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

TL;DR

This paper investigates the ill-posedness of the Euler-Oldroyd-B system, introduces a Voigt regularization to stabilize it, and proves convergence to Navier-Stokes equations as the regularization parameter vanishes.

Contribution

The paper demonstrates ill-posedness of the Euler-Oldroyd-B system, introduces a regularization that ensures long-time well-posedness, and establishes the high Weissenberg number limit convergence to Navier-Stokes.

Findings

Euler-Oldroyd-B system is ill-posed in Sobolev spaces.

Voigt regularization stabilizes the system and ensures long-time well-posedness.

Solutions of the regularized model converge to Navier-Stokes solutions as regularization vanishes.

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.