## Global regularity of $$3D$$ rotating Navier-Stokes equations for resonant domains.(English)Zbl 0932.35160

Summary: We prove existence on infinite time intervals of regular solutions to the 3D rotating Navier-Stokes equations for strong rotation (large Coriolis parameter $$\Omega$$). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear “2 1/2- dimensional” limit equations for $$\Omega\to+\infty$$; smoothness assumptions are the same as for local existence theorems. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D rotating Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit equations and convergence theorems. In generic cases, sharper regularity results are derived from the algebraic geometry of resonant Poincaré curves.

### MSC:

 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs 76U05 General theory of rotating fluids 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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