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The self-consistent chemotaxis-Navier–Stokes system with nonlinear diffusion Formula: see text is considered in a bounded domain Formula: see text with smooth boundary. Compared to the previously most-studied chemotaxis-fluid system proposed in I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA. 102 (2005) 2277–2282, the coupling in this system is stronger and more nonlinear. When the system is accompanied by homogeneous boundary conditions of no-flux type for Formula: see text and Formula: see text, and of Dirichlet type for Formula: see text, a quasi-Lyapunov structure provides sufficient regularity features to facilitate a basic existence theory. However, if we change the boundary condition of the signal to Formula: see text with a given non-negative function Formula: see textFormula: see text, then the Dirichlet boundary condition imposed here seems to destroy the quasi-Lyapunov structure. Despite this, we shall find a new energy structure and prove that for suitably regular initial data, the assumption Formula: see text is sufficient for the global existence and boundedness of the weak solution. To the best of our knowledge, this is the first work on the global well-posedness problem of the self-consistent chemotaxis-fluid system involving Dirichlet boundary conditions for the signal.
Dong et al. (Thu,) studied this question.