On the regularity of the Q-Tensor depending on the data

The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals, an intermediate phase between crystalline solids and isotropic fluids. These equations model the dynamics of the fluid via velocity and pressure (u, p) and the orientation of the molecules via a tensor Q. A review on the existence of weak solutions, maximum principle and a uniqueness criteria can be seen in [Guillen-González, F., and Rodríguez-Bellido, M. Á. Some properties on the Q-tensor system. Monogr. Mat. García Galdeano 39 (2014), 133–145] (the corresponding Cauchy problem in the whole R3 is analyzed by Zarnescu. However, the regularity of such solutions is only analyzed under some restrictive conditions: large viscosity or periodic boundary conditions. In this work, we study two different types of regularity for the Q-Tensor model: one inherited from the usual strong solution for the Navier-Stokes equations, and another one where (u, Q) and (∂tu, ∂tQ) have weak regularity (weak-t). This latter regularity is introduced due to the impossibility of obtaining local in time strong estimates for nonperiodic boundary conditions, where only the existence (and uniqueness) of local weak-t solution is obtained. Some regularity criteria for (u, Q) will also be given. In the particular case of Neumann boundary conditions for Q, the regularity criteria only must be imposed for the velocity u).