Arbitrary-Size Global Regularity for a Reduced Oldroyd-B Active-Line Model

Abstract

We study a one-dimensional active-line equation motivated by thin stress-sheet dynamics in the high-Weissenberg Oldroyd-B regime. A positive periodic line density $\ddot{I}=m+\hat{I}\cdot$ satisfies $\ddot{I}{}_t+c{P}_0\{\ddot{I}\hat{I}›\ddot{I}-(\mathcal{H}\ddot{I})\ddot{I}{}_s\}+{\hat{I}}^3(\ddot{I}-m)=0$, where $\hat{I}›=\mathcal{H}{\partial }_s$ and $P_{0}f=f-⟨f⟩$. We prove that every strictly positive smooth initial density of arbitrary size generates a unique global smooth solution. The key is the pointwise cancellation obtained after one differentiation: for $w=\ddot{I}{}_s$, $w_t+c\ddot{I}\hat{I}›w-c(\mathcal{H}\ddot{I})w_s+{\hat{I}}^{3}w=0$. Its maximum principle controls the slope globally, while critical-drift Hölder and Schauder estimates close all higher derivatives. We also prove quantitative small-oscillation stability. Independently, an exact fourth-difference sum-of-squares identity gives ${\int }_{\mathbb{T}}\ddot{I}{}^2\hat{I}{›}^3\ddot{I}ds\ge 0$ for every smooth nonnegative density. The stronger derivative-energy sign leads, in its sharp phase-opposed form, to the cubic convolution inequality $\mathcal{C}(x)\le 2A{E}_3$, where $A={\sum }_{n\ge 1}{x}_n$, $E_3={\sum }_{n\ge 1}{n}^3{x}_n^2$, and $\mathcal{C}(x)={\sum }_{a,b\ge 1}(a+b)(a^2+ab+b^2)x_a{x}_b{x}_{a+b}$. The constant $2$ is sharp along critical $n^{-3/2}$ plateaux. We prove the inequality for several cutoff-uniform and infinite-support classes, including selected Schur, dyadic-layer, Mellin, and moment families. The unrestricted inequality remains open, but it is not needed for the global regularity theorem.

arXiv admin note: substantial text overlap with arXiv:2606.25309