Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids

We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic (as well as zero vertical flux) boundary conditions are imposed, the ratios of domain periods are assumed to be generic (non-resonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x₁, x₂, x₃) = Ũ(t, x₁, x₂) +V(t, x₁, x₂, x₃) + r, where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e₃). The vector field V(t, x₁, x₂, x₃) is exactly solved in terms of the phases Ωt, τ₁ (t) and τ₂(t). The phases τ₁(t) and τ₂(t) explicitly expressed in terms of vertically averaged vertical voracity curl Ū(t) · e₃ and velocity Ū³(t). The remainder r is uniformly estimated from above by a majorant of order a₃/Ω, a₃ is the vertical aspect ratio (shallowness) and Ω is non-dimensional rotation parameter based on horizontal scales. The resolution of resonances and a non-standard small divisor problem for 3D rotating Euler are the basis for error estimates. Contribution of 3-wave resonances is estimated in terms of the measure of almost resonant aspect ratios. Global solvability of the limit equations and estimates of the error r are used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* → +∞, as 1/Ω → 0); and existence on infinite time interval of regular solutions to 3D Navier-Stokes equations with smooth arbitrary initial data in the case of small 1/Ω.