We establish a global existence result for the rotating Navier-Stokes equations with nondecaying initial data in a critical space which includes a large class of almost periodic functions. The scaling invariant function space we introduce is given as the divergence of the space of 3 × 3 fields of Fourier transformed finite Radon measures. The smallness condition on initial data for global existence is explicitly given in terms of the Reynolds number. The condition is independent of the size of the angular velocity of rotation.
Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter Ω and initial data nondecreasing at infinity. In contrast to the nonrotating 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, uniform in Ω unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space L ∞ σ,a (R 3 ) which consists of L ∞ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov spaceḂ 0 ∞,1 which is smaller than L ∞ but still contains various periodic and almost periodic functions. This restriction of initial data to L ∞ σ,a (R 3 ) * Present Address : Graduate School of Mathematics Sciences, University of Tokyo, Komaba 3-8-1 Meguro, Tokyo 153-8914, Japan 1 which is a subspace of L ∞ σ (R 3 ) is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. The proof of uniform in Ω local in time unique solvability requires detailed study of the symbol of this semigroup and obtaining uniform in Ω estimates of the corresponding operator norms in Banach spaces. Using the rotation transformation, we also obtain local in time, uniform in Ω solvability of the classical 3D Navier-Stokes equations in R 3 with initial velocity and vorticity of the form V(0) =Ṽ 0 (y) + Ω 2 e 3 × y, curlV(0) = curlṼ 0 (y) + Ωe 3 whereṼ 0 (y) ∈ L ∞ σ,a (R 3 ).
This is a survey article of the results obtained in [8] by Y. Giga, K. Inui, A. Mahalov, S. Matsui, and me. There, existence and uniquness of local-in-time solutions for the Ekman boundary layer problem is proved.
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