Global regularity for 3D generalized Hall magnetohydrodynamics equations 

Keywords:Hall magnetohydrodynamics equations, global regularity, hyperdissipation, LittlewoodPaley decomposition. 
Abstract: 
For the 3D incompressible Hall magnetohydrodynamics equations, global regularity of the weak solutions
is not established so far.
The major difficulty is that the dissipation given by the
Laplacian operator is insufficient to control the nonlinearity.
Wan obtained the global regularity
of the 3D generalized HallMHD equations with critical and subcritical
hyperdissipation regimes $m_{1}(\xi)=\xi^{\alpha}$, $m_{2}(\xi)=\xi^{\beta}$
for $\alpha\geq\frac{5}{4}$, $\beta\geq\frac{7}{4}$.
We improve this slightly by making logarithmic reductions in the dissipation and
still obtain the global regularity. More precisely, the hyperdissipation regimes in our system are
$m_{1}(\xi)\geq\frac{\xi^{\alpha}}{g_{1}(\xi)}$, and $m_{2}(\xi)\geq\frac{\xi^{\beta}}{g_{2}(\xi)}$
for some nondecreasing functions $g_{1}$ and $g_{2}$: $\mr^{+}\rightarrow\mr^{+}$
such that $\int_{1}^{\infty}\frac{1}{s(g_{1}^{2}(s)+g_{2}^{2}(s))^{2}}\md s=+\infty.$ 



