Long-time behavior to the 3D isentropic compressible Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909375254233088 |
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| author | Wu, Guochun Zhong, Xin |
| author_facet | Wu, Guochun Zhong, Xin |
| contents | We are concerned with the long-time behavior of classical solutions to the isentropic compressible Navier-Stokes equations in $\mathbb R^3$. Our main results and innovations can be stated as follows: Under the assumption that the density $ρ({\bf{x}}, t)$ verifies $ρ({\bf{x}},0)\geq c>0$ and $\sup_{t\geq 0}\|ρ(\cdot,t)\|_{L^\infty}\leq M$, we establish the optimal decay rates of the solutions. This greatly improves the previous result (Arch. Ration. Mech. Anal. 234 (2019), 1167--1222), where the authors require an extra hypothesis $\sup_{t\geq 0}\|ρ(\cdot,t)\|_{C^α}\leq M$ with $α$ arbitrarily small. We prove that the vacuum state will persist for any time provided that the initial density contains vacuum and the far-field density is away from vacuum, which extends the torus case obtained in (SIAM J. Math. Anal. 55 (2023), 882--899) to the whole space. We derive the decay properties of the solutions with vacuum as far-field density. This in particular gives the first result concerning the $L^\infty$-decay with a rate $(1+t)^{-1}$ for the pressure to the 3D compressible Navier-Stokes equations in the presence of vacuum. The main ingredient of the proof relies on the techniques involving blow-up criterion, a key time-independent positive upper and lower bounds of the density, and a regularity interpolation trick. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time behavior to the 3D isentropic compressible Navier-Stokes equations Wu, Guochun Zhong, Xin Analysis of PDEs We are concerned with the long-time behavior of classical solutions to the isentropic compressible Navier-Stokes equations in $\mathbb R^3$. Our main results and innovations can be stated as follows: Under the assumption that the density $ρ({\bf{x}}, t)$ verifies $ρ({\bf{x}},0)\geq c>0$ and $\sup_{t\geq 0}\|ρ(\cdot,t)\|_{L^\infty}\leq M$, we establish the optimal decay rates of the solutions. This greatly improves the previous result (Arch. Ration. Mech. Anal. 234 (2019), 1167--1222), where the authors require an extra hypothesis $\sup_{t\geq 0}\|ρ(\cdot,t)\|_{C^α}\leq M$ with $α$ arbitrarily small. We prove that the vacuum state will persist for any time provided that the initial density contains vacuum and the far-field density is away from vacuum, which extends the torus case obtained in (SIAM J. Math. Anal. 55 (2023), 882--899) to the whole space. We derive the decay properties of the solutions with vacuum as far-field density. This in particular gives the first result concerning the $L^\infty$-decay with a rate $(1+t)^{-1}$ for the pressure to the 3D compressible Navier-Stokes equations in the presence of vacuum. The main ingredient of the proof relies on the techniques involving blow-up criterion, a key time-independent positive upper and lower bounds of the density, and a regularity interpolation trick. |
| title | Long-time behavior to the 3D isentropic compressible Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.17439 |