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Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary
Mathematical Models and Methods in Applied Sciences ( IF 3.5 ) Pub Date : 2021-12-24 , DOI: 10.1142/s021820252250004x
Tobias Black 1 , Michael Winkler 1
Affiliation  

An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes model with porous medium type diffusion is considered. Previous related literature has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip boundary conditions for the density of cells, the chemical concentration and the fluid velocity field, respectively; in particular, available qualitative results strongly indicate that only trivial solution behavior can be expected on large time scales. In line with refined modeling approaches to oxygen evolution near fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant concentration on the boundary. Despite an apparent loss of mathematically favorable energy structures thereby induced, by means of an alternative variational approach a basic theory of global existence is developed in a natural framework of weak solvability. Beyond this, some additional qualitative information on the large time behavior of these solutions is derived by identifying a certain global relaxation property. Specifically, a second result asserts, within a suitable topological setting, the existence of a bounded set which eventually absorbs each individual of the obtained trajectories, and the diameter of which is bounded only by the physically relevant quantities of total population size and prescribed boundary concentration of the chemical signal.

中文翻译:

边界上具有规定信号浓度的趋化-纳维-斯托克斯系统中的全局弱解和吸收集

考虑了具有多孔介质型扩散的耦合趋化性-Navier-Stokes 模型的初始边界值问题。以前的相关文献已经为系统分别增加了细胞密度、化学浓度和流体速度场的无通量/无通量/无滑移边界条件的情况提供了深刻的知识;特别是,可用的定性结果强烈表明,在大时间尺度上只能预期微不足道的解决方案行为。根据流体 - 空气界面附近氧气释放的改进建模方法,本研究现在关注涉及边界上固定化学引诱剂浓度的情况。尽管由此导致数学上有利的能量结构明显丧失,通过另一种变分方法,全球存在的基本理论在弱可解性的自然框架中得到发展。除此之外,通过识别某种全局松弛特性,可以得出一些关于这些解决方案的大时间行为的额外定性信息。具体而言,第二个结果断言,在合适的拓扑设置内,存在一个最终吸收所获得轨迹的每个个体的有界集,其直径仅受总人口规模和规定边界浓度的物理相关量的限制的化学信号。通过识别某种全局松弛特性,可以得出关于这些解的大时间行为的一些额外的定性信息。具体而言,第二个结果断言,在合适的拓扑设置内,存在一个最终吸收所获得轨迹的每个个体的有界集,其直径仅受总人口规模和规定边界浓度的物理相关量的限制的化学信号。通过识别某种全局松弛特性,可以得出关于这些解的大时间行为的一些额外的定性信息。具体而言,第二个结果断言,在合适的拓扑设置内,存在一个最终吸收所获得轨迹的每个个体的有界集,其直径仅受总人口规模和规定边界浓度的物理相关量的限制的化学信号。
更新日期:2021-12-24
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