Neumann problems for quasi-linear parabolic systems modeling polydisperse suspensions

We discuss the well-posedness of a class of Neumann problems for n x n quasi-linear parabolic systems arising from models of sedimentation of polydisperse suspensions in engineering applications. This class of initial-boundary value problems includes the standard (zero-flux) Neumann condition in the limit as a positive perturbation parameter theta goes to 0. We call, in general, the problem associated with theta >= 0 the theta-flux Neumann problem. The Neumann boundary conditions, although natural and usually convenient for integration by parts, are nonlinear and couple the different components of the system. An important aspect of our analysis is a time stepping procedure that considers linear boundary conditions for each time step in order to circumvent the difficulties arising from the nonlinear coupling in the original boundary conditions. We prove the well-posedness of the flux Neumann problems for theta > 0 and obtain a solution of the standard (zero-flux) Neumann problem as the limit for theta --> 0 of solutions of the theta-flux Neumann problems. Concerning applications, the analysis developed here supports a new model for the settling of polydisperse suspensions forming compressible sediments.