GLOBAL WEAK SOLUTIONS IN A SELF-CONSISTENT CHEMOTAXIS-FLUID SYSTEM WITH PRESCRIBED SIGNAL CONCENTRATION ON THE BOUNDARY

In this article, we investigate an incompressible chemotaxis-Stokessystem with nonlinear diffusion and rotational flux {n(t)+u<middle dot>del n= triangle n(m)-del<middle dot>(nS(x,n,c)<middle dot>del c) +del<middle dot>(n del phi), x is an element of ohm, t >0, c(t)+u<middle dot>del c= triangle c-nc, x is an element of ohm, t > 0, u(t )+ del P = triangle u-n del phi+nS(x,n,c)<middle dot>del c,del<middle dot>u= 0, x is an element of ohm, t > 0 in a bounded domain ohm subset of R-3 with smooth boundary partial derivative ohm. The corresponding boundary conditions satisfy (del n(m)-nS(x,n,c)<middle dot>del c+n del phi)<middle dot>nu= 0, c=c & lowast;(x,t),u=0, x is an element of partial derivative ohm, t >0,with m > 1 and a given nonnegative function c & lowast;(x,t)is an element of C2,1(ohm x(0,infinity)). The chemotatic sensitivity S is a given tensor-valued function fulfilling |S(x,n,c)|76. In the homogeneous Dirichlet signal boundary condition (i.e., c & lowast;(x,t) equivalent to 0) case, we further prove that the solutions will stabilize to the mass-preserving spatial equilibrium (n0,0,0), wheren0:=1|ohm|R ohm n0(x)dx.