Well-Posedness of a Multiscale Model for Concentrated Suspensions

In a previous work [E. Cance s, I. Catto, and Y. Gati, SIAM J. Math. Anal., 37 (2005), pp. 60--82], three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles which are subjected to a given time-dependent shear rate. In the present work we extend the model to a more physically relevant situation where the shear rate actually depends on the macroscopic velocity of the fluid. As a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker--Planck-type equation with nonhomogeneous, nonlocal, and possibly degenerate coefficients. We show the existence and the uniqueness of the global-in-time weak solutionto such a system.