Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Ma, Yansheng Pang, Peter Y. H. Wang, Yifu |
| author_facet | Ma, Yansheng Pang, Peter Y. H. Wang, Yifu |
| contents | This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=Δ(nϕ(v))+μn(1-n),\\ v_t+u\cdot\nabla v=Δv-nv,\\ u_t+ κ(u\cdot\nabla) u=Δu+n\nablaΦ-\nabla P, \quad\nabla\cdot u=0, \end{cases} \end{equation*}
accounting for signal-dependent motilities of microbial populations
interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function $ϕ$ satisfies $ϕ>0$ on $(0,\infty)$ with $ϕ(0)=0$ and $ϕ'(0)>0$, and the parameter $μ\geq 0$. For all reasonably regular initial data, if $μ=0$, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on $\int_Ωn_0$; whereas if $μ>0$, this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on the initial data $v_0$. These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion Ma, Yansheng Pang, Peter Y. H. Wang, Yifu Analysis of PDEs This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=Δ(nϕ(v))+μn(1-n),\\ v_t+u\cdot\nabla v=Δv-nv,\\ u_t+ κ(u\cdot\nabla) u=Δu+n\nablaΦ-\nabla P, \quad\nabla\cdot u=0, \end{cases} \end{equation*} accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function $ϕ$ satisfies $ϕ>0$ on $(0,\infty)$ with $ϕ(0)=0$ and $ϕ'(0)>0$, and the parameter $μ\geq 0$. For all reasonably regular initial data, if $μ=0$, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on $\int_Ωn_0$; whereas if $μ>0$, this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on the initial data $v_0$. These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment. |
| title | Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.17073 |