In this talk, we are concerned with the co-rotational Beris-Edwards system, which models incompressible liquid crystal flows with the Landau-de Gennes bulk potential. This system is not scale-invariant due to the presence of the bulk potential, and it involves a loss of regularity for both the velocity field and the Q-tensor through the stress term and the highly nonlinear coupling term. Motivated by quantitative results for the incompressible Navier-Stokes equation established by Tao and Barker-Prange, we derive triple-logarithmic quantitative blow-up rates for critically bounded solutions of the co-rotational Beris-Edwards system. To address the loss of regularity, we utilize a cancellation property and a linearization technique for the Q-tensor. Moreover, we establish a refined quantitative Carleman estimate based on the vorticity-Hessian formulation to overcome the bulk potential. Furthermore, with the help of partial regularity for perturbed co-rotational Beris-Edwards system, we establish local-in-space smoothing near the initial time. The results are partly based on joint work with Prof. HU Xianpeng.

学术报告海报20251123叶诗怡.pdf