Mathematical Justification of a Baer$-$Nunziato Model for a Compressible Viscous Fluid with Phase Transition
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2025
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| author | Rohde, Christian Wendt, Florian |
| author_facet | Rohde, Christian Wendt, Florian |
| contents | In this work, we justify a Baer$-$Nunziato system including appropriate closure terms as the macroscopic description of a compressible viscous fluid that can occur in a liquid or a vapor phase in the isothermal framework. As a mathematical model for the two-phase fluid on the detailed scale we chose a non-local version of the Navier$-$Stokes$-$Korteweg equations in the one-dimensional and periodic setting. Our justification relies on anticipating the macroscopic description of the two-phase fluid as the limit system for a sequence of solutions with highly oscillating initial densities. Interpreting the density as a parametrized measure, we extract a limit system consisting of a kinetic equation for the parametrized measure and a momentum equation for the velocity. Under the assumption that the initial density distributions converge in the limit to a convex combination of Dirac-measures, we show by a uniqueness result that the parametrized measure also has to be a convex combination of Dirac-measures and, that the limit system reduces to the Baer$-$Nunziato system. This work extends existing results concerning the justification of Baer$-$Nunziato models as the macroscopic description of multi-fluid models in the sense, that we allow for phase transition effects on the detailed scale. This work also includes a new global-in-time well-posedness result for the Cauchy problem of the non-local Navier$-$Stokes$-$Korteweg equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mathematical Justification of a Baer$-$Nunziato Model for a Compressible Viscous Fluid with Phase Transition Rohde, Christian Wendt, Florian Analysis of PDEs 35Q35, 76M50, 76N06, 76T10 In this work, we justify a Baer$-$Nunziato system including appropriate closure terms as the macroscopic description of a compressible viscous fluid that can occur in a liquid or a vapor phase in the isothermal framework. As a mathematical model for the two-phase fluid on the detailed scale we chose a non-local version of the Navier$-$Stokes$-$Korteweg equations in the one-dimensional and periodic setting. Our justification relies on anticipating the macroscopic description of the two-phase fluid as the limit system for a sequence of solutions with highly oscillating initial densities. Interpreting the density as a parametrized measure, we extract a limit system consisting of a kinetic equation for the parametrized measure and a momentum equation for the velocity. Under the assumption that the initial density distributions converge in the limit to a convex combination of Dirac-measures, we show by a uniqueness result that the parametrized measure also has to be a convex combination of Dirac-measures and, that the limit system reduces to the Baer$-$Nunziato system. This work extends existing results concerning the justification of Baer$-$Nunziato models as the macroscopic description of multi-fluid models in the sense, that we allow for phase transition effects on the detailed scale. This work also includes a new global-in-time well-posedness result for the Cauchy problem of the non-local Navier$-$Stokes$-$Korteweg equations. |
| title | Mathematical Justification of a Baer$-$Nunziato Model for a Compressible Viscous Fluid with Phase Transition |
| topic | Analysis of PDEs 35Q35, 76M50, 76N06, 76T10 |
| url | https://arxiv.org/abs/2504.10161 |