Well-posedness of 3D Navier-Stokes and MHD equations with partial fractional hyper-diffusion

It is well-known that by substituting standard velocity and magnetic diffusion with fractional hyper-diffusion, the Navier-Stokes and magnetohydrodynamic equations are well-posed. We examine the 3D Navier-Stokes and magnetohydrodynamic equations with partial fractional hyper-dissipation. It is proved that when each component of the velocity and magnetic field lacks dissipation along certain directions, the existence and conditional uniqueness of the solution are maintained.