Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Mass Transport: Cross Diffusion versus Logistic Degradation
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arXiv
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| Format: | Preprint |
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2024
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| author | Giorgini, Andrea He, Jingning Wu, Hao |
| author_facet | Giorgini, Andrea He, Jingning Wu, Hao |
| contents | We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows influenced by interactions with a soluble chemical substance, encompassing the chemotaxis effect, mass transport, and reactions. In the resulting coupled evolutionary system, the macroscopic fluid velocity field $\boldsymbol{v}$ satisfies a Navier--Stokes system driven by a capillary force, the phase field variable $φ$ is governed by a convective Cahn--Hilliard equation incorporating a mass source and a singular potential (e.g., the Flory--Huggins type), and the chemical concentration $σ$ obeys an advection-reaction-diffusion equation with logistic degradation, exhibiting a cross-diffusion structure akin to the Keller--Segel model for chemotaxis. Under general structural assumptions, we establish the existence of global weak solutions to the initial boundary value problem within a bounded smooth domain $Ω\subset \mathbb{R}^d$, $d=2,3$. The proof hinges on a novel semi-Galerkin scheme for a suitably regularized system, featuring a non-standard approximation of the singular potential. Moreover, with more restrictive assumptions on coefficients and data, we establish regularity properties and uniqueness of global weak solutions in the two-dimensional case. Our analysis contributes to a further understanding of phase separation processes under the interplay of fluid dynamics and chemotaxis, in particular, the influence of cross diffusion and logistic degradation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Mass Transport: Cross Diffusion versus Logistic Degradation Giorgini, Andrea He, Jingning Wu, Hao Analysis of PDEs We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows influenced by interactions with a soluble chemical substance, encompassing the chemotaxis effect, mass transport, and reactions. In the resulting coupled evolutionary system, the macroscopic fluid velocity field $\boldsymbol{v}$ satisfies a Navier--Stokes system driven by a capillary force, the phase field variable $φ$ is governed by a convective Cahn--Hilliard equation incorporating a mass source and a singular potential (e.g., the Flory--Huggins type), and the chemical concentration $σ$ obeys an advection-reaction-diffusion equation with logistic degradation, exhibiting a cross-diffusion structure akin to the Keller--Segel model for chemotaxis. Under general structural assumptions, we establish the existence of global weak solutions to the initial boundary value problem within a bounded smooth domain $Ω\subset \mathbb{R}^d$, $d=2,3$. The proof hinges on a novel semi-Galerkin scheme for a suitably regularized system, featuring a non-standard approximation of the singular potential. Moreover, with more restrictive assumptions on coefficients and data, we establish regularity properties and uniqueness of global weak solutions in the two-dimensional case. Our analysis contributes to a further understanding of phase separation processes under the interplay of fluid dynamics and chemotaxis, in particular, the influence of cross diffusion and logistic degradation. |
| title | Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Mass Transport: Cross Diffusion versus Logistic Degradation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.05751 |