Nonlinear stability of chemotactic clustering with discontinuous advection

Abstract: 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 Poincar\'e 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.