Global existence results for the Navier-Stokes equations in the rotational framework

TL;DR

This paper proves the existence and uniqueness of global solutions to the Navier-Stokes equations with Coriolis force in three and two dimensions, under small and non-small initial data respectively, using Fourier-Besov space analysis.

Contribution

It establishes global existence results for the Navier-Stokes equations in the rotational framework with new conditions on initial data size.

Findings

Global mild solutions exist for small initial data in 3D Fourier-Besov spaces.

In 2D, solutions exist for non-small initial data in L^p spaces.

The results extend understanding of Navier-Stokes with Coriolis force in different dimensions.

Abstract

Consider the equations of Navier-Stokes in $\R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided the initial data is small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(\R^3)$, where $p \in (1,\infty]$ and $r \in [1,\infty]$. In the two-dimensional setting, a unique, global mild solution to this set of equations exists for {\em non-small} initial data $u_0 \in L^p_\sigma(\R^2)$ for $p \in [2,\infty)$.