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Global regularity for 3D generalized Hall magneto-hydrodynamics equations
Keywords:Hall magneto-hydrodynamics equations, global regularity, hyperdissipation, Littlewood-Paley decomposition.
      For the 3D incompressible Hall magneto-hydrodynamics 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 Hall-MHD 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 non-decreasing functions $g_{1}$ and $g_{2}$: $\mr^{+}\rightarrow\mr^{+}$ such that $\int_{1}^{\infty}\frac{1}{s(g_{1}^{2}(s)+g_{2}^{2}(s))^{2}}\md s=+\infty.$