Rotating Navier-Stokes equations in ℝ₊³ with initial data nondecreasing at infinity: The Ekman boundary layer problem

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 Ḃ_∞,1,σ⁰ (ℝ²; L ^p (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 Ḃ_∞,1⁰ 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 Ḃ _∞,1⁰(ℝⁿ; E) for a general Banach space E.