Projection-Lift Equivalence and Dissipation-Tight Compactness for Continuum-State FENE-Markov Fluids

Abstract

We consider incompressible FENE dumbbells coupled to a reversible Markov operator on a compact continuum of internal states. At zero centre-of-mass diffusion we prove that state averaging is an exact factor map and that a Lagrangian state lift is its unique inverse on the natural energy-solution classes. Thus existence, multiplicity, and uniqueness of the state-resolved system are precisely those of its scalar FENE projection; the internal-state dynamics creates no additional large-data nonuniqueness. The lift is driven by a trace-free matrix fibre evolution and permits nonlinear local activities, including rates with linear dependence on the singular Kramers stress.We also prove a sequential form of this structure. For state-resolved regularizations with relative-entropy-prepared initial fibres and no viscous dissipation defect, the complete densities converge strongly without reconstructing them after the scalar limit. Consequently the full drag,state-dependent activity, singular stress, and non-atomic Jeffreys production all pass to the limit. The argument combines stability of regular Lagrangian flows with a relative-entropy estimate for simultaneously varying matrix drifts and jump rates. A stationary oscillation shows that the preparation cannot follow from the natural entropy bounds alone. With positive centre-of-mass diffusion we independently construct global large-data weak solutions and identify the same nonlinear terms. An explicit infinite-rank kernel proves that these results are not finite-species reductions.