Small-mass solutions in a two-dimensional logarithmic Chemotaxis–Navier–Stokes system with indirect nutrient consumption

This paper is concerned with a two-dimensional singular chemotaxis–Navier–Stokes system under the no-flux/Neumann/Neumann/Dirichlet boundary condition which models the movement of a bacteria population with indirect nutrient consumption in a fluid environment. In this model, we consider a general tensor-valued chemotactic sensitivity, where the bacteria movement orients toward higher concentration of nutrients under the Weber–Fechner law of stimulus perception, resulting in a logarithmic singularity in the chemotactic sensitivity tensor when the nutrient concentration drops to 0. We demonstrate that, when the initial bacteria population is suitably small, the system possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state where the bacteria concentration settles on an even distribution of the initial population. This rigorously confirms that, at least in the two-dimensional setting, in comparison with the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity in the solutions to the chemotaxis–fluid system even with a singular tensor-valued sensitivity.