论文标题
边界上的规定信号浓度：趋化性降解系统中的弱溶解度
Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation
论文作者
论文摘要
我们研究了一个具有信号消耗和逻辑源术语的趋化性 - 启动系统\ noindent \ begin {align*} \ left \ {\ stray {array} {r@{r@{\} l@{\ Quad} l@{ n＆=Δn-\ nabla \！\ cdot（n \ nabla c）+κn-μn^{2}，\＆x \inΩ，＆t> 0，\\ c_ {t}+u \ cdot \ cdot \ cdot \！ u_ {t}＆=ΔU+\ nabla p+n \ n \ nabla ϕ，\＆x \inΩ，＆t> 0，\\ \ \ \ \ \ \ \ \ \ cdot u＆= 0，\＆x \inΩ，＆t> 0，＆t> 0，\ \ \ \ big（ c = c _ {\ star}（x），\ quad u = 0，＆x \ in \partialΩ，＆t> 0，\ end {array} \ right。 \end{align*} where $κ\geq0$, $μ>0$ and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $Ω\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribed time-independent nonnegative function $ c _ {\ star} \ in C^{2} \！\ big（\overlineΩ\ big）$。利用第一个方程式的二次衰减项所需的界限信息，我们将表明上面的系统至少具有一个全球弱解决方案，用于任何适当的常规初始数据三重态。
We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=Δn-\nabla\!\cdot(n\nabla c)+κn-μn^{2},\ &x\inΩ,& t>0,\\ c_{t}+u\cdot\!\nabla c&=Δc-nc,\ &x\inΩ,& t>0,\\ u_{t}&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ \nabla\cdot u&=0,\ &x\inΩ,& t>0,\\ \big(\nabla n-n\nabla c\big)\cdotν&=0,\quad c=c_{\star}(x),\quad u=0, &x\in\partialΩ,& t>0, \end{array}\right. \end{align*} where $κ\geq0$, $μ>0$ and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $Ω\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribed time-independent nonnegative function $c_{\star}\in C^{2}\!\big(\overlineΩ\big)$. Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.
