On Stability of the Spatially Inhomogeneous Navier–Stokes–Boussinesq System with General Nonlinearity

This paper considers L²-asymptotic stability of the spatially inhomogeneous Navier–Stokes–Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes–Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal L^p-in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal L²-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal L^p-in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear.