Global regularity and incompressible limit of 2D compressible Navier-Stokes equations with large bulk viscosity
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arXiv
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2025
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| _version_ | 1866918136293359616 |
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| author | Liu, Shengquan Zhang, Jianwen |
| author_facet | Liu, Shengquan Zhang, Jianwen |
| contents | In this paper, we study the global regularity of large solutions with vacuum to the two-dimensional compressible Navier-Stokes equations on $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$, when the volume (bulk) viscosity coefficient $ν$ is sufficiently large. It firstly fixes a flaw in \cite[Proposition 3.3]{Danchin2023}, which concerns the $ν$-independent global $t$-weighted estimates of the solutions. Amending the proof requires non-trivially mathematical analysis. As a by-product, the incompressible limit with an explicit rate of convergence is shown, when the volume viscosity tends to infinity. In contrast to \cite[Theorem 1.3]{Danchin2019} and \cite[Corollary 1.1]{DM2017} where vacuum was excluded, the convergence rate of the incompressible limit is obtained for the global solutions with vacuum, based on some $t$-growth and singular $t$-weighted estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global regularity and incompressible limit of 2D compressible Navier-Stokes equations with large bulk viscosity Liu, Shengquan Zhang, Jianwen Analysis of PDEs 35Q35, 76N10, 35B65 In this paper, we study the global regularity of large solutions with vacuum to the two-dimensional compressible Navier-Stokes equations on $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$, when the volume (bulk) viscosity coefficient $ν$ is sufficiently large. It firstly fixes a flaw in \cite[Proposition 3.3]{Danchin2023}, which concerns the $ν$-independent global $t$-weighted estimates of the solutions. Amending the proof requires non-trivially mathematical analysis. As a by-product, the incompressible limit with an explicit rate of convergence is shown, when the volume viscosity tends to infinity. In contrast to \cite[Theorem 1.3]{Danchin2019} and \cite[Corollary 1.1]{DM2017} where vacuum was excluded, the convergence rate of the incompressible limit is obtained for the global solutions with vacuum, based on some $t$-growth and singular $t$-weighted estimates. |
| title | Global regularity and incompressible limit of 2D compressible Navier-Stokes equations with large bulk viscosity |
| topic | Analysis of PDEs 35Q35, 76N10, 35B65 |
| url | https://arxiv.org/abs/2506.22235 |