Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914115573776384 |
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| author | He, Jingning Wu, Hao |
| author_facet | He, Jingning Wu, Hao |
| contents | We analyze a diffuse interface model that describes the dynamics of incompressible viscous two-phase flows, incorporating mechanisms such as chemotaxis, active transport, and long-range interactions of Oono's type. The evolution system couples the Navier--Stokes equations for the volume-averaged fluid velocity $\bm{v}$, a convective Cahn--Hilliard equation for the phase-field variable $φ$, and an advection-diffusion equation for the density of a chemical substance $σ$. For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\bm{v}, φ, σ)$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable $φ$ transitions into a strong solution, whereas the chemical density $σ$ only partially regularizes. Subsequently, the velocity field $\bm{v}$ becomes regular after a sufficiently large time, followed by a further regularization of the chemical density $σ$, which in turn enhances the spatial regularity of $φ$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\to +\infty$. Our analysis uncovers the influence of chemotaxis, active transport, and long-range interactions on the propagation of regularity at different stages of time. The proof relies on several key points, including a novel regularity result for a convective Cahn--Hilliard--diffusion system with a velocity field $\bm{v}$ of Leray type, the strict separation property of $φ$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(φ, σ)$ with a given velocity, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport He, Jingning Wu, Hao Analysis of PDEs 35A01, 35A02, 35K35, 35Q92, 76D05 We analyze a diffuse interface model that describes the dynamics of incompressible viscous two-phase flows, incorporating mechanisms such as chemotaxis, active transport, and long-range interactions of Oono's type. The evolution system couples the Navier--Stokes equations for the volume-averaged fluid velocity $\bm{v}$, a convective Cahn--Hilliard equation for the phase-field variable $φ$, and an advection-diffusion equation for the density of a chemical substance $σ$. For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\bm{v}, φ, σ)$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable $φ$ transitions into a strong solution, whereas the chemical density $σ$ only partially regularizes. Subsequently, the velocity field $\bm{v}$ becomes regular after a sufficiently large time, followed by a further regularization of the chemical density $σ$, which in turn enhances the spatial regularity of $φ$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\to +\infty$. Our analysis uncovers the influence of chemotaxis, active transport, and long-range interactions on the propagation of regularity at different stages of time. The proof relies on several key points, including a novel regularity result for a convective Cahn--Hilliard--diffusion system with a velocity field $\bm{v}$ of Leray type, the strict separation property of $φ$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(φ, σ)$ with a given velocity, respectively. |
| title | Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport |
| topic | Analysis of PDEs 35A01, 35A02, 35K35, 35Q92, 76D05 |
| url | https://arxiv.org/abs/2408.09514 |