Weak time regularity and uniqueness for a Q-Tensor model

Abstract

The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals. The existence of weak solutions and a uniqueness criteria have been already studied (see [11] Marius Paicu and Arghir Zarnescu. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal., 203(1):45–67, 2012) for a Cauchy problem in the whole R3 and [7] F. Guillén-Gonz´alez and M. A. Rodríguez-Bellido. Weak solutions for an initialboundary Q-tensor problem related to liquid crystals. Submitted, 2014 for a initial-boundary problem in a bounded domain Ω). Nevertheless, results on strong regularity have only been treated in [11] for a Cauchy problem in the whole R3. In this paper, imposing Dirichlet or Neumann boundary conditions, we show the existence and uniqueness of a local in time weak solution with weak regularity for the time derivative of the velocity and the tensor variables (u, Q). Moreover, we gives a regularity criteria implying that this solution is global in time. Note that the regularity furnished by the weak regularity for (u, Q) and the weak regularity for (∂tu, ∂tQ) is not equivalent to the strong regularity. Finally, when large enough viscosity is imposed, we obtain the existence (and uniqueness) of global in time strong solution. In fact, if non-homogeneous Dirichlet condition for Q is imposed, the strong regularity needs to be obtained together with the weak regularity for (∂tu, ∂tQ).