On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910486254059520 |
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| author | Chiodaroli, Elisabetta Feireisl, Eduard |
| author_facet | Chiodaroli, Elisabetta Feireisl, Eduard |
| contents | We consider the Navier-Stokes-Fourier system on an unbounded domain in the Euclidean space $R^3$, supplemented by the far field conditions for the phase variables, specifically: $ρ\to 0,\ \vartheta \to \vartheta_\infty, \ u \to 0$ as $\ |x| \to \infty$. We study the long time behaviour of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium $ρ_s = 0,\ \vartheta_s = \vartheta_\infty,\ u_s = 0$ in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09587 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains Chiodaroli, Elisabetta Feireisl, Eduard Analysis of PDEs We consider the Navier-Stokes-Fourier system on an unbounded domain in the Euclidean space $R^3$, supplemented by the far field conditions for the phase variables, specifically: $ρ\to 0,\ \vartheta \to \vartheta_\infty, \ u \to 0$ as $\ |x| \to \infty$. We study the long time behaviour of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium $ρ_s = 0,\ \vartheta_s = \vartheta_\infty,\ u_s = 0$ in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved. |
| title | On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.09587 |