## Abstract

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, x_{1}, x_{2}, x_{3}) = Ũ(t, x_{1}, x_{2}) +V(t, x_{1}, x_{2}, x_{3}) + 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 e_{3}). The vector field V(t, x_{1}, x_{2}, x_{3}) is exactly solved in terms of the phases Ωt, τ_{1} (t) and τ_{2}(t). The phases τ_{1}(t) and τ_{2}(t) explicitly expressed in terms of vertically averaged vertical voracity curl Ū(t) · e_{3} and velocity Ū^{3}(t). The remainder r is uniformly estimated from above by a majorant of order a_{3}/Ω, a_{3} 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.

Original language | English (US) |
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Pages (from-to) | 103-150 |

Number of pages | 48 |

Journal | Asymptotic Analysis |

Volume | 15 |

Issue number | 2 |

State | Published - Oct 1 1997 |

## ASJC Scopus subject areas

- General Mathematics