Global weak entropy solution to Doi–Saintillan–Shelley model for active and passive rod-like and ellipsoidal particle suspensions

We prove the existence of the global weak entropy solution to the Doi–Saintillan–Shelley model for active and passive rod-like particle suspensions, which couples a Fokker–Planck equation with the incompressible Navier–Stokes or Stokes equation, under the no-flux boundary conditions, L2(Ω;L1(Sd−1)) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L2(Ω×Sd−1) weak solution in two and three dimensions and the uniqueness in two dimension.