Small-convection limit for two-dimensional chemotaxis-Navier-Stokes system with logarithmic sensitivity and logistic-type source

In this paper, we consider the small-convection limit of chemotaxis-Navier-Stokes system with logarithmic sensitivity and logistic-type source {n(t)(kappa) + u(kappa).del n(kappa) = Delta n(kappa)-chi del.(n(kappa)del log c(kappa)) + f(n(kappa)), x is an element of Omega, t > 0, c(t)(kappa)+u(kappa).del c(kappa) = Delta c(kappa)-c(kappa) + n(kappa), x is an element of Omega, t > 0, u(t)(kappa)+kappa(u(kappa).del)u(kappa) = Delta u(kappa) + del P-kappa+n(kappa)del phi, x is an element of Omega, t > 0, del.u(kappa) = 0, x is an element of Omega, t > 0, in a bounded convex domain Omega subset of R-2 with smooth boundary, where kappa is an element of R, f(s) = mu(1)s - mu(2)s(lambda), lambda > 1, and : phi: Omega -> R is a given smooth potential with second-order partial derivatives. When the chemotaxis sensitivity chi satisfies the appropriate conditions, it is proved that the unique global classical solutions (n(kappa), c(kappa), u(kappa)) will stabilize to (n(0), c(0), u(0)) as kappa -> 0.