Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence
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| Format: | Preprint |
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2026
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| author | Cheng, Fan Lasarzik, Robert Thomas, Marita |
| author_facet | Cheng, Fan Lasarzik, Robert Thomas, Marita |
| contents | In [Cheng, Lasarzik, Thomas 2025 ARXIV-Preprint 2509.25508], we studied a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids in the framework of dissipative solutions using a logarithmic potential for the phase-field variable. This choice of potential has the effect that the fluid mixture cannot fully separate into two pure phases. The notion of dissipative solutions is based on a relative energy-dissipation inequality featuring a suitable regularity weight. In this way, this is a very weak solution concept. In the present work, we study the well-posedness of the geodynamical two-phase flow in the notion of energy-variational solutions. They feature an additional scalar energy variable that majorizes the system energy along solutions and they are further characterized by a variational inequality that combines an energy-dissipation estimate with the weak formulation of the system adding an error term that accounts for the mismatch between the energy variable and the system energy multiplied by a suitable regularity weight. We give a comparison of these two concepts. We further study different phase-field potentials for the geodynamical two-phase flow model. In particular, we address the variational limit from a potential with a logarithmic contribution to a double-obstacle potential, then also allowing for the emergence of pure phases. This study underlines that, thanks to its structure, the energy-variational solution is better suited for variational convergence methods than the dissipative solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_08625 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence Cheng, Fan Lasarzik, Robert Thomas, Marita Analysis of PDEs 35A15, 35A35, 35D99, 35M86, 35Q86, 76Txx, 76M30 In [Cheng, Lasarzik, Thomas 2025 ARXIV-Preprint 2509.25508], we studied a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids in the framework of dissipative solutions using a logarithmic potential for the phase-field variable. This choice of potential has the effect that the fluid mixture cannot fully separate into two pure phases. The notion of dissipative solutions is based on a relative energy-dissipation inequality featuring a suitable regularity weight. In this way, this is a very weak solution concept. In the present work, we study the well-posedness of the geodynamical two-phase flow in the notion of energy-variational solutions. They feature an additional scalar energy variable that majorizes the system energy along solutions and they are further characterized by a variational inequality that combines an energy-dissipation estimate with the weak formulation of the system adding an error term that accounts for the mismatch between the energy variable and the system energy multiplied by a suitable regularity weight. We give a comparison of these two concepts. We further study different phase-field potentials for the geodynamical two-phase flow model. In particular, we address the variational limit from a potential with a logarithmic contribution to a double-obstacle potential, then also allowing for the emergence of pure phases. This study underlines that, thanks to its structure, the energy-variational solution is better suited for variational convergence methods than the dissipative solution. |
| title | Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence |
| topic | Analysis of PDEs 35A15, 35A35, 35D99, 35M86, 35Q86, 76Txx, 76M30 |
| url | https://arxiv.org/abs/2601.08625 |