Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics(n(1))t + u . n(1) =n(1) - chi backward difference center dot (n1 backward difference w) +n1( lambda 1-mu 1n1 +an2), x is an element of,t > 0, (n2)t+ u center dot backward difference n2 = n2+ xi backward difference center dot (n2 backward difference z) + n2(lambda 2 - mu 2n2 - bn1), x is an element of , t > 0, wt +u center dot backward difference w= w -w +n2, x is an element of , t > 0, u center dot backward difference z= z -z + n1, x is an element of , t > 0, ut + backward difference P = u + (n1 + n2) backward difference phi, x is an element of ,t > 0, backward difference center dot u = 0, x is an element of ,t >0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is partial differential n1 = partial differential nu partial differential n2 = partial differential nu partial differential w = partial differential nu partial differential z =0, u=0, x is an element of partial differential ,t > 0, partial differential nu where chi > 0, xi> 0, mu 1 >= 0, mu 2 >= 0, lambda 1 >= 0, lambda 2 >= 0, a >= 0, b >= 0 and phi is an element of W2,infinity(). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.