On the initial-boundary value problem for the 2D partially dissipative Oldroyd-B model: global well-posedness and large time stability
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929755594424320 |
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| author | Nong, Zhenrong Wang, Yinghui Yao, Huancheng Zhang, Shihao |
| author_facet | Nong, Zhenrong Wang, Yinghui Yao, Huancheng Zhang, Shihao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the initial-boundary value problem for the 2D partially dissipative Oldroyd-B model: global well-posedness and large time stability Nong, Zhenrong Wang, Yinghui Yao, Huancheng Zhang, Shihao Analysis of PDEs 35Q35, 76A10, 76A05, 76B03 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. |
| title | On the initial-boundary value problem for the 2D partially dissipative Oldroyd-B model: global well-posedness and large time stability |
| topic | Analysis of PDEs 35Q35, 76A10, 76A05, 76B03 |
| url | https://arxiv.org/abs/2503.09067 |