Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth

In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier-Stokes equations. This paper introduces an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by {n(t) + u . del(n) = Delta n - (n del c) + n - n(2), x is an element of Omega, t > 0, c(t) + u . del c = Delta c - nc, x is an element of Omega, t > 0, u(t) = Delta(u) + u . del u + del p - n del phi, x is an element of Omega, t > 0, del . u = 0, x is an element of Omega, t > 0 in conjunction with the initial data (n, c, u)(., 0) = (n(0), c(0), u(0)) and the boundary conditions partial derivative c/partial derivative nu = 1 -c, partial derivative n/partial derivative nu = n partial derivative c/partial derivative nu, u = 0, x is an element of partial derivative Omega, t > 0. Here, the fluid drop is described by Omega subset of R-N being a bounded convex domain with smooth boundary. Moreover, phi is a given smooth gravitational potential. Requiring sufficiently smooth initial data, the existence of a unique global classical solution for N = 2 is proved, where parallel to n parallel to L-p(Omega) is bounded in time for all p < infinity, as well as the existence of a global weak solution for N = 3. (C) 2016 Elsevier Masson SAS. All rights reserved. .