REMARKS ON OLDROYD-B AND RELATED COMPLEX FLUID MODELS

We prove global existence and uniqueness of solutions of Oldroyd-B systems with relatively small data in R d , in a large functional setting (C � \ L 1 ). This is a stability result, solutions select an equilibrium and converge exponentially to it. Large spatial derivatives of the initial density and stress are allowed, provided the L 1 norm of the density and stress are small enough. We prove global regularity for large data for a model in which the potential responds to high rates of strain in the fluid. We also prove global existence for a class of large data for a didactic scalar model which attempts to capture in the simplest way the essence of the dissipative nature of the coupling to fluid. This latter model has an unexpected cone invariance in function space that is crucial for the global existence. 1. Introduction The complex fluids models we consider treat the interaction of a normal fluid (incompressible in our case) with insoluble matter suspended in it. Models have been devised to deal with microscopic elastic thread-like objects such as polymers ((12), (26)). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by few representative ones. In the models we consider here, an end-to-end vector in R 3 (m in this paper) represents the orientation of the polymer and it is the sole variable retained to describe the microscopic object. The models that are most studied with this degree of simplification include the kinetic description of the Oldroyd B model, the FENE model and variants. The probability distribution of the vectors m depends on time and physical space, and it is assumed to be absolutely continuous with respect to the usual Lebesgue measure dm, so the chance of finding m at time t and location x 2 R 3 in the volume element dm is f(x;m;t)dm. The aim is to describe the evolution of f in space and time. When the polymer concentration is so dilute that the polymers do not interact, and when the fluid is smooth, then the equation of f is a linear Fokker-Planck equation. The effect of the fluid can be understood perturbatively: because the objects suspended are assumed to have faster time scales and shorter length scales than the scales on which solvent varies, it is then justified to treat the fluid as varying little as far as the microscopic suspended objects are concerned. This macro-micro interaction can be rationally discussed, and leads to a kinetic description of the particles, given a fluid flow. On the other hand, the collective effect that the particles have on the flow itself is a macroscopic effect of microscopic insertions: a micro- macro interaction. These interactions are much more mysterious. There have been attempts to produce systematic upscalings based on non-equilibrium thermodynamics formalisms ((2), (13)). When the microscopic insertions have a larger number of degrees of freedom, m 2 M, where M is a Riemannian manifold representing a finite number of degrees of freedom with constrains, then the kinetic equation is a Smoluchowski equation on the manifold. The upscaling principle advocated by one of us ((5), (6)) is easiest formulated as the requirement that the sum of the energy of the fluid