On the Doi model for the suspensions of rod-like molecules: global-in-time existence

The Doi model for the suspensions of rod-like molecules in a dilute regime describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained (cf.[M.Doi and S.F.Edwards, Oxford University Press, 1986]).The orientation distribution of the rods on the microscopic level is described by a Fokker-Planck-type equation on the sphere, while the fluid flow is given by the Navier-Stokes equations, which are now enhanced by an additional macroscopic stress σ reflecting the orientation of the rods on the molecular level.Prescribing arbitrarily the initial velocity and the initial orientation distribution in suitable spaces we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in three dimensional space.The proof relies on a quasi-compressible approximation of the pressure, the construction of a sequence of approximate solutions, and the establishment of compactness.