The unsteady Navier–Stokes–Boussinesq system with non-zero boundary conditions

The paper deals with an initial-boundary value problem for the 3D Navier–Stokes–Boussinesq equations under non-zero Dirichlet boundary conditions. We introduce the concept of strong solutions assuming that boundary data for velocity and temperature belong to suitable fractional Sobolev spaces. The existence of small-data strong solutions is proved. To establish this result, we reformulate the original problem as a nonlinear operator equation and apply to it a new abstract theorem on the local solvability of an operator equation involving an isomorphism between Banach spaces with linear and quadratic compact perturbations. The proposed approach is quite universal and can be applied to the solvability proving for many other nonlinear partial differential equations that arise in the modelling of both Newtonian and non-Newtonian fluids.