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 $\rho=m+\eta$ satisfies $\rho_t+cP_0\{\rho\Lambda\rho-(\mathcal H\rho)\rho_s\}+\gamma(\rho-m)=0$, where $\Lambda=\mathcal H\partial_s$ and $P_0f=f-\langle f\rangle$. 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 $w=\rho_s$, $w_t+c\rho\Lambda w-c(\mathcal H\rho)w_s+\gamma w=0$. Its maximum principle controls the slope globally, while critical-drift H\"older and Schauder estimates close all higher derivatives. We also prove quantitative small-oscillation stability. Independently, an exact fourth-difference sum-of-squares identity gives $\int_{\mathbb T}\rho^2\Lambda^3\rho\,ds\geq0$ for every smooth nonnegative density. The stronger derivative-energy sign leads, in its sharp phase-opposed form, to the cubic convolution inequality $\mathcal C(x)\leq2AE_3$, where $A=\sum_{n\geq1}x_n$, $E_3=\sum_{n\geq1}n^3x_n^2$, and $\mathcal C(x)=\sum_{a,b\geq1}(a+b)(a^2+ab+b^2)x_ax_bx_{a+b}$. 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.