Some results on the three-dimensional Prandtl equations and boundary layer theory for MHD
City University of Hong Kong
In this talk, I will introduce some results on the 3D Prandtl equations and boundary layer
theory for MHD. First, I give some results on the well-posedness of 3D Prandtl equations under
some constraint on its flow structure, where the complicated secondary flow can be avoided.
Moreover, it is shown that this structured flow is linearly stable for any 3D perturbation, and
the global existence of weak solutions is studied with some favorable condition on pressure.
Then, as a supplement to the perious well-posedness results, we show that the shear flow is
linear unstable for 3D Prandtl equations if such flow structure isn`t satised, which is quite
different from the 2D case.
Next, we are concerned with the validity of the Prandtl boundary layer theory in the
inviscid limit for MHD equations in the half plane with no-slip boundary conditions on the
velocity vector and the perfect conducting boundary conditions on the magnetic field. In
the case that viscosity and magnetic diusion tend to zero at the same rate, we derive the
boundary layer problem and establish the well-posedness result under the assumption of
nonzero initial tangential magnetic field. Then, when the initial tangential magnetic field
of MHD doesn`t vanish at the boundary, we justify the validity of corresponding Prandtl
boundary layer expansions. This justifies the physical understanding that the magnetic field
has a stabilizing effect on MHD boundary layer in rigorous mathematics.