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Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter 0 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 L 1 (R 3 ). Moreover. the corresponding integral operator is unbounded in the space L ∝ σ (R 3 ) of solenoidal vector fields in R 3 and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in L ∞ σ (R 3 ). Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space L ∝ σ,α (R 3 ) which consists of L ∞ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space B 0 ∞,1 which is smaller than L ∞ but still contains various periodic and almost periodic functions. This restriction of initial data to L ∞ σ,α (R 3 ) which is a subspace of L ∞ σ (R 3 ) 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 R 3 with initial velocity and vorticity of the form V(0) = V 0 (y) + (Ω/2)e 3 x y, curl V(0) = curl V 0 (y) + Ωe 3 where V 0(y) ∈ L ∞ σ,α (R 3 ).

Navier-Stokes equations in a rotating frame in ${\mathbb R}^3$ with initial data nondecreasing at infinity