Solutions of incompressible hydrodynamic flow of liquid crystals

Local existence and uniqueness of strong solutions to the Navier–Stokes equations with nonnegative density

“…Specifically, one can weaken the regularity assumptions and drop the compatibility conditions on the initial data stated in [6,8,13,14,16,21,32,33], without destroying the existence and uniqueness of strong solutions; however, the definitions of the strong solutions in those papers should be modified accordingly.…”

“…Since the work [7], conditions (1.4)-(1.5) and their necessary modifications are widely used, as the standard assumptions, in many papers concerning the studies of the existence and uniqueness of strong solutions, with initial vacuum allowed, to the nonhomogeneous Navier-Stokes equations and some related systems, such as the magnetohydrodynamics (MHD) and liquid crystals, see, e.g., [6,8,13,14,16,21,32,33].…”

BKM's criterion for the 3D nematic liquid crystal flows in Besov spaces of negative regular index

“…It is well-known that the strong solutions of heat flow of harmonic maps must be blowing up at finite time (see [2]), therefore we can not expect that the system (1.1) has a global smooth solution with general initial data. However, Wen and Ding [29] …”

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

“…During the past several decades, there have been many researches on system (1.1)-(1.4), see, for example, [4]- [8], [12]- [19], [28], [29] and the references therein. In what follows, we only briefly recall some related results.…”