The mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions.

: In a previous work [1], three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and 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 solution to such a system.

SUMMARYWe deal here with the numerical simulation of a multiscale model of concentrated suspensions. We compare the deterministic solution procedure for the Fokker Planck equation with the Monte Carlo simulation of the stochastic di erential equation. In particular, we examine questions of variance reduction.

For a real degree d polynomial P with all nonvanishing coefficients, with c sign changes and p sign preservations in the sequence of its coefficients (c + p = d), Descartes' rule of signs says that P has pos ≤ c positive and neg ≤ p negative roots, where pos ≡ c( mod 2) and neg ≡ p( mod 2). For 1 ≤ d ≤ 3, for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair (pos, neg) satisfying these conditions there exists a polynomial P with exactly pos positive and neg negative roots (all of them simple); that is, all these cases are realizable. This is not true for d ≥ 4, yet for 4 ≤ d ≤ 8 (for these degrees the exhaustive answer to the question of realizability is known) in all nonrealizable cases either pos = 0 or neg = 0. It was conjectured that this is the case for any d ≥ 4. For d = 9, we show a counterexample to this conjecture: for the sign pattern (+, −, −, −, −, +, +, +, +, −) and the pair (1, 6) there exists no polynomial with 1 positive, 6 negative simple roots and a complex conjugate pair and, up to equivalence, this is the only case for d = 9.