Uniform global well-posedness of the Navier–Stokes–Coriolis system in a new critical space

Abstract

We prove global well-posedness for the Navier–Stokes–Coriolis system (NSC) in a critical space whose definition is based on Fourier transform, namely the Fourier–Besov–Morrey space $\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$ with $0<\mu <3$. The smallness condition on the initial data is uniform with respect to the angular velocity $\omega$. Our result provides a new class for the uniform global solvability of (NSC) and covers some previous ones. For $\mu =0$, (NSC) is ill-posedness in $\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$ which shows the optimality of the results with respect to the space parameter $\mu >0$. The lack of Hausdorff–Young inequality in Morrey spaces suggests that there are no inclusions between $\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$ and the largest previously known existence classes of Kozono–Yamazaki (Besov–Morrey space) and Koch–Tataru ($\textit{BMO}^{-1}$) for Navier–Stokes equations (3DNS). So, taking in particular $\omega =0$, we obtain a critical initial data class that seems to be new for global existence of solutions of (3DNS).