In a bounded planar domain \(\varOmega \) with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t} + \nabla \cdot (nu) = \Delta n - \nabla \cdot (n\nabla c), & x\in \varOmega , \ t>0, \\ 0 = \Delta c -c+n, & x\in \varOmega , \ t>0, \end{array}\displaystyle \right . \end{aligned}$$ is considered, where \(u\) is a given sufficiently smooth velocity field on \(\overline {\varOmega }\times [0,\infty )\) that is tangential on \(\partial \varOmega \) but not necessarily solenoidal.It is firstly shown that for any choice of \(n_{0}\in C^{0}(\overline {\varOmega })\) with \(\int _{\varOmega}n_{0}<4\pi \), this problem admits a global classical solution with \(n(\cdot ,0)=n_{0}\), and that this solution is even bounded whenever \(u\) is bounded and \(\int _{\varOmega}n_{0}<2\pi \). Secondly, it is seen that for each \(m>4\pi \) one can find a classical solution with \(\int _{\varOmega}n(\cdot ,0)=m\) which blows up in finite time, provided that \(\varOmega \) satisfies a technical assumption requiring \(\partial \varOmega \) to contain a line segment.In particular, this indicates that the value \(4\pi \) of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.

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流体相互作用会影响有界平面域中出租车驱动的爆炸的临界质量吗？

在具有光滑边界的有界平面域\（\ varOmega \）中，Keller-Segel流体系统的齐次Neumann型初边界值问题$$ \ begin {aligned} \ left \ {\ textstyle \ begin {array } {l @ {\ quad} l} n_ {t} + \ nabla \ cdot（nu）= \ Delta n-\ nabla \ cdot（n \ nabla c），＆x \ in \ varOmega，\ t> 0， \\ 0 = \ Delta c -c + n，＆x \ in \ varOmega，\ t> 0，\ end {array} \ displaystyle \ right。考虑\ end {aligned} $$，其中\（u \）是\（\ overline {\ varOmega} \ times [0，\ infty} \）上给定的足够平滑的速度场，该速度场在\（\ partial上是切向的\ varOmega \），但不一定是螺线管。首先显示出对于\（n_ {0} \ in C ^ {0}（\ overline {\ varOmega}）\）的任何选择对于\（\ int _ {\ varOmega} n_ {0} <4 \ pi \），此问题允许使用\（n（\ cdot，0）= n_ {0} \）的全局经典解，并且该解决方案每当\（u \）有界且\（\ int _ {\ varOmega} n_ {0} <2 \ pi \）时，甚至有界。其次，可以看到对于每个\（m> 4 \ pi \）可以找到一个经典解决方案，其中\（\ int _ {\ varOmega} n（\ cdot，0）= m \）在有限的时间内爆炸，前提是\（\ varOmega \）满足一个技术假设，要求\（\ partial \ varOmega \）包含一个线段。特别是，这表明值\（4 \ pi \） 相应的无流体Keller-Segel系统的临界质量的任何相关因素在所考虑类型的任何流体相互作用下均保持不变，因此与最近的结果形成了鲜明的对比，该结果揭示了流体在相关过程中临界爆破质量的某些增加整个飞机上的柯西问题。