Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

Abstract

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solutions. The global-in-time existence and uniqueness of a small-data strong solution is proved. For the proof of this result, we propose a new approach. Our approach is based on the operator treatment of the problem with the consequent application of a theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations.

Keywords

• generalized Navier–Stokes equations
• non-homogeneous Dirichlet boundary condition
• fractional Sobolev spaces
• trace operator
• divergence-free lifting operator
• strong solutions