Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case (Q738714)

The authors extend results for the initial-boundary value problem for the coupled Keller-Segel-Stokes system with a matrix-valued chemotactic sensitivity from [\textit{Z. Shen} and \textit{C. Qian}, Z. Angew. Math. Phys. 66, No. 2, 399--415 (2015; Zbl 1323.35017)] to the case of three-dimensional domains. Using a new approach they establish the existence and boundedness of global classical solutions for arbitrarily large initial data. The crucial idea in this method is to establish the general \(L^p\) regularity of the fluid velocity from a rather low \(L^p\) regularity of the bacteria density, which is obtained using a new combinational functional.