论文标题
与不连续的对流的趋化聚类的非线性稳定性
Nonlinear stability of chemotactic clustering with discontinuous advection
论文作者
论文摘要
假设细菌对化学信号的反应良好，我们对细菌自组织的趋化模型进行了非线性稳定性分析。结果不连续的对流速度代表了稳定分析的关键挑战。我们遵循一种扰动方法，其中细胞轮廓的形状显然与其全局运动分开，从而使我们能够规避不连续性问题。此外，问题的同质性导致了两个保护定律，这些定律在加权的功能空间不同。权重之间的这种差异代表了另一个关键的方法论挑战。我们得出了改进的庞加莱不平等，该不平等现象允许将保护法中编码的信息转移到适当加权的空间。结果，我们以明确的速率获得指数式放松。数值调查说明了我们的结果。
We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results.
