Sharp decay characterization for the Oldroyd-B system with only velocity dissipation

Abstract

We study the incompressible Oldroyd-B system on ${\mathbb{R}}^d$, $d\ge 2$, in the regime in which the velocity is the only dissipated quantity: the stress tensor undergoes neither relaxation nor diffusion. In this regime the operator $\mathbb{P}\mathrm{div}$ acting on symmetric stresses has an infinite-dimensional kernel consisting of exact stationary states, so no decay estimate can hold for the full stress tensor and the existing sharp-decay theory for dissipative systems does not apply. We identify the active pair $U=(u,{\Lambda }^{-1}\mathbb{P}\mathrm{div}\tau )$ as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For $-d/2<{\sigma }_1<\sigma \le d/2-1$ we prove that $\Vert U(t){\Vert }_{{\mathcal{H}}^{\sigma }}=O(t^{-(\sigma -{\sigma }_1)/2})$ holds if and only if the low-frequency part of $U_0$ belongs to ${\dot{B}}_{2,\infty }^{{\sigma }_1}$, and, for $\sigma <d/2-1$, that the two-sided estimate $\Vert U(t){\Vert }_{{\mathcal{H}}^{\sigma }}\simeq t^{-(\sigma -{\sigma }_1)/2}$ holds if and only if the low-frequency part of $U_0$ belongs to a nondegenerate subclass ${\mathfrak{B}}_{2,\infty }^{{\sigma }_1}$ of ${\dot{B}}_{2,\infty }^{{\sigma }_1}$.

arXiv comment42 pages