Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows
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arXiv
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2025
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| author | Fei, Mingwen Fei, Xiang Han, Daozhi Liu, Yadong |
| author_facet | Fei, Mingwen Fei, Xiang Han, Daozhi Liu, Yadong |
| contents | We study a quasi-incompressible Navier--Stokes/Cahn--Hilliard coupled system which describes the motion of two macroscopically immiscible incompressible viscous fluids with partial mixing in a small interfacial region and long-range interactions. The case of unmatched densities with mass-averaged velocity is considered so that the velocity field is no longer divergence-free, and the pressure enters the equation of the chemical potential. We first prove the existence of global weak solutions to the model in a three-dimensional periodic domain, for which the implicit time discretization together with a fixed-point argument to the approximate system is employed. In particular, we obtain a new regularity estimate of the order parameter by exploiting the partial damping effect of the capillary force. Then utilizing the relative entropy method, we establish the incompressible limit -- the quasi-incompressible two-phase model converges to model H as the density difference tends to zero. Crucial to the passage of the incompressible limit, due to the lack of regularity of the pressure, are some non-standard uniform-in-density difference controls of the pressure, which are derived from the structure of the momentum equations and the improved regularity of the order parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows Fei, Mingwen Fei, Xiang Han, Daozhi Liu, Yadong Analysis of PDEs 35Q35, 76T06, 76T99, 35D30, 35B25, 35Q30 We study a quasi-incompressible Navier--Stokes/Cahn--Hilliard coupled system which describes the motion of two macroscopically immiscible incompressible viscous fluids with partial mixing in a small interfacial region and long-range interactions. The case of unmatched densities with mass-averaged velocity is considered so that the velocity field is no longer divergence-free, and the pressure enters the equation of the chemical potential. We first prove the existence of global weak solutions to the model in a three-dimensional periodic domain, for which the implicit time discretization together with a fixed-point argument to the approximate system is employed. In particular, we obtain a new regularity estimate of the order parameter by exploiting the partial damping effect of the capillary force. Then utilizing the relative entropy method, we establish the incompressible limit -- the quasi-incompressible two-phase model converges to model H as the density difference tends to zero. Crucial to the passage of the incompressible limit, due to the lack of regularity of the pressure, are some non-standard uniform-in-density difference controls of the pressure, which are derived from the structure of the momentum equations and the improved regularity of the order parameter. |
| title | Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows |
| topic | Analysis of PDEs 35Q35, 76T06, 76T99, 35D30, 35B25, 35Q30 |
| url | https://arxiv.org/abs/2508.08090 |