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It is well-known that if one replaces standard velocity and magneticdissipation by $(-\Delta)^\alpha u$ and $(-\Delta)^\beta b$ respectively, themagnetohydrodynamic equations are well-posed for $\alpha\ge\frac{5}{4}$ and$\alpha + \beta \ge \frac{5}{2}$. This paper considers the 3D Navier-Stokes andmagnetohydrodynamic equations with partial fractional hyper-dissipation. It isproved that when each component of the velocity and magnetic field lacksdissipation along some direction, the existence and conditional uniqueness ofthe solution still hold. This paper extends the previous results in (Yang, Jiuand Wu J. Differential Equations 266(1): 630-652, 2019) to a more general case.

The well-posedness of three-dimensional Navier-Stokes and magnetohydrodynamic equations with partial fractional dissipation