Active-Line Dynamics and Residual Work in a Reduced Oldroyd-B Mechanism

We study a one-dimensional active-line equation arising as a thin-sheet reduced mechanism for high-Weissenberg Oldroyd-B dynamics.

The unknown is a positive periodic line density rho=m+eta satisfying rho_t+c rho Lambda rho-c(H rho)rho_s+gamma(rho-m)=0, where m>0 and Lambda=H partial_s.

We prove positive-density local well-posedness, a no-rupture principle, small-oscillation global stability, finite-band relaxation, and a one-tail alternative for any remaining positive-density obstruction.

We also separate the reduced scalar mechanism from the two-dimensional positive-cone Oldroyd-B admissibility problem.

For finite-thickness lifts written in log-conformation variables A=exp(B), we prove the residual-work inequality W_E <= (alpha/2) eta_E L_E S_E, which excludes structured active-line lifts whose pressure-free work is too large relative to the lever, residual size, and alignment.

Under an active-line compactness extraction, an admissible noncompactness profile must either retain a non-vanishing active one-tail clock or satisfy this residual-work budget.

Finite-thickness active-line concentrations with a positive residual-work gap are therefore ruled out.

The compactness theorem that would place every two-dimensional noncompactness scenario in this class is left as a separate problem.