Further study on the global existence and boundedness of the weak solution in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

We consider an initial-value problem for the incompressible chemotaxis-Stokes equations generalizing the porous-medium-type diffusion model with tensor-valued sensitivity {nt +u center dot del n = Delta nm - del center dot (nS(x, n, c) center dot del c), x is an element of Omega, t > 0, ct +u center dot del c = Delta c -nc, x is an element of Omega, t > 0, ut + del P = Delta u + n del phi, x is an element of Omega, t > 0, (CNF) del center dot u = 0, x is an element of Omega, t > 0, in a bounded domain Omega subset of R-3 with a smooth boundary, where phi is an element of W-2,W-infinity(Omega) is a given gravitational potential function. Problems of this type have been used to describe the mutual interaction between swimming aerobic bacteria populations and the surrounding fluid. The main feature is that the chemotactic sensitivity S is a given parameter matrix on Omega x [0,infinity)(2), whose Frobenius norm satisfies |S(x, n, c)| <= S-0(c) for all (x, n, c) is an element of (Omega) over bar x [0,infinity) x [0,infinity) with S-0(c) nondecreasing on [0,infinity). Based on a new energy-type argument combined with a new estimation technique, we show that if m > 11/4 - root 3, an associated initial-boundary value problem admits at least one globally bounded weak solution. The above mentioned results significantly improved and extended previous results of several authors. (c) 2022 Elsevier B.V. All rights reserved.