Hydrodynamics of Anisotropic Liquid Crystals in an Applied Magnetic Field

We consider a magnetic effect on hydrodynamic flows of nematic liquid crystals with unequal elastic constants in dimension two. Under a planar ansatz for the director field and velocity, we derive a reduced Ericksen--Leslie system which consists of a quasilinear parabolic equation and a Navier--Stokes equation involving critical nonlinearity due to elastic anisotropy. When unequal elastic constants are sufficiently close, we establish the global existence of smooth solutions to the reduced system and show convergence to an equilibrium as $t$ tends to $\infty$. A magnetic field-induced instability of the aligned state under the hydrodynamic flow is discussed.