Navier-Stokes equations in a rotating frame in R³ with initial data nondecreasing at infinity

Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter Ω and initial data nondecreasing at infinity. In contrast to the non-rotating case (Ω = 0), it is shown for the problem with rotation (Ω ≠ 0) that Green’s function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to L¹(R³). Moreover, the corresponding integral operator is unbounded in the space [formula] of solenoidal vector fields in R³ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in [formula]. Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space [formula] which consists of L^∞ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space [formula], 1 which is smaller than L^∞ but still contains various periodic and almost periodic functions. This restriction of initial data to [formula] which is a subspace of [formula] is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time solvability of the classical 3D Navier-Stokes equations in R³ with initial velocity and vorticity of the form V(0) = Ṽ⁰(y) + (Ω/2)e₃ × y, curl V(0) = curl Ṽ₀(y) + Ωe₃ where V₀(y) ∈ [formula].