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Large global solutions to the Oldroyd-B model with dissipation

Tao Liang, Yongsheng Li, Xiaoping Zhai

TL;DR

The paper proves two principal results for the Oldroyd-B model with stress diffusion: (i) in $ ext{R}^d$, global strong solutions exist for stress data of arbitrary size provided the initial velocity is sufficiently small in a Besov-frame sense, with the smallness depending exponentially on the divergence-free part of the initial stress; (ii) on the torus $ ext{T}^d$, global well-posedness holds for small data in $H^3$ with solutions that decay exponentially in time. The analysis combines a weighted Chemin-Lerner Besov framework with a detailed frequency-decomposed energy method, decoupling the system via the Leray projection into incompressible and stress-driven parts, and exploiting a novel high-frequency damping mechanism through an effective velocity $G=- ext{Q} au- frac12 abla riangle^{-1}u$. The main contributions include allowing arbitrarily large initial stress in critical Besov spaces, minimal velocity constraints, and a unified treatment that covers the full constitutive law $g_{eta}( au, abla u)$ and three-dimensional global regularity, contrasted against finite-time singularity in Euler flows. These results illuminate the stabilizing influence of polymeric stresses and provide robust long-time behavior descriptions for viscoelastic flows in both unbounded and periodic domains.

Abstract

In the first part of this work, we investigate the Cauchy problem for the $d$-dimensional incompressible Oldroyd-B model with dissipation in the stress tensor equation. By developing a weighted Chemin-Lerner framework combined with a refined energy argument, we prove the existence and uniqueness of global solutions for the system under a mild constraint on the initial velocity field, while allowing a broad class of large initial data for the stress tensor. Notably, our analysis accommodates general divergence-free initial stress tensors ( $\mathrm{div}τ_0=0$) and significantly relaxes the requirements on initial velocities compared to classical fluid models. This stands in sharp contrast to the finite-time singularity formation observed in the incompressible Euler equations, even for small initial data, thereby highlighting the intrinsic stabilizing role of the stress tensor in polymeric fluid dynamics. The second part of this paper focuses on the small-data regime. Through a systematic exploitation of the perturbative structure of the system, we establish global well-posedness and quantify the long-time behavior of solutions in Sobolev spaces $H^3(\mathbb{T}^d)$. Specifically, we derive exponential decay rates for perturbations, demonstrating how the dissipative mechanisms inherent to the Oldroyd-B model govern the asymptotic stability of the system.

Large global solutions to the Oldroyd-B model with dissipation

TL;DR

The paper proves two principal results for the Oldroyd-B model with stress diffusion: (i) in , global strong solutions exist for stress data of arbitrary size provided the initial velocity is sufficiently small in a Besov-frame sense, with the smallness depending exponentially on the divergence-free part of the initial stress; (ii) on the torus , global well-posedness holds for small data in with solutions that decay exponentially in time. The analysis combines a weighted Chemin-Lerner Besov framework with a detailed frequency-decomposed energy method, decoupling the system via the Leray projection into incompressible and stress-driven parts, and exploiting a novel high-frequency damping mechanism through an effective velocity . The main contributions include allowing arbitrarily large initial stress in critical Besov spaces, minimal velocity constraints, and a unified treatment that covers the full constitutive law and three-dimensional global regularity, contrasted against finite-time singularity in Euler flows. These results illuminate the stabilizing influence of polymeric stresses and provide robust long-time behavior descriptions for viscoelastic flows in both unbounded and periodic domains.

Abstract

In the first part of this work, we investigate the Cauchy problem for the -dimensional incompressible Oldroyd-B model with dissipation in the stress tensor equation. By developing a weighted Chemin-Lerner framework combined with a refined energy argument, we prove the existence and uniqueness of global solutions for the system under a mild constraint on the initial velocity field, while allowing a broad class of large initial data for the stress tensor. Notably, our analysis accommodates general divergence-free initial stress tensors ( ) and significantly relaxes the requirements on initial velocities compared to classical fluid models. This stands in sharp contrast to the finite-time singularity formation observed in the incompressible Euler equations, even for small initial data, thereby highlighting the intrinsic stabilizing role of the stress tensor in polymeric fluid dynamics. The second part of this paper focuses on the small-data regime. Through a systematic exploitation of the perturbative structure of the system, we establish global well-posedness and quantify the long-time behavior of solutions in Sobolev spaces . Specifically, we derive exponential decay rates for perturbations, demonstrating how the dissipative mechanisms inherent to the Oldroyd-B model govern the asymptotic stability of the system.

Paper Structure

This paper contains 22 sections, 2 theorems, 207 equations.

Table of Contents

  1. Introduction and the main results
  2. Model and synopsis of related studies
  3. Main results
  4. Large global solutions in Besov spaces
  5. Global well-posedness on the Torus
  6. Organization of the paper
  7. Notations
  8. Preliminaries
  9. Littlewood-Paley decomposition and Besov spaces
  10. Analysis tools in Besov spaces
  11. The proof of Theorem \ref{['theorem1.1']}
  12. Estimates for the incompressible part $\mathbb{P}\tau_F$
  13. Estimates for the incompressible part $\mathbb{P} \bar{\tau}$
  14. Estimates of $( \mathbf{u} _{\lambda}, \delta_{\lambda})$ in Low-frequency
  15. Estimates of $( \mathbf{u} _{\lambda}, \delta_{\lambda})$ in high-frequency
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $d\geq 2$ and $1 \le p \le 2d.$ For any $\mathbf{u} ^\ell_0 \in \dot B^{\frac{d}{p}-1}_{p,1}({\mathbb R}^d)$, $\mathbf{u} ^h_0 \in \dot B^{\frac{d}{p} + 1}_{p,1}({\mathbb R}^d)$, $\tau_0\in \dot B^{\frac{d}{p}}_{p,1}({\mathbb R}^d)$, there exist a small positive constant $c_0$ and a large pos then the system modle with initial admits a unique global-in-time solution $( \mathbf{u} , \tau)$

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • proof
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