On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914401349533696 |
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| author | Bulíček, Miroslav Kaplický, Petr Wintrová, Lucie |
| author_facet | Bulíček, Miroslav Kaplický, Petr Wintrová, Lucie |
| contents | We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a~bounded two-dimensional domain, subject to Dirichlet boundary conditions for velocity and temperature. The fluid obeys a power-law constitutive relation for the Cauchy stress with exponent~$p$. For $p\geq 2$ and finite-energy initial data, we establish the existence of a global-in-time weak solution that satisfies the entropy equality. The novelty of this work is the rigorous proof of time continuity of the temperature in $L^1(Ω)$, a property not previously established in this setting.
Furthermore, we prove regularity and time continuity for a weak solution of the entropy equation with a convective term and an $L^1$ right-hand side under minimal assumptions on the velocity regularity, in arbitrary spatial dimensions. We show that this continuity is equivalently described by vanishing dissipation on high level sets, a truncated variational inequality for admissible test functions, or the associated equality. This reveals the connection between energy dissipation, weak stability, and temporal regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system Bulíček, Miroslav Kaplický, Petr Wintrová, Lucie Analysis of PDEs 35Q30, 35K61, 35K92, 37L15, 76D03, 76E30 We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a~bounded two-dimensional domain, subject to Dirichlet boundary conditions for velocity and temperature. The fluid obeys a power-law constitutive relation for the Cauchy stress with exponent~$p$. For $p\geq 2$ and finite-energy initial data, we establish the existence of a global-in-time weak solution that satisfies the entropy equality. The novelty of this work is the rigorous proof of time continuity of the temperature in $L^1(Ω)$, a property not previously established in this setting. Furthermore, we prove regularity and time continuity for a weak solution of the entropy equation with a convective term and an $L^1$ right-hand side under minimal assumptions on the velocity regularity, in arbitrary spatial dimensions. We show that this continuity is equivalently described by vanishing dissipation on high level sets, a truncated variational inequality for admissible test functions, or the associated equality. This reveals the connection between energy dissipation, weak stability, and temporal regularity. |
| title | On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system |
| topic | Analysis of PDEs 35Q30, 35K61, 35K92, 37L15, 76D03, 76E30 |
| url | https://arxiv.org/abs/2510.21218 |