On a three-dimensional quasilinear Keller-Segel-Stokes system with indirect signal production

This paper is concerned with the effect of the indirect signal production mechanism on global boundedness of solutions for the following Keller-Segel-Stokes system {n(t) + u . del n = del . (D(n)del n - S(n)c), (x, t) is an element of Omega x (0, infinity), c(t) + u . del c = Delta c - c + v, (x,t) is an element of Omega x (0,infinity), v(t) + u . del v = Delta v - v + n, (x,t) is an element of Omega x (0,infinity ), u(t) = Delta u + del P + n del phi, (x, t) is an element of Omega x (0, infinity), del . u = 0, (x,t) is an element of Omega x (0, infinity), in a smoothly bounded domain Omega subset of R-3 under zero-flux boundary conditions for n, c, v and no-slip boundary conditions for u, where D(n) and S(n) denote the nonlinear diffusion and sensitivity, respectively, u represents the velocity of the fluid, P is the pressure within the fluid, and phi is the gravitational potential. By means of the novel conditional estimates for del c and u, it is proved that for all appropriately regular nonnegative initial data, this model has a globally bounded classical solution provided that the functions D, S is an element of C-2 ([0, infinity)) satisfy D(n) >= K-1(n + 1)(-m) and S(n) <= K(2)n (n + 1)(alpha-1) with K-1, K-2 > 0 and alpha + m < 8/9, which improves the previous subcritical exponent alpha+m < 2/3 in the direct signaling Keller-Segel-Stokes system by Winkler (Appl Math Lett 112:106785, 2021). It is shown that the indirect signal production mechanism can be beneficial to the global boundedness of solutions for the three-dimensional quasilinear Keller-Segel-Stokes system.