Weak Time Regularity and Uniqueness for a $Q$-Tensor Model

Source abstract

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 $\mathbb{R}^3$, and [F. Guillén-González and M. Á. Rodríguez-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 $\mathbb{R}^3$. 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 $({\it u} ,Q)$. Moreover, we gives a regularity criterion implying that this solution is global in time. Note that the regularity furnished by the weak regularity for $({\it u},Q)$ and for $(\partial_t {\it u}, \partial_t Q)$ 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 $(\partial_t {\it u}, \partial_t Q)$.