Pressure-Free Anisotropic Cancellation and Endpoint Continuation for the Non-Diffusive Oldroyd-B System

Abstract

This paper proves an endpoint continuation criterion for the two-dimensional stress-diffusion-free Oldroyd-B system. The conformation tensor is transported and stretched without spatial diffusion, while the elastic stress enters the viscous velocity equation through one derivative. The key structural observation is that the isotropic conformation mode is invisible to the velocity after pressure renormalization: writing $A = aI + Y$, the force div($aI$) is absorbed into the pressure, and only the traceless anisotropic stress $Y$ produces vorticity. A weighted anisotropic energy then gives an exact top-order cancellation between the velocity equation and the anisotropic equation, before any smallness or absorption is used. This pressure-free cancellation closes the continuation criterion under $\nabla u \in L^1(0,T; B^0_{\infty,1})$ and $\log A \in L^2(0,T; H^{1+ε})$, with no independent compact-spectral-window hypothesis, since the endpoint velocity clock propagates the upper and lower eigenvalue bounds of A by a Lagrangian comparison argument. Thus any finite-time singularity must force loss of the endpoint velocity clock or concentration of the logarithmic conformation field. We also record the corresponding geometric extension to FENE-P, where finite extensibility adds the trace-gap barrier $ϕ_b(C) = -\log(b - {\rm tr}\, C)$.