论文标题
趋化的密度约束模型
A density-constrained model for Chemotaxis
论文作者
论文摘要
我们考虑了趋化性的拥塞动力学模型：细胞的密度遵循其产生的化学信号，同时受到不可压缩性约束。不可压缩的约束导致斑块的形成，描述了达到最大密度的区域。这些斑块的动力学可以通过Hele-Shaw或Richards方程类型流量来描述（取决于我们是使用扩散的模型还是具有纯对流的模型）。我们在本文中的重点是通过JKO类型的各种离散时间方案为该问题构建弱解决方案。我们还建立了这些解决方案的独特性。此外，我们更加严格地使这种不可压缩的趋化模型与描述斑块运动的自由边界问题之间的联系，以密度和相关的压力变量为单位。特别是，我们获得了将压力变量描述为障碍物问题的解决方案的新结果，并证明在纯对流案例中，动态保留了斑块。
We consider a model of congestion dynamics with chemotaxis: The density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incompressibility constraint results in the formation of patches, describing regions where the maximal density has been reached. The dynamics of these patches can be described by either Hele-Shaw or Richards equation type flow (depending on whether we consider the model with diffusion or the model with pure advection). Our focus in this paper is on the construction of weak solutions for this problem via a variational discrete time scheme of JKO type. We also establish the uniqueness of these solutions. In addition, we make more rigorous the connection between this incompressible chemotaxis model and the free boundary problems describing the motion of the patches in terms of the density and associated pressure variable. In particular, we obtain new results characterizing the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.
