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Global existence of weak solutions to inhomogeneous Doi-Onsager equations

In this paper, we study the inhomogeneous Doi-Onsager equations with a special viscous stress. We prove the global existence of weak solutions in the case of periodic regions without considering the effect of the constraint force arising from the rigidity of the rods. The key ingredient is to show the convergence of the nonlinear terms, which can be reduced to proving the strong compactness of the moment of the family of number density functions. The proof is based on the propagation of strong compactness by studying a transport equation for some defect measure, L2-estimates for a family of number density functions, and energy dissipation estimates.