Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the initial-boundary value problem for the 2D partially dissipative Oldroyd-B model: global well-posedness and large time stability

Zhenrong Nong, Yinghui Wang, Huancheng Yao, Shihao Zhang

TL;DR

This work analyzes the 2D Oldroyd-B model with purely horizontal viscosity, proving global well-posedness for arbitrarily large initial data on domains $[0,1]\times\mathbb{R}$ and $[0,1]\times\mathcal{T}$, and establishing long-time stability in the periodic-strip setting for small data. The authors employ energy methods combined with anisotropic Sobolev inequalities and maximum-principle structure to control the stress-velocity coupling and to compensate for the lack of vertical dissipation. Key contributions include a hierarchical set of a priori estimates (from $L^{2}$ to $H^{2}$ and time-derivatives) that close the global existence argument, and, in the periodic case, uniform-in-time bounds and exponential decay to equilibrium for small perturbations. The results highlight how domain geometry and anisotropic regularization enhance dissipation in viscoelastic flows and provide rigorous long-time behavior for this partially dissipative system.

Abstract

This paper establishes the global well-posedness of solutions to the Oldroyd-B model with purely horizontal viscosity and arbitrarily large initial data in two-dimensional settings, including the full space $\mathbb{R}^2$, the partially periodic domain $\mathcal{T}\times\mathbb{R}$ and the fully periodic torus $\mathcal{T}^2$, where $\mathcal{T}$ represents the one-dimensional periodic torus. Our analysis relies on energy methods to derive key {\it a priori} estimates that capture the anisotropic regularization induced by horizontal viscosity. Furthermore, for the cases of spatial domains $\mathcal{T}\times\mathbb{R}$ and $\mathcal{T}^2$, we further investigate the long-time behavior of solutions with small initial data. The compactness along the horizontal direction plays a pivotal role in constructing uniform-in-time estimates, ultimately leading to exponential decay of solutions as $t\to\infty$. This decay mechanism reveals how geometric constraints enhance the dissipation in viscoelastic flows.

On the initial-boundary value problem for the 2D partially dissipative Oldroyd-B model: global well-posedness and large time stability

TL;DR

This work analyzes the 2D Oldroyd-B model with purely horizontal viscosity, proving global well-posedness for arbitrarily large initial data on domains and , and establishing long-time stability in the periodic-strip setting for small data. The authors employ energy methods combined with anisotropic Sobolev inequalities and maximum-principle structure to control the stress-velocity coupling and to compensate for the lack of vertical dissipation. Key contributions include a hierarchical set of a priori estimates (from to and time-derivatives) that close the global existence argument, and, in the periodic case, uniform-in-time bounds and exponential decay to equilibrium for small perturbations. The results highlight how domain geometry and anisotropic regularization enhance dissipation in viscoelastic flows and provide rigorous long-time behavior for this partially dissipative system.

Abstract

This paper establishes the global well-posedness of solutions to the Oldroyd-B model with purely horizontal viscosity and arbitrarily large initial data in two-dimensional settings, including the full space , the partially periodic domain and the fully periodic torus , where represents the one-dimensional periodic torus. Our analysis relies on energy methods to derive key {\it a priori} estimates that capture the anisotropic regularization induced by horizontal viscosity. Furthermore, for the cases of spatial domains and , we further investigate the long-time behavior of solutions with small initial data. The compactness along the horizontal direction plays a pivotal role in constructing uniform-in-time estimates, ultimately leading to exponential decay of solutions as . This decay mechanism reveals how geometric constraints enhance the dissipation in viscoelastic flows.

Paper Structure

This paper contains 14 sections, 12 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main results
  4. Main ideas
  5. Preliminaries
  6. Global well-posedness
  7. The estimates of (u,tau,eta)
  8. The first-order estimates of (u,tau,eta)
  9. The second-order estimates of u, tau, eta
  10. Proof of Theorem \ref{['thm4']}
  11. Proof of Theorem \ref{['thm5']} and Theorem \ref{['thm6']}.
  12. The uniform estimates of u, tau, bar eta
  13. The proof of Theorem \ref{['thm5']}.
  14. The proof of Theorem \ref{['thm6']}.

Key Result

Theorem 1.1

Assume that $u_0\in H^2(\Omega)$, $\eta_0$, $\tau_0 = \mathbb{T}_0 - \eta_0\mathbb{I}\in L^{1}(\Omega)\cap H^2(\Omega)$ with $\mathop{\mathrm{\mathrm{div}}}\nolimits u=0$, $\eta_0$ non-negative, and $\mathbb{T}_0$ non-negative definite. For arbitrary given $T>0$, problem EQ-t1h--OB_as_boundary_1 adm

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.3
  • Lemma 2.1: Lemma 2.3 in Dong-Wu-Xu2021
  • Remark 2.1
  • Proposition 2.1: Local well-posedness
  • Remark 2.2
  • ...and 15 more