WEAK TIME REGULARITY AND UNIQUENESS FOR A Q-TENSOR MODEL

The coupled Navier-Stokes and Q-tensor system is one of the models used to describe the behavior of nematic liquid crystals. The existence of weak solutions and a uniqueness criterion have been already studied (see [M. Paicu and A. Zarnescu, Arch. Ration. Mech. Anal., 203 (2012), pp. 45-67] for a Cauchy problem in the whole R-3, and [F. Guillen-Gonzalez and M. A. Rodriguez-Bellido, Nonlinear Anal., 112 (2015), pp. 84-104] for an initial-boundary problem in a bounded domain Omega). Nevertheless, results on strong regularity have been treated only in Paicu and Zarnescu's paper for a Cauchy problem in the whole R-3. 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 criterion implying that this solution is global in time. Note that the regularity furnished by the weak regularity for (u, Q) and for (partial derivative(t)u, partial derivative(t)Q) is not equivalent to the strong regularity. Finally, when large enough viscosity is imposed, we obtain the existence (and uniqueness) of a global in time strong solution. In fact, if a nonhomogeneous Dirichlet condition for Q is imposed, the strong regularity needs to be obtained together with the weak regularity for (partial derivative(t)u, partial derivative(t)Q).