Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912434665553920 |
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| author | Black, Tobias |
| author_facet | Black, Tobias |
| contents | We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\nabla c=Δc-cn,\ &x\inΩ,& t>0,\\ u_{t}+(u\cdot\nabla)u=Δu+\nabla P+n\nablaΦ,\quad \nabla\cdot u=0,\ &x\inΩ,& t>0,\\ \big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c)\cdotν=\nabla c\cdotν=0,\ u=0,\ &x\in\partialΩ,& t>0,\\ n(\cdot,0)=n_0,\ c(\cdot,0)=c_0,\ u(\cdot,0)=u_0,\ &x\inΩ. \end{array}\right. \end{align*} in a smoothly bounded domain $Ω\subset\mathbb{R}^2$. Assuming $S:\overlineΩ\times[0,\infty)\times(0,\infty)\rightarrow \mathbb{R}^{2\times 2}$ to be sufficiently regular and such that with $γ\in[0,\frac56]$ and some non-decreasing $S_0:(0,\infty)\to(0,\infty)$, we have \begin{align*} \big|S(x,n,c)\big|\leq \frac{S_0(c)}{c^γ}\quad\text{for all }(x,n,c)\in\overlineΩ\times[0,\infty)\times(0,\infty), \end{align*} we show that if $D:[0,\infty)\to[0,\infty)$ is suitably regular and positive throughout $(0,\infty)$, then for all $M>0$ one can find $L(M)>0$ such that whenever $$\liminf_{n\to\infty} D(n)>L\quad\text{and}\quad \liminf_{n\searrow0}\frac{D(n)}{n}>0$$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy $\|c_0\|_{L^{\infty}(Ω)}\leq M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D(0)>0$ this solution is moreover a classical solution of the same problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system Black, Tobias Analysis of PDEs 35D30, 35A01, 35K65, 35Q35, 35Q92, 92C17 We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\nabla c=Δc-cn,\ &x\inΩ,& t>0,\\ u_{t}+(u\cdot\nabla)u=Δu+\nabla P+n\nablaΦ,\quad \nabla\cdot u=0,\ &x\inΩ,& t>0,\\ \big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c)\cdotν=\nabla c\cdotν=0,\ u=0,\ &x\in\partialΩ,& t>0,\\ n(\cdot,0)=n_0,\ c(\cdot,0)=c_0,\ u(\cdot,0)=u_0,\ &x\inΩ. \end{array}\right. \end{align*} in a smoothly bounded domain $Ω\subset\mathbb{R}^2$. Assuming $S:\overlineΩ\times[0,\infty)\times(0,\infty)\rightarrow \mathbb{R}^{2\times 2}$ to be sufficiently regular and such that with $γ\in[0,\frac56]$ and some non-decreasing $S_0:(0,\infty)\to(0,\infty)$, we have \begin{align*} \big|S(x,n,c)\big|\leq \frac{S_0(c)}{c^γ}\quad\text{for all }(x,n,c)\in\overlineΩ\times[0,\infty)\times(0,\infty), \end{align*} we show that if $D:[0,\infty)\to[0,\infty)$ is suitably regular and positive throughout $(0,\infty)$, then for all $M>0$ one can find $L(M)>0$ such that whenever $$\liminf_{n\to\infty} D(n)>L\quad\text{and}\quad \liminf_{n\searrow0}\frac{D(n)}{n}>0$$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy $\|c_0\|_{L^{\infty}(Ω)}\leq M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D(0)>0$ this solution is moreover a classical solution of the same problem. |
| title | Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system |
| topic | Analysis of PDEs 35D30, 35A01, 35K65, 35Q35, 35Q92, 92C17 |
| url | https://arxiv.org/abs/2411.18336 |