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 L ^p-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 ${\mathsf{E}}$.