Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916785675042816 |
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| author | Basarić, Danica Giorgini, Andrea |
| author_facet | Basarić, Danica Giorgini, Andrea |
| contents | We study a Navier-Stokes/Cahn-Hilliard system modeling the evolution of a compressible binary mixture of viscous fluids undergoing phase separation. The novelty of this work is a free energy potential including the physically relevant Flory-Huggins (logarithmic) entropy, as opposed to previous studies in the literature, which only consider regular potentials with polynomial growth. Our main result establishes the existence of global-in-time weak solutions in three-dimensional bounded domains for arbitrarily large initial data. The core contribution is the derivation of new estimates for the chemical potential and the Flory-Huggins entropy arising from a density-dependent Cahn-Hilliard equation under minimal assumptions: non-negative $γ$-integrable density with $γ>\frac32$. In addition, we prove that the phase variable, which represents the difference of the mass concentrations, takes value within the physical interval $[-1,1]$ almost everywhere on the set where the density is positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing Basarić, Danica Giorgini, Andrea Analysis of PDEs We study a Navier-Stokes/Cahn-Hilliard system modeling the evolution of a compressible binary mixture of viscous fluids undergoing phase separation. The novelty of this work is a free energy potential including the physically relevant Flory-Huggins (logarithmic) entropy, as opposed to previous studies in the literature, which only consider regular potentials with polynomial growth. Our main result establishes the existence of global-in-time weak solutions in three-dimensional bounded domains for arbitrarily large initial data. The core contribution is the derivation of new estimates for the chemical potential and the Flory-Huggins entropy arising from a density-dependent Cahn-Hilliard equation under minimal assumptions: non-negative $γ$-integrable density with $γ>\frac32$. In addition, we prove that the phase variable, which represents the difference of the mass concentrations, takes value within the physical interval $[-1,1]$ almost everywhere on the set where the density is positive. |
| title | Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.07835 |