Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier-Stokes system with logistic proliferation

We consider chemotaxis-Navier-Stokes systems with logistic proliferation and signal consumption of the form n(t) + u center dot del n = Delta n - del center dot (n del c) + kappa n - mu n(2), x is an element of Omega, t > 0, c(t) + u center dot del C = Delta C - Delta c - nc, x is an element of Omega, t > 0, u(t) + (u center dot del)u = Delta u + del P + 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 - n del c) center dot v = 0, c = c(star)(x), u = 0, x is an element of partial derivative Omega, t > 0 for parameter choices kappa >= 0 and mu > 0. Herein, we moreover impose a nonnegative and time-constant prescribed concentration c(star) is an element of C-2(Omega) for the signal chemical on the boundary of the domain Omega subset of R-N with N is an element of {2, 3}. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier-Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of c(star) being also constant in space. We show that sufficiently strong logistic influence, in the sense that for omega > 0 and mu(0) > 0 there is some eta = eta(omega, mu(0), c(star)) > 0 with the property that whenever mu(0) <= mu and kappa/min{mu, mu N+6/6 + omega} < eta are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on omega, mu(0), eta, c(star) and the initial data.