## Abstract

Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter Ω 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 [formula] of solenoidal vector fields in R^{3} and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in [formula]. Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space [formula] which consists of L^{∞} solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space [formula], 1 which is smaller than L^{∞} but still contains various periodic and almost periodic functions. This restriction of initial data to [formula] which is a subspace of [formula] 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) = Ṽ^{0}(y) + (Ω/2)e_{3} × y, curl V(0) = curl Ṽ_{0}(y) + Ωe_{3} where V_{0}(y) ∈ [formula].

Original language | English (US) |
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Pages (from-to) | 321-364 |

Number of pages | 44 |

Journal | Hokkaido Mathematical Journal |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

## Keywords

- Homogeneous besov spaces
- Nondecreasing initial data
- Riesz operators
- Rotating Navier-Stokes equations

## ASJC Scopus subject areas

- Mathematics(all)