On asymptotic isotropy for a hydrodynamic model of liquid crystals (Q2814642)

Let \(\mathbb Q\) denote a symmetric traceless three-dimensional matrix. The tensor \(\mathbb Q = \mathbb Q(x,t)(x\in \mathbb R^3, t \geq 0)\) expresses the local configuration of the crystal. Its time evolution is described by the equation \(\partial_t \mathbb Q+ \operatorname{div}_x (\mathbb Q \mathbf u) - \mathbb S(\nabla_x \mathbf u,\mathbb Q)=\Delta_x \mathbb Q- L[\partial F(\mathbb Q)]\) where \(L\) is the projection onto the space of traceless matrices, \(\mathbf u\) is the velocity field satisfying the Navier-Stokes equations, \(\mathbb S\) is the special tensor. Using the method of Fourier splitting, the authors show that \(\mathbb Q\) and \(\mathbf u\) tend to the isotropic state at the rate \((1+t)^{-3/2}\) as \(t \to \infty\).