A Good-Vorticity Low-Regularity Integrator for the Inviscid Oldroyd--B System with Stress Diffusion
Abstract
We establish sharp convergence results for a sequential semi-implicit good-vorticity integrator for the two-dimensional inviscid Oldroyd--B system with stress diffusion on the torus. For exact solutions satisfying u∈ CtB1+α∞,1 and τ∈ CtBα∞,1∩ L1tB2+α∞,1, 0≤α≤2, we prove convergence in the critical norms B1∞,1× B0∞,1, uniformly over the time nodes on every finite interval where these regularity assumptions hold, including the critical endpoint α=0. For 0<α≤2, we obtain the rate O(hα/2) without any additional time Hölder assumption, reaching first order at α=2. These algebraic exponents are optimal for this Eulerian scheme over the stated solution classes, as shown by fixed-data lower bounds in the case a=b=0 with nonzero velocity mean. For a fixed shear solution at critical regularity, we establish matching logarithmic upper and lower bounds on the numerical error, ruling out every positive algebraic convergence rate. We also prove that the gain of one stress derivative at the time nodes is optimal when the velocity is controlled in B1∞,1, even for smooth initial stress.
Related
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BibTeX
@misc{cheng2026goodvorticity,
title = {{A Good-Vorticity Low-Regularity Integrator for the Inviscid Oldroyd--B System with Stress Diffusion}},
author = {Xinyu Cheng and Zhaonan Luo and Zhaojie Yang and Cheng Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41250},
eprint = {2610.41250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41250},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Good-Vorticity Low-Regularity Integrator for the Inviscid Oldroyd--B System with Stress Diffusion
AU - Xinyu Cheng
AU - Zhaonan Luo
AU - Zhaojie Yang
AU - Cheng Yuan
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41250
UR - https://arxiv.org/abs/2610.41250
AB - We establish sharp convergence results for a sequential semi-implicit good-vorticity integrator for the two-dimensional inviscid Oldroyd--B system with stress diffusion on the torus. For exact solutions satisfying u∈ C_tB^1+α_∞,1 and τ∈ C_tB^α_∞,1∩ L^1_tB^2+α_∞,1, 0≤α≤2, we prove convergence in the critical norms B^1_∞,1× B^0_∞,1, uniformly over the time nodes on every finite interval where these regularity assumptions hold, including the critical endpoint α=0. For 0<α≤2, we obtain the rate O(h^α/2) without any additional time Hölder assumption, reaching first order at α=2. These algebraic exponents are optimal for this Eulerian scheme over the stated solution classes, as shown by fixed-data lower bounds in the case a=b=0 with nonzero velocity mean. For a fixed shear solution at critical regularity, we establish matching logarithmic upper and lower bounds on the numerical error, ruling out every positive algebraic convergence rate. We also prove that the gain of one stress derivative at the time nodes is optimal when the velocity is controlled in B^1_∞,1, even for smooth initial stress.
ER -