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We perform the nonlinear stability analysis of a chemotaxis model ofbacterial self-organization, assuming that bacteria respond sharply to chemicalsignals. The resulting discontinuous advection speed represents the keychallenge for the stability analysis. We follow a perturbative approach, wherethe shape of the cellular profile is clearly separated from its global motion,allowing us to circumvent the discontinuity issue. Further, the homogeneity ofthe problem leads to two conservation laws, which express themselves indifferently weighted functional spaces. This discrepancy between the weightsrepresents another key methodological challenge. We derive an improvedPoincar\'e inequality that allows to transfer the information encoded in theconservation laws to the appropriately weighted spaces. As a result, we obtainexponential relaxation to equilibrium with an explicit rate. A numericalinvestigation illustrates our results.

Nonlinear stability of chemotactic clustering with discontinuous advection