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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.14914 |
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| _version_ | 1866910054103384064 |
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| author | Liang, Chen Luo, Zhaonan Yin, Zhaoyang |
| author_facet | Liang, Chen Luo, Zhaonan Yin, Zhaoyang |
| contents | In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^β$ with $\frac{1}{2}<β<1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\leβ<\frac{3}{4}$. We find a phenomenon that $\|(a,u)\|_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\leβ<\frac{3}{4}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations Liang, Chen Luo, Zhaonan Yin, Zhaoyang Analysis of PDEs In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^β$ with $\frac{1}{2}<β<1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\leβ<\frac{3}{4}$. We find a phenomenon that $\|(a,u)\|_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\leβ<\frac{3}{4}$. |
| title | Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.14914 |