Global existence and optimal time-decay rates of the compressible Navier-Stokes equations with density-dependent viscosities
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| Format: | Preprint |
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2025
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| _version_ | 1866912496589209600 |
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| author | Fan, Jie Huang, Xiangdi Ni, Anchun |
| author_facet | Fan, Jie Huang, Xiangdi Ni, Anchun |
| contents | This paper is devoted to studying the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosities given by $μ=ρ^α,λ=ρ^α(α>0)$. We establish the global existence and optimal decay rates of classical solutions under the assumptions of small initial data in $L^1(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ and the viscosity constraint $|α-1|\ll 1$. The key idea of our proof lies in the combination of Green's function method, energy method and a time-decay regularity criterion. In contrast to previous works, the Sobolev norms of the spatial derivatives of the initial data may be arbitrarily large in our analysis |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global existence and optimal time-decay rates of the compressible Navier-Stokes equations with density-dependent viscosities Fan, Jie Huang, Xiangdi Ni, Anchun Analysis of PDEs 35Q30, 76N10, 35B45 This paper is devoted to studying the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosities given by $μ=ρ^α,λ=ρ^α(α>0)$. We establish the global existence and optimal decay rates of classical solutions under the assumptions of small initial data in $L^1(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ and the viscosity constraint $|α-1|\ll 1$. The key idea of our proof lies in the combination of Green's function method, energy method and a time-decay regularity criterion. In contrast to previous works, the Sobolev norms of the spatial derivatives of the initial data may be arbitrarily large in our analysis |
| title | Global existence and optimal time-decay rates of the compressible Navier-Stokes equations with density-dependent viscosities |
| topic | Analysis of PDEs 35Q30, 76N10, 35B45 |
| url | https://arxiv.org/abs/2507.16436 |