Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity

This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity: {n(t) + u . del n = del . (D(n)del n) - del . (nS(x, n, c) . del c), x is an element of Omega, t > 0, c(t) + u . del c = Delta c - nc, x is an element of Omega, t > 0, u(t) + del P = Delta u + n del Phi, del . u = 0, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-N with 2 <= N <= 3, where D is an element of C-0([0, infinity)) boolean AND C-2((0, infinity)) and S is an element of C-2((Omega) over bar x [0, infinity)(2); R-NxN). The global solvability of the system in a natural weak sense is obtained under the conditions that D(n) >= k(D)n(m-1) and vertical bar S(x, n, c)vertical bar <= S-0(c)/c(alpha), for all (x, n, c) is an element of Omega x (0, infinity)(2) with some k(D) > 0, m > 3N-2/2N, alpha is an element of [0, 1) and some nondecreasing S-0 : (0, infinity) -> (0, infinity). Moreover, in the case that m = 3N-2/2N and alpha is an element of [0, 1), we also get the global weak solutions under smallness assumptions on the initial data parallel to n(0)parallel to(L1(Omega)) and parallel to c(0)parallel to(L infinity(Omega)).