A new result for the global existence and boundedness of weak solutions to a chemotaxis-Stokes system with rotational flux term

In this paper, we consider a three-dimensional chemotaxis-Stokes system {n(t)+u center dot del n=Delta n(m)- del center dot(nS(x,n,c)center dot del c), x is an element of omega,t>0, c(t)+u center dot del c=Delta c-nf(c),x is an element of omega,t>0, u(t)+ del P=Delta u+n del phi,x is an element of omega,t>0, del center dot u=0,x is an element of omega,t>0, (del n(m)-nS(x,n,c)center dot del c)center dot nu= partial derivative c/partial derivative nu=0,u=0,x is an element of partial derivative Omega,t>0, n(x,0)=n(0)(x),c(x,0)=c(0)(x),u(x,0)=u(0)(x),x is an element of Omega in a bounded domain omega subset of R-3 with smooth boundary, where phi\, f and S are given functions with values in Omega, [0,infinity) and R-3x3, respectively, under the no-flux boundary condition for n, c and Dirichlet boundary condition for u. It is also required that f is an element of C-1([0,infinity)) is locally bounded in [0,infinity), that S satisfies |S(x,n,c)|<= n(l-2)S(0)(c) with some nondecreasing S0:[0,infinity)->[0,infinity), and that m fulfills m>5/3l-20/9 and m>-3/4l+21/8 with 25/12 >= l>2. (0.2) Then, with any reasonably regular initial data, the corresponding initial-boundary problem for (0.1) processes a global in time and bounded solution. This result extends the previous global boundedness result with m>m*(l), where m*(l)={l-5/6, if31/12 >= l>2, 7/5l-28/15,if l>31/12, ([15]) since both l-5/6 >= 5/3l - 20/9 and l - 5/6 >= -3/4l+21/8 hold on 25/12 >= l>2.