  A three-dimensional chemotaxis-Navier-Stokes system is considered. It is
known that for all suitably regular initial data, a corresponding
initial-boundary value problem admits at least one global weak solution which
can be obtained as the pointwise limit of a sequence of solutions to
appropriately regularized problems. The present paper shows that after some
relaxation time, this solution enjoys further regularity properties and thereby
complies with the concept of eventual energy solutions which is newly
introduced here, and which inter alia requires that two quasi-dissipative
inequalities are ultimately satisfied. Moreover, it is shown that actually for
any such eventual energy solution there exists a waiting time beyond which the
solution is smooth and classical, and that a spatially homogeneous equilibrium
is approached in the large time limit.
