Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2021-03-22 , DOI: 10.1007/s00033-021-01508-8 Michael Winkler
The flux-limited Keller–Segel–Navier–Stokes system
$$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$is considered in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^2\). It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that
$$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$holds for all \(\xi \ge 0\) with some \(K_f>0\) and \(\alpha >0\), for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in \(L^\infty (\Omega \times (0,\infty ))\) with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of (\(\star \)), this confirms that with regard to the possible emergence of blow-up phenomena, the choice \(f\equiv const.\) retains some criticality also in the presence of fluid interaction.
中文翻译：
在平面Keller–Segel–Navier–Stokes系统中通过梯度相关的通量限制来抑制爆炸
通量受限的Keller–Segel–Navier–Stokes系统
$$ \ begin {aligned} \ left \ {\ begin {array} {lcl} n_t + u \ cdot \ nabla n＆{} =＆{} \ Delta n-\ nabla \ cdot \ Big（nf（| \ nabla c | ^ 2）\ nabla c \ Big），\\ c_t + u \ cdot \ nabla c＆{} =＆{} \ Delta c-c + n，\\ u_t +（u \ cdot \ nabla）u＆ {} =＆{} \ Delta u + \ nabla P + n \ nabla \ Phi，\ qquad \ nabla \ cdot u = 0，\ end {array} \ right。\ qquad \ qquad（\ star）\ end {aligned} $$被认为是在有界的域\（\ Omega \ subset {\ mathbb {R}} ^ 2 \）中。结果表明，只要适当的光滑函数f建模交叉扩散通量的任何渐近代数型饱和度，就意味着
$$ \ begin {aligned} | f（\ xi）| \ le K_f \ cdot（\ xi +1）^ {-\ frac {\ alpha} {2}} \ end {aligned} $$对于具有\（K_f> 0 \）和\（\ alpha> 0 \）的所有\（\ xi \ ge 0 \）成立，对于所有合理的常规初始数据，对应的无通量/无通量/ Dirichlet问题接受全局定义的经典解决方案，该解决方案除其他因素外，以\（L ^ \ infty（\ Omega \ times（0，\ infty））\）为界。通过扩展已知的（\（\ star \））的无液对应物的对应结果，这证实了关于爆炸现象的可能出现，选择\（f \ equiv const。\）保留了一些。在流体相互作用的情况下也具有临界性。
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