On global existence and large-time behaviour of weak solutions to the compressible barotropic Navier--Stokes Equations on $\mathbb{T}^2$ with density-dependent bulk viscosity: beyond the Va\uıgant--Kazhikhov regime
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| Format: | Preprint |
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2024
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| _version_ | 1866916814111375360 |
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| author | Li, Siran Yang, Jianing |
| author_facet | Li, Siran Yang, Jianing |
| contents | We are concerned with the compressible barotropic Navier--Stokes equations for a $γ$-law gas with density-dependent bulk viscosity coefficient $λ=λ(ρ)=ρ^β$ on the two-dimensional periodic domain $\mathbb{T}^2$. The global existence of weak solutions with initial density bounded away from zero and infinity for $β>3$, $γ>1$ has been established by Va\uıgant--Kazhikhov [\textit{Sib. Math. J.} 36 (1995), 1283--1316]. When $γ=β>3$, the large-time behaviour of the weak solutions and, in particular, the absence of formation of vacuum and concentration of density as $t \to \infty$, has been proved by Perepelitsa [\textit{SIAM J. Math. Anal.} 39 (2007/08), 1344--1365]. Huang--Li [\textit{J. Math. Pures Appl.} 106 (2016), 123--154] extended these results by establishing the global existence of weak solutions and large-time behaviour under the assumptions $β>3/2$, $1< γ<4β-3$, and that the initial density stays away from infinity (but may contain vacuum).
Improving upon the works listed above, we prove that in the regime of parameters as in Huang--Li, namely that $β>3/2$ and $1< γ<4β-3$, if the density has no vacuum or concentration at $t=0$, then it stays away from zero and infinity at all later time $t \in ]0,\infty[$. Moreover, under the mere assumption that $β>1$ and $γ>1$, we establish the global existence of weak solutions, thus pushing the global existence theory of the barotropic Navier--Stokes equations on $\mathbb{T}^2$ to the most general setting to date. One of the key ingredients of our proof is a novel application -- motivated by the recent work due to Danchin--Mucha [\textit{Comm. Pure Appl. Math.} 76 (2023), 3437--3492] -- of Desjardins' logarithmic interpolation inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On global existence and large-time behaviour of weak solutions to the compressible barotropic Navier--Stokes Equations on $\mathbb{T}^2$ with density-dependent bulk viscosity: beyond the Va\uıgant--Kazhikhov regime Li, Siran Yang, Jianing Analysis of PDEs Fluid Dynamics We are concerned with the compressible barotropic Navier--Stokes equations for a $γ$-law gas with density-dependent bulk viscosity coefficient $λ=λ(ρ)=ρ^β$ on the two-dimensional periodic domain $\mathbb{T}^2$. The global existence of weak solutions with initial density bounded away from zero and infinity for $β>3$, $γ>1$ has been established by Va\uıgant--Kazhikhov [\textit{Sib. Math. J.} 36 (1995), 1283--1316]. When $γ=β>3$, the large-time behaviour of the weak solutions and, in particular, the absence of formation of vacuum and concentration of density as $t \to \infty$, has been proved by Perepelitsa [\textit{SIAM J. Math. Anal.} 39 (2007/08), 1344--1365]. Huang--Li [\textit{J. Math. Pures Appl.} 106 (2016), 123--154] extended these results by establishing the global existence of weak solutions and large-time behaviour under the assumptions $β>3/2$, $1< γ<4β-3$, and that the initial density stays away from infinity (but may contain vacuum). Improving upon the works listed above, we prove that in the regime of parameters as in Huang--Li, namely that $β>3/2$ and $1< γ<4β-3$, if the density has no vacuum or concentration at $t=0$, then it stays away from zero and infinity at all later time $t \in ]0,\infty[$. Moreover, under the mere assumption that $β>1$ and $γ>1$, we establish the global existence of weak solutions, thus pushing the global existence theory of the barotropic Navier--Stokes equations on $\mathbb{T}^2$ to the most general setting to date. One of the key ingredients of our proof is a novel application -- motivated by the recent work due to Danchin--Mucha [\textit{Comm. Pure Appl. Math.} 76 (2023), 3437--3492] -- of Desjardins' logarithmic interpolation inequality. |
| title | On global existence and large-time behaviour of weak solutions to the compressible barotropic Navier--Stokes Equations on $\mathbb{T}^2$ with density-dependent bulk viscosity: beyond the Va\uıgant--Kazhikhov regime |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2412.01226 |