Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation
Abstract
The data-to-solution map of the incompressible Euler equations is continuous but not uniformly continuous on Sobolev spaces, whereas viscosity makes the Navier--Stokes solution map Lipschitz continuous on small data, uniformly in time. This paper identifies the behavior of the solution map for a hybrid system sitting between these two extremes: the incompressible Oldroyd--B model in ℝd, d=2,3, in the high-Reynolds-number, high-Weissenberg-number regime. The velocity equation is a forced Euler equation and the only dissipation is a fractional diffusion η(−Δ)β, 1/2≤β≤1, acting on stress. We prove that, on the ball of small Hs data (s>1+d/2) on which global solutions exist, the solution map is Lipschitz continuous with respect to the Hr-norm, r=s−(2β−1), uniformly for all time, and Lipschitz into the natural dissipation spaces; the constant is explicit and tends to √3 as the size of the data tends to zero. Consequently the map is Hölder continuous into Cb([0,∞);Hs') for every s'<s, and is Lipschitz in Hs itself when β=1/2. Thus, although the velocity field is only transported, the solution map behaves like that of the Navier--Stokes equations rather than that of the Euler equations. The proof exploits the damped wave structure generated by the coupling between u and ∇·τ through a Lyapunov functional with a cross term (δu,∇·δτ)Hr−β, which transfers the stress dissipation to the velocity difference, together with a low-frequency interpolation argument that keeps all Gronwall coefficients integrable on (0,∞). The loss of 2β−1 derivatives is exactly the gap between the hidden velocity dissipation of order 2(1−β) and the dissipation required by the Lipschitz estimate, and we show that it cannot be reduced by the energy method.
Related
- A Good-Vorticity Low-Regularity Integrator for the Inviscid Oldroyd--B System with Stress Diffusion
- Global Well-Posedness for the 3-D Inviscid Oldroyd-B Model with Fractional Stress Diffusion
- Sharp decay characterization for the Oldroyd-B system with only velocity dissipation
- Global regularity for the unidirectional Euler-alignment system with supercritical dissipation
- Regularising Flows and Singular Fractional SDEs with Superlinear Dissipation
BibTeX
@misc{misiolek2026regularity,
title = {{Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation}},
author = {Gerard Misiołek and Jiahong Wu and Mingyu Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34498},
eprint = {2610.34498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34498},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation
AU - Gerard Misiołek
AU - Jiahong Wu
AU - Mingyu Yu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34498
UR - https://arxiv.org/abs/2610.34498
AB - The data-to-solution map of the incompressible Euler equations is continuous but not uniformly continuous on Sobolev spaces, whereas viscosity makes the Navier--Stokes solution map Lipschitz continuous on small data, uniformly in time. This paper identifies the behavior of the solution map for a hybrid system sitting between these two extremes: the incompressible Oldroyd--B model in ℝ^d, d=2,3, in the high-Reynolds-number, high-Weissenberg-number regime. The velocity equation is a forced Euler equation and the only dissipation is a fractional diffusion η(−Δ)^β, 1/2≤β≤1, acting on stress. We prove that, on the ball of small H^s data (s>1+d/2) on which global solutions exist, the solution map is Lipschitz continuous with respect to the H^r-norm, r=s−(2β−1), uniformly for all time, and Lipschitz into the natural dissipation spaces; the constant is explicit and tends to √3 as the size of the data tends to zero. Consequently the map is Hölder continuous into C_b([0,∞);H^s') for every s'<s, and is Lipschitz in H^s itself when β=1/2. Thus, although the velocity field is only transported, the solution map behaves like that of the Navier--Stokes equations rather than that of the Euler equations. The proof exploits the damped wave structure generated by the coupling between u and ∇·τ through a Lyapunov functional with a cross term (δu,∇·δτ)_H^r−β, which transfers the stress dissipation to the velocity difference, together with a low-frequency interpolation argument that keeps all Gronwall coefficients integrable on (0,∞). The loss of 2β−1 derivatives is exactly the gap between the hidden velocity dissipation of order 2(1−β) and the dissipation required by the Lipschitz estimate, and we show that it cannot be reduced by the energy method.
ER -