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Abstract:This paper proves geometric continuation criteria for two-dimensional stress-diffusion-free Oldroyd--B and FENE-P flows. In both models the conformation tensor is transported and stretched without spatial diffusion, while the elastic stress enters the viscous velocity equation through one derivative. The positive-cone geometry is encoded by the logarithmic variable B=Log C. For Oldroyd--B this leads to two possible continuation obstructions: loss of the endpoint velocity-gradient Besov modulus and concentration of logarithmic conformation. For FENE-P the state space is smaller, D_b={C in S_{++}^2: tr C<b}, and the same argument reveals a third, finite-extensibility obstruction measured by the barrier variable phi_b(C)=-log(b-tr C). The main theorem gives a closed continuation criterion involving the velocity-gradient Besov modulus, the logarithmic conformation norm, and, in the FENE-P case, the finite-extensibility barrier norm. The proof is organized around a logarithmic good unknown which removes the top matrix-exponential commutator and a separate trace-gap equation which isolates the FENE boundary. On bounded trace windows the FENE-P criterion reduces to the Oldroyd--B criterion in the Hookean limit b->infinity; at fixed b, however, the finite-extensibility barrier is an independent high-frequency obstruction.

Submission history

From: Sai Peng [view email]
[v1] Wed, 24 Jun 2026 00:33:41 UTC (19 KB)