Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

Abstract

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space \(\dot{{\mathcal B}}_{\infty,1,\sigma}^0 ({\mathbb R}^2; L^p({\mathbb R}_+))\) for 2 < p < ∞. Here the Lp-integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space \(\dot{{\mathcal B}}_{\infty,1}^0\) contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H∞-calculus for the Laplacian in \(\dot{{\mathcal B}}_{\infty,1}^0({\mathbb R}^n; {\mathsf{E}})\) for a general Banach space \({\mathsf{E}}\).