Existence And Uniqueness Of Global Weak Solution To A Kinetic Model For The Sedimentation Of Rod-Like Particles

Passive rod-like and ellipsoidal particle suspensions in fluid are com-mon in nature, such as liquid crystal molecules moving in a solvent, and bacteria swimming in the water. The particles suspensions can be effective-ly modeled by a coupled microscopic Fokker-Planck equation and macro-scopic (Navier-) Stokes equation. More and more mathematicians, biolo-gists and physicists pay attention to these models.We investigate a kinetic model for the sedimentation of dilute suspen-sions of rodlike particles under gravity, deduced by C. Helzel, F. Otto and A. E. Tzavaras (2011). First, we establish the existence and uniqueness of global weak solution with large initial data to the two dimensional model involving a Stokes equation; then with similar method, we can deal with the counterpart with a Navier-Stokes equation and small initial data. In the proof of existence, a semi-implicit scheme and cut-off functions are used to construct a approximate problem, which is solved by employing Leray-Schauder fixed point theorem. Then the Aubin-Lion lemma with piecewise constant functions of time is applied in the compactness argument.