Global weak solvability in a self-consistent chemotaxis-Navier–Stokes system involving Dirichlet boundary conditions for the signal

Abstract

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 text][Formula: 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.