On the local and global existence of the Hall equations with fractional Laplacian and related equations

In this paper, we deal with the Hall equations with fractional Laplacian

$$B_t+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left((\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}B)\times B\right)+\mathrm{\Lambda }B=0.$$

We begin to prove the existence of unique global in time solutions with sufficiently small initial data in $H^k$, $k>\frac{5}{2}$. By correcting $\mathrm{\Lambda }B$ logarithmically, we then show the existence of unique local in time solutions. We also deal with the two dimensional systems closely related to the $2\frac{1}{2}$ dimensional version of the above Hall equations. In this case, we show the existence of unique local and global in time solutions depending on whether the damping term is present or not.