Continuity of velocity gradients in suspensions of rod-like molecules

We investigate the Doi model for suspensions of rod-like molecules in the dilute regime. For certain parameter values, the velocity gradient vs. stress relation defined by the stationary and homogeneous flow is not rank-one monotone. We then consider the evolution of possibly large perturbations of stationary flows. We prove that, even in the absence of a microscopic cut-off, discontinuities in the velocity gradient cannot occur in finite time. The proof relies on a novel type of estimate for the Smoluchowski equation.