Global smooth solutions to 3D MHD with mixed partial dissipation and magnetic diffusion (Q2437674)

The authors consider 3D incompressible magnetohydrodynamics equations when the kinematic viscosity \(\mu\) and the magnetic diffusion \(\eta\) appear only in the horizontal directions (i.e., \((x,y)\)-directions). Then the corresponding equations are of the following form: \[ u_t + u\cdot\nabla u = -\nabla p + \mu u_{xx} + \mu u_{yy} + b\cdot \nabla b,\;\;b_t + u\cdot\nabla b = \eta b_{xx} + \eta b_{yy} + b\cdot \nabla u, \] together with \(\mathrm{div}\!\;u=0,\;\mathrm{div}\!\;b=0\) and the initial data \(u(0,x)=u_0 ,\;b(0,x)=b_0\). Here \(u(t,x,y,z)\) is the velocity field, \(b(t,x,y,z)\) is the magnetic field, and \(p(t,x,y,z)\) is the scalar pressure. They prove the existence of the global solution to this Cauchy problem in the Sobolev space, when the initial conditions are suitably small. For this purpose, they first give some a priori estimates. Then they consider modified equations adding small kinematic viscosity and magnetic diffusion also in the vertical direction (i.e., \(z\)-direction). The existence of the solution to this modified problem is already known, and one can prove prove the main result using the method of vanishing viscosities.