Remarks on the three and two and a half dimensional Hall-magnetohydrodynamics system: deterministic and stochastic cases

Abstract

The Hall-magnetohydrodynamics system has a long history of various applications in physics and engineering, in particular magnetic reconnection, star formation and the study of the sun. Mathematically, the additional Hall term elevates the level of non-linearity to the quasi-linear type while similar systems of equations such as the Navier–Stokes equations, as well as the viscous and magnetically diffusive magnetohydrodynamics system are of the semi-linear type. Consequently, the mathematical analysis on the Hall-magnetohydrodynamics system had been relatively absent for more than half a century, and only started seeing rapid developments in the last few years. In this manuscript, we survey recent progress on the mathematical analysis of the Hall-magnetohydrodynamics system, both in the deterministic and stochastic cases, elaborating on the structure of the Hall term. We list some remaining open problems, and discuss their difficulty. As a byproduct of our discussion, it is deduced that the two and a half dimensional Hall-magnetohydrodynamics system with viscous diffusion in the form of a Laplacian and the magnetic diffusion in the form of a fractional Laplacian with power $\frac{3}{2}$ admits a unique smooth solution for all time.

Appendix

The following inequality is a standard product estimate:

Lemma 5.1

(e.g. Lemma A.2 [28]) Suppose that the spatial dimension is ${\mathbb{R}}^{N}$ for any $N \ge 2.$ Let

$$f \in W^{\delta , p_{1}} \cap L^{q_{2}}, g \in W^{\delta , p_{2}} \cap L^{q_{1}}$$

where $\delta \ge 0, 1< p_{k}< \infty , 1 < q_{k} \le \infty ,$ and

$$\frac{1}{p} = \frac{1}{p_{k}} + \frac{1}{q_{k}}$$

for both $k = 1, 2.$ Then

$$||fg||_{{\dot{W}}^{\delta , p}} \lesssim ||f||_{{\dot{W}}^{\delta , p_{1}}}||g||_{L^{q_{1}}} + ||f||_{L^{q_{2}}} ||g||_{{\dot{W}}^{\delta , p_{2}}}.$$

We also recall the following Kato-Ponce commutator estimate:

Lemma 5.2

([29]) Suppose that the spatial dimension is ${\mathbb{R}}^{N}$ for any $N \ge 2.$ Let f, g be smooth such that $\nabla f \in L^{p_{1}},$ $\Lambda^{s-1}g \in L^{p_{2}},$ $\Lambda^{s}f \in L^{p_{3}},\;g \in L^{p_{4}}$ where $p_{2}, p_{3} \in (1, \infty )$ and $s > 0.$ Moreover, suppose that $p \in (1,\infty )$ satisfies

$$\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}} = \frac{1}{p_{3}} + \frac{1}{p_{4}}.$$

Then

$$||\Lambda^{s}(fg) - f\Lambda^{s}g||_{L^{p}} \lesssim ||\nabla f||_{L^{p_{1}}}||\Lambda^{s-1}g||_{L^{p_{2}}} + ||\Lambda^{s}f||_{L^{p_{3}}}||g||_{L^{p_{4}}}.$$

Lemma 5.3

([39, Lemma 2.3]) Suppose that the spatial dimension is ${\mathbb{R}}^{2}.$ Let $f \in L^{2}\cap H^{s},\quad s > 2$ satisfy $\nabla \cdot f =0$ and $\nabla \times f \in L^{\infty}.$ Then

$$||\nabla f||_{L^{\infty }} \lesssim ||f||_{L^{2}} + ||\nabla \times f||_{L^{\infty }} \log_{2}(2+ ||f||_{H^{s}}) + 1.$$