Stabilization of the chemotaxis–Navier–Stokes equations: Maximal regularity approach

Consider the chemotaxis–Navier–Stokes equations in a bounded smooth domain Ω⊂R^d for d≥3. We show that any solution starting close to an equilibrium exists globally and converges exponentially fast to the equilibrium as time tends to infinity, provided that the initial density n₀ of amoebae satisfies ∫_Ωn₀dx<2|Ω|, where |Ω| stands for the Lebesgue measure of Ω. First, we prove the existence of a local strong solution for large initial data. Then, the global existence result is obtained assuming that the initial data are close to the equilibrium in their natural norm. In particular, we show the strong solution in the maximal L^p−L^q-regularity class with (p,q)∈(2,∞)×(d,∞) satisfying 2/p+d/q<1. Furthermore, the solution is real analytic in space and time.