Global dynamics for the generalised chemotaxis-Navier-Stokes system in R3

We consider the chemotaxis-Navier-Stokes system with generalised fluid dissipation in R-3: {partial derivative(t)n + u . del n = Delta n- del . (chi (c)n del c), partial derivative(t)c + u . del c = Delta c - nf(c), partial derivative(t)t + u . del u + del P = -(-Delta)(alpha) u - n del phi, del.u = 0, which models the motion of swimming bacteria in water flows. First, we prove blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin-type criterion for alpha > 3/4 and the Beirao da Veiga-type criterion for alpha > 1/2. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for alpha >= 5/4. Furthermore, in the scenario of 3/4 < alpha < 5/4, we establish uniform regularity estimates and optimal time-decay rates of global solutions if only the L-2-norm of initial data is small. To our knowledge, this work provides the first result concerning the global existence and large-time behaviour of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations.