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

Abstract

<p style='text-indent:20px;'>In this paper, we deal with the Hall equations with fractional Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda B = 0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We begin to prove the existence of unique global in time solutions with sufficiently small initial data in <inline-formula><tex-math id="M1">\begin{document}$ H^{k} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ k&gt;\frac{5}{2} $\end{document}</tex-math></inline-formula>. By correcting <inline-formula><tex-math id="M3">\begin{document}$ \Lambda B $\end{document}</tex-math></inline-formula> logarithmically, we then show the existence of unique local in time solutions. We also deal with the two dimensional systems closely related to the <inline-formula><tex-math id="M4">\begin{document}$ 2\frac{1}{2} $\end{document}</tex-math></inline-formula> 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.</p>