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2D incompressible inviscid Oldroyd-B equations: ill-posedness, long time existence, and high Weissenberg number limit

Xin Liu, Weinan Wang

Abstract

In this paper, we consider the high-Weissenberg number limit of a Voigt-regularized two-dimensional Oldroyd-B model for viscoelastic fluids. We first demonstrate that the Euler-Oldroyd-B system is both linearly and nonlinearly ill-posed in Sobolev spaces, exhibiting Hadamard instability. Then, we introduce a Voigt-type regularization on the stress tensor, which stabilizes the system. For the regularized model, we establish long time ($T \sim \mathcal O(\varepsilon^{-2/3})$) well-posedness and uniform energy estimates with respect to the relaxation parameter $\varepsilon>0$. Lastly, we prove that, as $\varepsilon \to 0$, the solutions converge to a solution of the 2-d incompressible Navier-Stokes equations over time intervals of size $\mathcal O(\varepsilon^{-2/3})$. The proof relies on a decomposition of the stress tensor, high-order energy estimates, and a detailed analysis of the nonlinear coupling terms. Our results provide a mathematical justification for the Newtonian limit of a regularized viscoelastic fluid model that is otherwise ill-posed.

2D incompressible inviscid Oldroyd-B equations: ill-posedness, long time existence, and high Weissenberg number limit

Abstract

In this paper, we consider the high-Weissenberg number limit of a Voigt-regularized two-dimensional Oldroyd-B model for viscoelastic fluids. We first demonstrate that the Euler-Oldroyd-B system is both linearly and nonlinearly ill-posed in Sobolev spaces, exhibiting Hadamard instability. Then, we introduce a Voigt-type regularization on the stress tensor, which stabilizes the system. For the regularized model, we establish long time () well-posedness and uniform energy estimates with respect to the relaxation parameter . Lastly, we prove that, as , the solutions converge to a solution of the 2-d incompressible Navier-Stokes equations over time intervals of size . The proof relies on a decomposition of the stress tensor, high-order energy estimates, and a detailed analysis of the nonlinear coupling terms. Our results provide a mathematical justification for the Newtonian limit of a regularized viscoelastic fluid model that is otherwise ill-posed.
Paper Structure (14 sections, 3 theorems, 103 equations)

This paper contains 14 sections, 3 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Euler-Oldroyd-B system
  3. Main contribution and highlights
  4. Two-tier energy perspective.
  5. Ill-posedness of the inviscid Euler--Oldroyd--B system in Sobolev spaces.
  6. Voigt regularization
  7. Main theorems
  8. Ill-posedness of the first order Oldroyd-b system \ref{['sys:oldroyd-b']}
  9. Linear ill-posedness
  10. Nonliear ill-posedness
  11. Low norm estimate for the tightened variables
  12. High norm estimate for the relaxed variables
  13. Long time existence
  14. High Weissenberg number imit

Key Result

Theorem 1

System sys:oldroyd-b with arbitrary fixed $\varepsilon$ is ill-posed in any Sobolev space $H^s$, $s \geq 3$, in the sense of Hadamard.

Theorems & Definitions (6)

  • Theorem 1: Nonlinear ill-posedness
  • Theorem 2: Long time existence
  • Theorem 3: High Weissenberg number limit
  • proof
  • proof : Proof of Theorem \ref{['thm:long-time']}
  • proof