Three-Dimensional Positive-Cone Oldroyd-B Flows:Geometric Continuation and Residual-Work Criteria

Abstract

We prove a three-dimensional positive-cone continuation criterion for the stress-diffusion-free Oldroyd-B system on the periodic torus. Writing the positive conformation tensor as A = exp(B), we show that finite-time breakdown of a strong H^s solution, s > 5/2, can occur only through loss of the logarithmic spectral envelope of A or divergence of the endpoint vorticity clock given by the time integral of the B^0_{infty,1} norm of curl u. The proof combines compact positive-cone envelopes, endpoint Biot-Savart estimates, and high-order logarithmic conformation estimates, without using stress diffusion. We also derive a positive-cone Reynolds admissibility criterion with an exact residual-work cost. The least L^2 conformation residual needed to pay positive pressure-free residual work is determined by the entropy-dual lever G = I - A^{-1}, and this cost degenerates quantitatively near the equilibrium A = I. Together, the two criteria identify the same positive-cone obstruction in the strong and relaxed regimes: before breakdown one must control the endpoint flow clock on a compact logarithmic cone, while after passage to a relaxed description positive residual work must be paid for by an exact entropy-dual conformation defect.