Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains

Abstract

Consider the Stokes resolvent system in general unbounded domains $\Omega \subset {\mathbb{R}^n}$, $n\geq 2$, with boundary of uniform class $C^{3}$, and Navier slip boundary condition. The main result is the resolvent estimate in function spaces of the type ${\tilde{L}^q}$ defined as $L^q\cap L^2$ when $q\geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain. As a consequence, we get that the Stokes operator generates an analytic semigroup on a solenoidal subspace ${\tilde{L}^q}_\sigma(\Omega)$ of ${\tilde{L}^q}(\Omega)$.