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

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.

25 pages