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

  • Zhi Chen
  • Mingwen Fei
  • Lvqiao Liu
  • Jiahong Wu
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Abstract

We study the incompressible Oldroyd-B system on ℝd, d≥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 ℙdiv 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,Λ−1ℙdivτ) as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For −d/2<σ1<σ≤ d/2−1 we prove that |U(t)|ℋσ=O(t−(σ−σ1)/2) holds if and only if the low-frequency part of U0 belongs to Ḃσ12,∞, and, for σ<d/2−1, that the two-sided estimate |U(t)|ℋσ≃ t−(σ−σ1)/2 holds if and only if the low-frequency part of U0 belongs to a nondegenerate subclass Bσ12,∞ of Ḃσ12,∞.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp decay characterization for the Oldroyd-B system with only velocity dissipation}},
  author = {Zhi Chen and Mingwen Fei and Lvqiao Liu and Jiahong Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16228},
  eprint = {2610.16228},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16228},
}
RIS
TY  - UNPB
TI  - Sharp decay characterization for the Oldroyd-B system with only velocity dissipation
AU  - Zhi Chen
AU  - Mingwen Fei
AU  - Lvqiao Liu
AU  - Jiahong Wu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16228
UR  - https://arxiv.org/abs/2610.16228
AB  - We study the incompressible Oldroyd-B system on ℝ^d, d≥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 ℙdiv 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,Λ^−1ℙdivτ) as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For −d/2<σ_1<σ≤ d/2−1 we prove that |U(t)|_ℋ^σ=O(t^−(σ−σ_1)/2) holds if and only if the low-frequency part of U_0 belongs to Ḃ^σ_1_2,∞, and, for σ<d/2−1, that the two-sided estimate |U(t)|_ℋ^σ≃ t^−(σ−σ_1)/2 holds if and only if the low-frequency part of U_0 belongs to a nondegenerate subclass B^σ_1_2,∞ of Ḃ^σ_1_2,∞.
ER  -