in $\mathbb{R}^3$ as well as the local well-posedness of smooth solution is proved in [2], where the global existence of smooth solution for small initial data is also established. A refined form of the blow-up criteria and small data global existence is obtained in [3]. Temporal decay estimateof the global small solutions is deduced in [4]. In the case of zero resistivity we present finite time blow-up result for the solutions obtained in [5]. We note that this is quite rare case, as far as the authors know, where the blow-up result for the incompressible flows is proved.